/* Subroutine */ int sgbbrd_(char *vect, integer *m, integer *n, integer *ncc,
integer *kl, integer *ku, real *ab, integer *ldab, real *d__, real *
e, real *q, integer *ldq, real *pt, integer *ldpt, real *c__, integer
*ldc, real *work, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, c_dim1, c_offset, pt_dim1, pt_offset, q_dim1,
q_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
/* Local variables */
integer i__, j, l, j1, j2, kb;
real ra, rb, rc;
integer kk, ml, mn, nr, mu;
real rs;
integer kb1, ml0, mu0, klm, kun, nrt, klu1, inca;
logical wantb, wantc;
integer minmn;
logical wantq;
logical wantpt;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* SGBBRD reduces a real general m-by-n band matrix A to upper */
/* bidiagonal form B by an orthogonal transformation: Q' * A * P = B. */
/* The routine computes B, and optionally forms Q or P', or computes */
/* Q'*C for a given matrix C. */
/* Arguments */
/* ========= */
/* VECT (input) CHARACTER*1 */
/* Specifies whether or not the matrices Q and P' are to be */
/* formed. */
/* = 'N': do not form Q or P'; */
/* = 'Q': form Q only; */
/* = 'P': form P' only; */
/* = 'B': form both. */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix A. N >= 0. */
/* NCC (input) INTEGER */
/* The number of columns of the matrix C. NCC >= 0. */
/* KL (input) INTEGER */
/* The number of subdiagonals of the matrix A. KL >= 0. */
/* KU (input) INTEGER */
/* The number of superdiagonals of the matrix A. KU >= 0. */
/* AB (input/output) REAL array, dimension (LDAB,N) */
/* On entry, the m-by-n band matrix A, stored in rows 1 to */
/* KL+KU+1. The j-th column of A is stored in the j-th column of */
/* the array AB as follows: */
/* AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl). */
/* On exit, A is overwritten by values generated during the */
/* reduction. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array A. LDAB >= KL+KU+1. */
/* D (output) REAL array, dimension (min(M,N)) */
/* The diagonal elements of the bidiagonal matrix B. */
/* E (output) REAL array, dimension (min(M,N)-1) */
/* The superdiagonal elements of the bidiagonal matrix B. */
/* Q (output) REAL array, dimension (LDQ,M) */
/* If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q. */
/* If VECT = 'N' or 'P', the array Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. */
/* LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise. */
/* PT (output) REAL array, dimension (LDPT,N) */
/* If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'. */
/* If VECT = 'N' or 'Q', the array PT is not referenced. */
/* LDPT (input) INTEGER */
/* The leading dimension of the array PT. */
/* LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise. */
/* C (input/output) REAL array, dimension (LDC,NCC) */
/* On entry, an m-by-ncc matrix C. */
/* On exit, C is overwritten by Q'*C. */
/* C is not referenced if NCC = 0. */
/* LDC (input) INTEGER */
/* The leading dimension of the array C. */
/* LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0. */
/* WORK (workspace) REAL array, dimension (2*max(M,N)) */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* ===================================================================== */
/* Test the input parameters */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--d__;
--e;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
pt_dim1 = *ldpt;
pt_offset = 1 + pt_dim1;
pt -= pt_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1;
c__ -= c_offset;
--work;
/* Function Body */
wantb = lsame_(vect, "B");
wantq = lsame_(vect, "Q") || wantb;
wantpt = lsame_(vect, "P") || wantb;
wantc = *ncc > 0;
klu1 = *kl + *ku + 1;
*info = 0;
if (! wantq && ! wantpt && ! lsame_(vect, "N")) {
*info = -1;
} else if (*m < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ncc < 0) {
*info = -4;
} else if (*kl < 0) {
*info = -5;
} else if (*ku < 0) {
*info = -6;
} else if (*ldab < klu1) {
*info = -8;
} else if (*ldq < 1 || wantq && *ldq < max(1,*m)) {
*info = -12;
} else if (*ldpt < 1 || wantpt && *ldpt < max(1,*n)) {
*info = -14;
} else if (*ldc < 1 || wantc && *ldc < max(1,*m)) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGBBRD", &i__1);
return 0;
}
/* Initialize Q and P' to the unit matrix, if needed */
if (wantq) {
slaset_("Full", m, m, &c_b8, &c_b9, &q[q_offset], ldq);
}
if (wantpt) {
slaset_("Full", n, n, &c_b8, &c_b9, &pt[pt_offset], ldpt);
}
/* Quick return if possible. */
if (*m == 0 || *n == 0) {
return 0;
}
minmn = min(*m,*n);
if (*kl + *ku > 1) {
/* Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce */
/* first to lower bidiagonal form and then transform to upper */
/* bidiagonal */
if (*ku > 0) {
ml0 = 1;
mu0 = 2;
} else {
ml0 = 2;
mu0 = 1;
}
/* Wherever possible, plane rotations are generated and applied in */
/* vector operations of length NR over the index set J1:J2:KLU1. */
/* The sines of the plane rotations are stored in WORK(1:max(m,n)) */
/* and the cosines in WORK(max(m,n)+1:2*max(m,n)). */
mn = max(*m,*n);
/* Computing MIN */
i__1 = *m - 1;
klm = min(i__1,*kl);
/* Computing MIN */
i__1 = *n - 1;
kun = min(i__1,*ku);
kb = klm + kun;
kb1 = kb + 1;
inca = kb1 * *ldab;
nr = 0;
j1 = klm + 2;
j2 = 1 - kun;
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th column and i-th row of matrix to bidiagonal form */
ml = klm + 1;
mu = kun + 1;
i__2 = kb;
for (kk = 1; kk <= i__2; ++kk) {
j1 += kb;
j2 += kb;
/* generate plane rotations to annihilate nonzero elements */
/* which have been created below the band */
if (nr > 0) {
slargv_(&nr, &ab[klu1 + (j1 - klm - 1) * ab_dim1], &inca,
&work[j1], &kb1, &work[mn + j1], &kb1);
}
/* apply plane rotations from the left */
i__3 = kb;
for (l = 1; l <= i__3; ++l) {
if (j2 - klm + l - 1 > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab[klu1 - l + (j1 - klm + l - 1) *
ab_dim1], &inca, &ab[klu1 - l + 1 + (j1 - klm
+ l - 1) * ab_dim1], &inca, &work[mn + j1], &
work[j1], &kb1);
}
}
if (ml > ml0) {
if (ml <= *m - i__ + 1) {
/* generate plane rotation to annihilate a(i+ml-1,i) */
/* within the band, and apply rotation from the left */
slartg_(&ab[*ku + ml - 1 + i__ * ab_dim1], &ab[*ku +
ml + i__ * ab_dim1], &work[mn + i__ + ml - 1],
&work[i__ + ml - 1], &ra);
ab[*ku + ml - 1 + i__ * ab_dim1] = ra;
if (i__ < *n) {
/* Computing MIN */
i__4 = *ku + ml - 2, i__5 = *n - i__;
i__3 = min(i__4,i__5);
i__6 = *ldab - 1;
i__7 = *ldab - 1;
srot_(&i__3, &ab[*ku + ml - 2 + (i__ + 1) *
ab_dim1], &i__6, &ab[*ku + ml - 1 + (i__
+ 1) * ab_dim1], &i__7, &work[mn + i__ +
ml - 1], &work[i__ + ml - 1]);
}
}
++nr;
j1 -= kb1;
}
if (wantq) {
/* accumulate product of plane rotations in Q */
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4)
{
srot_(m, &q[(j - 1) * q_dim1 + 1], &c__1, &q[j *
q_dim1 + 1], &c__1, &work[mn + j], &work[j]);
}
}
if (wantc) {
/* apply plane rotations to C */
i__4 = j2;
i__3 = kb1;
for (j = j1; i__3 < 0 ? j >= i__4 : j <= i__4; j += i__3)
{
srot_(ncc, &c__[j - 1 + c_dim1], ldc, &c__[j + c_dim1]
, ldc, &work[mn + j], &work[j]);
}
}
if (j2 + kun > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 -= kb1;
}
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4) {
/* create nonzero element a(j-1,j+ku) above the band */
/* and store it in WORK(n+1:2*n) */
work[j + kun] = work[j] * ab[(j + kun) * ab_dim1 + 1];
ab[(j + kun) * ab_dim1 + 1] = work[mn + j] * ab[(j + kun)
* ab_dim1 + 1];
}
/* generate plane rotations to annihilate nonzero elements */
/* which have been generated above the band */
if (nr > 0) {
slargv_(&nr, &ab[(j1 + kun - 1) * ab_dim1 + 1], &inca, &
work[j1 + kun], &kb1, &work[mn + j1 + kun], &kb1);
}
/* apply plane rotations from the right */
i__4 = kb;
for (l = 1; l <= i__4; ++l) {
if (j2 + l - 1 > *m) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab[l + 1 + (j1 + kun - 1) * ab_dim1], &
inca, &ab[l + (j1 + kun) * ab_dim1], &inca, &
work[mn + j1 + kun], &work[j1 + kun], &kb1);
}
}
if (ml == ml0 && mu > mu0) {
if (mu <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i,i+mu-1) */
/* within the band, and apply rotation from the right */
slartg_(&ab[*ku - mu + 3 + (i__ + mu - 2) * ab_dim1],
&ab[*ku - mu + 2 + (i__ + mu - 1) * ab_dim1],
&work[mn + i__ + mu - 1], &work[i__ + mu - 1],
&ra);
ab[*ku - mu + 3 + (i__ + mu - 2) * ab_dim1] = ra;
/* Computing MIN */
i__3 = *kl + mu - 2, i__5 = *m - i__;
i__4 = min(i__3,i__5);
srot_(&i__4, &ab[*ku - mu + 4 + (i__ + mu - 2) *
ab_dim1], &c__1, &ab[*ku - mu + 3 + (i__ + mu
- 1) * ab_dim1], &c__1, &work[mn + i__ + mu -
1], &work[i__ + mu - 1]);
}
++nr;
j1 -= kb1;
}
if (wantpt) {
/* accumulate product of plane rotations in P' */
i__4 = j2;
i__3 = kb1;
for (j = j1; i__3 < 0 ? j >= i__4 : j <= i__4; j += i__3)
{
srot_(n, &pt[j + kun - 1 + pt_dim1], ldpt, &pt[j +
kun + pt_dim1], ldpt, &work[mn + j + kun], &
work[j + kun]);
}
}
if (j2 + kb > *m) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 -= kb1;
}
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4) {
/* create nonzero element a(j+kl+ku,j+ku-1) below the */
/* band and store it in WORK(1:n) */
work[j + kb] = work[j + kun] * ab[klu1 + (j + kun) *
ab_dim1];
ab[klu1 + (j + kun) * ab_dim1] = work[mn + j + kun] * ab[
klu1 + (j + kun) * ab_dim1];
}
if (ml > ml0) {
--ml;
} else {
--mu;
}
}
}
}
if (*ku == 0 && *kl > 0) {
/* A has been reduced to lower bidiagonal form */
/* Transform lower bidiagonal form to upper bidiagonal by applying */
/* plane rotations from the left, storing diagonal elements in D */
/* and off-diagonal elements in E */
/* Computing MIN */
i__2 = *m - 1;
i__1 = min(i__2,*n);
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&ab[i__ * ab_dim1 + 1], &ab[i__ * ab_dim1 + 2], &rc, &rs,
&ra);
d__[i__] = ra;
if (i__ < *n) {
e[i__] = rs * ab[(i__ + 1) * ab_dim1 + 1];
ab[(i__ + 1) * ab_dim1 + 1] = rc * ab[(i__ + 1) * ab_dim1 + 1]
;
}
if (wantq) {
srot_(m, &q[i__ * q_dim1 + 1], &c__1, &q[(i__ + 1) * q_dim1 +
1], &c__1, &rc, &rs);
}
if (wantc) {
srot_(ncc, &c__[i__ + c_dim1], ldc, &c__[i__ + 1 + c_dim1],
ldc, &rc, &rs);
}
}
if (*m <= *n) {
d__[*m] = ab[*m * ab_dim1 + 1];
}
} else if (*ku > 0) {
/* A has been reduced to upper bidiagonal form */
if (*m < *n) {
/* Annihilate a(m,m+1) by applying plane rotations from the */
/* right, storing diagonal elements in D and off-diagonal */
/* elements in E */
rb = ab[*ku + (*m + 1) * ab_dim1];
for (i__ = *m; i__ >= 1; --i__) {
slartg_(&ab[*ku + 1 + i__ * ab_dim1], &rb, &rc, &rs, &ra);
d__[i__] = ra;
if (i__ > 1) {
rb = -rs * ab[*ku + i__ * ab_dim1];
e[i__ - 1] = rc * ab[*ku + i__ * ab_dim1];
}
if (wantpt) {
srot_(n, &pt[i__ + pt_dim1], ldpt, &pt[*m + 1 + pt_dim1],
ldpt, &rc, &rs);
}
}
} else {
/* Copy off-diagonal elements to E and diagonal elements to D */
i__1 = minmn - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab[*ku + (i__ + 1) * ab_dim1];
}
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab[*ku + 1 + i__ * ab_dim1];
}
}
} else {
/* A is diagonal. Set elements of E to zero and copy diagonal */
/* elements to D. */
i__1 = minmn - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.f;
}
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab[i__ * ab_dim1 + 1];
}
}
return 0;
/* End of SGBBRD */
} /* sgbbrd_ */
int slasdq_(char *uplo, int *sqre, int *n, int *
ncvt, int *nru, int *ncc, float *d__, float *e, float *vt,
int *ldvt, float *u, int *ldu, float *c__, int *ldc, float *
work, int *info)
{
/* System generated locals */
int c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1,
i__2;
/* Local variables */
int i__, j;
float r__, cs, sn;
int np1, isub;
float smin;
int sqre1;
extern int lsame_(char *, char *);
extern int slasr_(char *, char *, char *, int *,
int *, float *, float *, float *, int *);
int iuplo;
extern int sswap_(int *, float *, int *, float *,
int *), xerbla_(char *, int *), slartg_(float *,
float *, float *, float *, float *);
int rotate;
extern int sbdsqr_(char *, int *, int *, int
*, int *, float *, float *, float *, int *, float *, int *
, float *, int *, float *, int *);
/* -- LAPACK auxiliary routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLASDQ computes the singular value decomposition (SVD) of a float */
/* (upper or lower) bidiagonal matrix with diagonal D and offdiagonal */
/* E, accumulating the transformations if desired. Letting B denote */
/* the input bidiagonal matrix, the algorithm computes orthogonal */
/* matrices Q and P such that B = Q * S * P' (P' denotes the transpose */
/* of P). The singular values S are overwritten on D. */
/* The input matrix U is changed to U * Q if desired. */
/* The input matrix VT is changed to P' * VT if desired. */
/* The input matrix C is changed to Q' * C if desired. */
/* See "Computing Small Singular Values of Bidiagonal Matrices With */
/* Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, */
/* LAPACK Working Note #3, for a detailed description of the algorithm. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* On entry, UPLO specifies whether the input bidiagonal matrix */
/* is upper or lower bidiagonal, and wether it is square are */
/* not. */
/* UPLO = 'U' or 'u' B is upper bidiagonal. */
/* UPLO = 'L' or 'l' B is lower bidiagonal. */
/* SQRE (input) INTEGER */
/* = 0: then the input matrix is N-by-N. */
/* = 1: then the input matrix is N-by-(N+1) if UPLU = 'U' and */
/* (N+1)-by-N if UPLU = 'L'. */
/* The bidiagonal matrix has */
/* N = NL + NR + 1 rows and */
/* M = N + SQRE >= N columns. */
/* N (input) INTEGER */
/* On entry, N specifies the number of rows and columns */
/* in the matrix. N must be at least 0. */
/* NCVT (input) INTEGER */
/* On entry, NCVT specifies the number of columns of */
/* the matrix VT. NCVT must be at least 0. */
/* NRU (input) INTEGER */
/* On entry, NRU specifies the number of rows of */
/* the matrix U. NRU must be at least 0. */
/* NCC (input) INTEGER */
/* On entry, NCC specifies the number of columns of */
/* the matrix C. NCC must be at least 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, D contains the diagonal entries of the */
/* bidiagonal matrix whose SVD is desired. On normal exit, */
/* D contains the singular values in ascending order. */
/* E (input/output) REAL array. */
/* dimension is (N-1) if SQRE = 0 and N if SQRE = 1. */
/* On entry, the entries of E contain the offdiagonal entries */
/* of the bidiagonal matrix whose SVD is desired. On normal */
/* exit, E will contain 0. If the algorithm does not converge, */
/* D and E will contain the diagonal and superdiagonal entries */
/* of a bidiagonal matrix orthogonally equivalent to the one */
/* given as input. */
/* VT (input/output) REAL array, dimension (LDVT, NCVT) */
/* On entry, contains a matrix which on exit has been */
/* premultiplied by P', dimension N-by-NCVT if SQRE = 0 */
/* and (N+1)-by-NCVT if SQRE = 1 (not referenced if NCVT=0). */
/* LDVT (input) INTEGER */
/* On entry, LDVT specifies the leading dimension of VT as */
/* declared in the calling (sub) program. LDVT must be at */
/* least 1. If NCVT is nonzero LDVT must also be at least N. */
/* U (input/output) REAL array, dimension (LDU, N) */
/* On entry, contains a matrix which on exit has been */
/* postmultiplied by Q, dimension NRU-by-N if SQRE = 0 */
/* and NRU-by-(N+1) if SQRE = 1 (not referenced if NRU=0). */
/* LDU (input) INTEGER */
/* On entry, LDU specifies the leading dimension of U as */
/* declared in the calling (sub) program. LDU must be at */
/* least MAX( 1, NRU ) . */
/* C (input/output) REAL array, dimension (LDC, NCC) */
/* On entry, contains an N-by-NCC matrix which on exit */
/* has been premultiplied by Q' dimension N-by-NCC if SQRE = 0 */
/* and (N+1)-by-NCC if SQRE = 1 (not referenced if NCC=0). */
/* LDC (input) INTEGER */
/* On entry, LDC specifies the leading dimension of C as */
/* declared in the calling (sub) program. LDC must be at */
/* least 1. If NCC is nonzero, LDC must also be at least N. */
/* WORK (workspace) REAL array, dimension (4*N) */
/* Workspace. Only referenced if one of NCVT, NRU, or NCC is */
/* nonzero, and if N is at least 2. */
/* INFO (output) INTEGER */
/* On exit, a value of 0 indicates a successful exit. */
/* If INFO < 0, argument number -INFO is illegal. */
/* If INFO > 0, the algorithm did not converge, and INFO */
/* specifies how many superdiagonals did not converge. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1;
c__ -= c_offset;
--work;
/* Function Body */
*info = 0;
iuplo = 0;
if (lsame_(uplo, "U")) {
iuplo = 1;
}
if (lsame_(uplo, "L")) {
iuplo = 2;
}
if (iuplo == 0) {
*info = -1;
} else if (*sqre < 0 || *sqre > 1) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ncvt < 0) {
*info = -4;
} else if (*nru < 0) {
*info = -5;
} else if (*ncc < 0) {
*info = -6;
} else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < MAX(1,*n)) {
*info = -10;
} else if (*ldu < MAX(1,*nru)) {
*info = -12;
} else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < MAX(1,*n)) {
*info = -14;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLASDQ", &i__1);
return 0;
}
if (*n == 0) {
return 0;
}
/* ROTATE is true if any singular vectors desired, false otherwise */
rotate = *ncvt > 0 || *nru > 0 || *ncc > 0;
np1 = *n + 1;
sqre1 = *sqre;
/* If matrix non-square upper bidiagonal, rotate to be lower */
/* bidiagonal. The rotations are on the right. */
if (iuplo == 1 && sqre1 == 1) {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (rotate) {
work[i__] = cs;
work[*n + i__] = sn;
}
/* L10: */
}
slartg_(&d__[*n], &e[*n], &cs, &sn, &r__);
d__[*n] = r__;
e[*n] = 0.f;
if (rotate) {
work[*n] = cs;
work[*n + *n] = sn;
}
iuplo = 2;
sqre1 = 0;
/* Update singular vectors if desired. */
if (*ncvt > 0) {
slasr_("L", "V", "F", &np1, ncvt, &work[1], &work[np1], &vt[
vt_offset], ldvt);
}
}
/* If matrix lower bidiagonal, rotate to be upper bidiagonal */
/* by applying Givens rotations on the left. */
if (iuplo == 2) {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (rotate) {
work[i__] = cs;
work[*n + i__] = sn;
}
/* L20: */
}
/* If matrix (N+1)-by-N lower bidiagonal, one additional */
/* rotation is needed. */
if (sqre1 == 1) {
slartg_(&d__[*n], &e[*n], &cs, &sn, &r__);
d__[*n] = r__;
if (rotate) {
work[*n] = cs;
work[*n + *n] = sn;
}
}
/* Update singular vectors if desired. */
if (*nru > 0) {
if (sqre1 == 0) {
slasr_("R", "V", "F", nru, n, &work[1], &work[np1], &u[
u_offset], ldu);
} else {
slasr_("R", "V", "F", nru, &np1, &work[1], &work[np1], &u[
u_offset], ldu);
}
}
if (*ncc > 0) {
if (sqre1 == 0) {
slasr_("L", "V", "F", n, ncc, &work[1], &work[np1], &c__[
c_offset], ldc);
} else {
slasr_("L", "V", "F", &np1, ncc, &work[1], &work[np1], &c__[
c_offset], ldc);
}
}
}
/* Call SBDSQR to compute the SVD of the reduced float */
/* N-by-N upper bidiagonal matrix. */
sbdsqr_("U", n, ncvt, nru, ncc, &d__[1], &e[1], &vt[vt_offset], ldvt, &u[
u_offset], ldu, &c__[c_offset], ldc, &work[1], info);
/* Sort the singular values into ascending order (insertion sort on */
/* singular values, but only one transposition per singular vector) */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Scan for smallest D(I). */
isub = i__;
smin = d__[i__];
i__2 = *n;
for (j = i__ + 1; j <= i__2; ++j) {
if (d__[j] < smin) {
isub = j;
smin = d__[j];
}
/* L30: */
}
if (isub != i__) {
/* Swap singular values and vectors. */
d__[isub] = d__[i__];
d__[i__] = smin;
if (*ncvt > 0) {
sswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[i__ + vt_dim1],
ldvt);
}
if (*nru > 0) {
sswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[i__ * u_dim1 + 1]
, &c__1);
}
if (*ncc > 0) {
sswap_(ncc, &c__[isub + c_dim1], ldc, &c__[i__ + c_dim1], ldc)
;
}
}
/* L40: */
}
return 0;
/* End of SLASDQ */
} /* slasdq_ */
/* Subroutine */ int slalsd_(char *uplo, integer *smlsiz, integer *n, integer
*nrhs, real *d__, real *e, real *b, integer *ldb, real *rcond,
integer *rank, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer b_dim1, b_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double log(doublereal), r_sign(real *, real *);
/* Local variables */
static integer difl, difr, perm, nsub, nlvl, sqre, bxst;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static integer c__, i__, j, k;
static real r__;
static integer s, u, z__;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static integer poles, sizei, nsize;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
static integer nwork, icmpq1, icmpq2;
extern doublereal sopbl3_(char *, integer *, integer *, integer *)
;
static real cs;
static integer bx;
static real sn;
static integer st;
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *);
extern doublereal slamch_(char *);
static integer vt;
extern /* Subroutine */ int xerbla_(char *, integer *), slalsa_(
integer *, integer *, integer *, integer *, real *, integer *,
real *, integer *, real *, integer *, real *, integer *, real *,
real *, real *, real *, integer *, integer *, integer *, integer *
, real *, real *, real *, real *, integer *, integer *), slascl_(
char *, integer *, integer *, real *, real *, integer *, integer *
, real *, integer *, integer *);
static integer givcol;
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *),
slacpy_(char *, integer *, integer *, real *, integer *, real *,
integer *), slartg_(real *, real *, real *, real *, real *
), slaset_(char *, integer *, integer *, real *, real *, real *,
integer *);
static real orgnrm;
static integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *);
static integer givptr, nm1, smlszp, st1;
static real eps;
static integer iwk;
static real tol;
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
/* -- LAPACK routine (instrumented to count ops, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
SLALSD uses the singular value decomposition of A to solve the least
squares problem of finding X to minimize the Euclidean norm of each
column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
are N-by-NRHS. The solution X overwrites B.
The singular values of A smaller than RCOND times the largest
singular value are treated as zero in solving the least squares
problem; in this case a minimum norm solution is returned.
The actual singular values are returned in D in ascending order.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': D and E define an upper bidiagonal matrix.
= 'L': D and E define a lower bidiagonal matrix.
SMLSIZ (input) INTEGER
The maximum size of the subproblems at the bottom of the
computation tree.
N (input) INTEGER
The dimension of the bidiagonal matrix. N >= 0.
NRHS (input) INTEGER
The number of columns of B. NRHS must be at least 1.
D (input/output) REAL array, dimension (N)
On entry D contains the main diagonal of the bidiagonal
matrix. On exit, if INFO = 0, D contains its singular values.
E (input) REAL array, dimension (N-1)
Contains the super-diagonal entries of the bidiagonal matrix.
On exit, E has been destroyed.
B (input/output) REAL array, dimension (LDB,NRHS)
On input, B contains the right hand sides of the least
squares problem. On output, B contains the solution X.
LDB (input) INTEGER
The leading dimension of B in the calling subprogram.
LDB must be at least max(1,N).
RCOND (input) REAL
The singular values of A less than or equal to RCOND times
the largest singular value are treated as zero in solving
the least squares problem. If RCOND is negative,
machine precision is used instead.
For example, if diag(S)*X=B were the least squares problem,
where diag(S) is a diagonal matrix of singular values, the
solution would be X(i) = B(i) / S(i) if S(i) is greater than
RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
RCOND*max(S).
RANK (output) INTEGER
The number of singular values of A greater than RCOND times
the largest singular value.
WORK (workspace) REAL array, dimension at least
(9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
IWORK (workspace) INTEGER array, dimension at least
(3 * N * NLVL + 11 * N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value while
working on the submatrix lying in rows and columns
INFO/(N+1) through MOD(INFO,N+1).
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--work;
--iwork;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < 1 || *ldb < *n) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SLALSD", &i__1);
return 0;
}
eps = slamch_("Epsilon");
/* Set up the tolerance. */
if (*rcond <= 0.f || *rcond >= 1.f) {
*rcond = eps;
}
*rank = 0;
/* Quick return if possible. */
if (*n == 0) {
return 0;
} else if (*n == 1) {
if (d__[1] == 0.f) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
} else {
*rank = 1;
latime_1.ops += (real) (*nrhs << 1);
slascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[
b_offset], ldb, info);
d__[1] = dabs(d__[1]);
}
return 0;
}
/* Rotate the matrix if it is lower bidiagonal. */
if (*(unsigned char *)uplo == 'L') {
latime_1.ops += (real) ((*n - 1) * 6);
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (*nrhs == 1) {
latime_1.ops += 6.f;
srot_(&c__1, &b_ref(i__, 1), &c__1, &b_ref(i__ + 1, 1), &c__1,
&cs, &sn);
} else {
work[(i__ << 1) - 1] = cs;
work[i__ * 2] = sn;
}
/* L10: */
}
if (*nrhs > 1) {
latime_1.ops += (real) ((*n - 1) * 6 * *nrhs);
i__1 = *nrhs;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n - 1;
for (j = 1; j <= i__2; ++j) {
cs = work[(j << 1) - 1];
sn = work[j * 2];
srot_(&c__1, &b_ref(j, i__), &c__1, &b_ref(j + 1, i__), &
c__1, &cs, &sn);
/* L20: */
}
/* L30: */
}
}
}
/* Scale. */
nm1 = *n - 1;
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
slaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
return 0;
}
latime_1.ops += (real) (*n + nm1);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1,
info);
/* If N is smaller than the minimum divide size SMLSIZ, then solve
the problem with another solver. */
if (*n <= *smlsiz) {
nwork = *n * *n + 1;
slaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
slasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, &
work[1], n, &b[b_offset], ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
latime_1.ops += 1.f;
tol = *rcond * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] <= tol) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b_ref(i__, 1), ldb);
} else {
latime_1.ops += (real) (*nrhs);
slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &
b_ref(i__, 1), ldb, info);
++(*rank);
}
/* L40: */
}
latime_1.ops += sopbl3_("SGEMM ", n, nrhs, n);
sgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, &
c_b6, &work[nwork], n);
slacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);
/* Unscale. */
latime_1.ops += (real) (*n + *n * *nrhs);
slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n,
info);
slasrt_("D", n, &d__[1], info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset],
ldb, info);
return 0;
}
/* Book-keeping and setting up some constants. */
nlvl = (integer) (log((real) (*n) / (real) (*smlsiz + 1)) / log(2.f)) + 1;
smlszp = *smlsiz + 1;
u = 1;
vt = *smlsiz * *n + 1;
difl = vt + smlszp * *n;
difr = difl + nlvl * *n;
z__ = difr + (nlvl * *n << 1);
c__ = z__ + nlvl * *n;
s = c__ + *n;
poles = s + *n;
givnum = poles + (nlvl << 1) * *n;
bx = givnum + (nlvl << 1) * *n;
nwork = bx + *n * *nrhs;
sizei = *n + 1;
k = sizei + *n;
givptr = k + *n;
perm = givptr + *n;
givcol = perm + nlvl * *n;
iwk = givcol + (nlvl * *n << 1);
st = 1;
sqre = 0;
icmpq1 = 1;
icmpq2 = 0;
nsub = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L50: */
}
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
++nsub;
iwork[nsub] = st;
/* Subproblem found. First determine its size and then
apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else {
/* A subproblem with E(NM1) small. This implies an
1-by-1 subproblem at D(N), which is not solved
explicitly. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
++nsub;
iwork[nsub] = *n;
iwork[sizei + nsub - 1] = 1;
scopy_(nrhs, &b_ref(*n, 1), ldb, &work[bx + nm1], n);
}
st1 = st - 1;
if (nsize == 1) {
/* This is a 1-by-1 subproblem and is not solved
explicitly. */
scopy_(nrhs, &b_ref(st, 1), ldb, &work[bx + st1], n);
} else if (nsize <= *smlsiz) {
/* This is a small subproblem and is solved by SLASDQ. */
slaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1],
n);
slasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[
st], &work[vt + st1], n, &work[nwork], n, &b_ref(st,
1), ldb, &work[nwork], info);
if (*info != 0) {
return 0;
}
slacpy_("A", &nsize, nrhs, &b_ref(st, 1), ldb, &work[bx + st1]
, n);
} else {
/* A large problem. Solve it using divide and conquer. */
slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
work[u + st1], n, &work[vt + st1], &iwork[k + st1], &
work[difl + st1], &work[difr + st1], &work[z__ + st1],
&work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum +
st1], &work[c__ + st1], &work[s + st1], &work[nwork],
&iwork[iwk], info);
if (*info != 0) {
return 0;
}
bxst = bx + st1;
slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b_ref(st, 1), ldb, &
work[bxst], n, &work[u + st1], n, &work[vt + st1], &
iwork[k + st1], &work[difl + st1], &work[difr + st1],
&work[z__ + st1], &work[poles + st1], &iwork[givptr +
st1], &iwork[givcol + st1], n, &iwork[perm + st1], &
work[givnum + st1], &work[c__ + st1], &work[s + st1],
&work[nwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
st = i__ + 1;
}
/* L60: */
}
/* Apply the singular values and treat the tiny ones as zero. */
tol = *rcond * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Some of the elements in D can be negative because 1-by-1
subproblems were not solved explicitly. */
if ((r__1 = d__[i__], dabs(r__1)) <= tol) {
slaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
} else {
++(*rank);
latime_1.ops += (real) (*nrhs);
slascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[
bx + i__ - 1], n, info);
}
d__[i__] = (r__1 = d__[i__], dabs(r__1));
/* L70: */
}
/* Now apply back the right singular vectors. */
icmpq2 = 1;
i__1 = nsub;
for (i__ = 1; i__ <= i__1; ++i__) {
st = iwork[i__];
st1 = st - 1;
nsize = iwork[sizei + i__ - 1];
bxst = bx + st1;
if (nsize == 1) {
scopy_(nrhs, &work[bxst], n, &b_ref(st, 1), ldb);
} else if (nsize <= *smlsiz) {
latime_1.ops += sopbl3_("SGEMM ", &nsize, nrhs, &nsize)
;
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n,
&work[bxst], n, &c_b6, &b_ref(st, 1), ldb);
} else {
slalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b_ref(st,
1), ldb, &work[u + st1], n, &work[vt + st1], &iwork[k +
st1], &work[difl + st1], &work[difr + st1], &work[z__ +
st1], &work[poles + st1], &iwork[givptr + st1], &iwork[
givcol + st1], n, &iwork[perm + st1], &work[givnum + st1],
&work[c__ + st1], &work[s + st1], &work[nwork], &iwork[
iwk], info);
if (*info != 0) {
return 0;
}
}
/* L80: */
}
/* Unscale and sort the singular values. */
latime_1.ops += (real) (*n + *n * *nrhs);
slascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
slasrt_("D", n, &d__[1], info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb,
info);
return 0;
/* End of SLALSD */
} /* slalsd_ */
int sbdsqr_(char *uplo, int *n, int *ncvt, int *
nru, int *ncc, float *d__, float *e, float *vt, int *ldvt, float *
u, int *ldu, float *c__, int *ldc, float *work, int *info)
{
/* System generated locals */
int c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1,
i__2;
float r__1, r__2, r__3, r__4;
double d__1;
/* Builtin functions */
double pow_dd(double *, double *), sqrt(double), r_sign(float *
, float *);
/* Local variables */
float f, g, h__;
int i__, j, m;
float r__, cs;
int ll;
float sn, mu;
int nm1, nm12, nm13, lll;
float eps, sll, tol, abse;
int idir;
float abss;
int oldm;
float cosl;
int isub, iter;
float unfl, sinl, cosr, smin, smax, sinr;
extern int srot_(int *, float *, int *, float *,
int *, float *, float *), slas2_(float *, float *, float *, float *,
float *);
extern int lsame_(char *, char *);
float oldcs;
extern int sscal_(int *, float *, float *, int *);
int oldll;
float shift, sigmn, oldsn;
int maxit;
float sminl;
extern int slasr_(char *, char *, char *, int *,
int *, float *, float *, float *, int *);
float sigmx;
int lower;
extern int sswap_(int *, float *, int *, float *,
int *), slasq1_(int *, float *, float *, float *, int *),
slasv2_(float *, float *, float *, float *, float *, float *, float *,
float *, float *);
extern double slamch_(char *);
extern int xerbla_(char *, int *);
float sminoa;
extern int slartg_(float *, float *, float *, float *, float *
);
float thresh;
int rotate;
float tolmul;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* January 2007 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SBDSQR computes the singular values and, optionally, the right and/or */
/* left singular vectors from the singular value decomposition (SVD) of */
/* a float N-by-N (upper or lower) bidiagonal matrix B using the implicit */
/* zero-shift QR algorithm. The SVD of B has the form */
/* B = Q * S * P**T */
/* where S is the diagonal matrix of singular values, Q is an orthogonal */
/* matrix of left singular vectors, and P is an orthogonal matrix of */
/* right singular vectors. If left singular vectors are requested, this */
/* subroutine actually returns U*Q instead of Q, and, if right singular */
/* vectors are requested, this subroutine returns P**T*VT instead of */
/* P**T, for given float input matrices U and VT. When U and VT are the */
/* orthogonal matrices that reduce a general matrix A to bidiagonal */
/* form: A = U*B*VT, as computed by SGEBRD, then */
/* A = (U*Q) * S * (P**T*VT) */
/* is the SVD of A. Optionally, the subroutine may also compute Q**T*C */
/* for a given float input matrix C. */
/* See "Computing Small Singular Values of Bidiagonal Matrices With */
/* Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, */
/* LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, */
/* no. 5, pp. 873-912, Sept 1990) and */
/* "Accurate singular values and differential qd algorithms," by */
/* B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics */
/* Department, University of California at Berkeley, July 1992 */
/* for a detailed description of the algorithm. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': B is upper bidiagonal; */
/* = 'L': B is lower bidiagonal. */
/* N (input) INTEGER */
/* The order of the matrix B. N >= 0. */
/* NCVT (input) INTEGER */
/* The number of columns of the matrix VT. NCVT >= 0. */
/* NRU (input) INTEGER */
/* The number of rows of the matrix U. NRU >= 0. */
/* NCC (input) INTEGER */
/* The number of columns of the matrix C. NCC >= 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, the n diagonal elements of the bidiagonal matrix B. */
/* On exit, if INFO=0, the singular values of B in decreasing */
/* order. */
/* E (input/output) REAL array, dimension (N-1) */
/* On entry, the N-1 offdiagonal elements of the bidiagonal */
/* matrix B. */
/* On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E */
/* will contain the diagonal and superdiagonal elements of a */
/* bidiagonal matrix orthogonally equivalent to the one given */
/* as input. */
/* VT (input/output) REAL array, dimension (LDVT, NCVT) */
/* On entry, an N-by-NCVT matrix VT. */
/* On exit, VT is overwritten by P**T * VT. */
/* Not referenced if NCVT = 0. */
/* LDVT (input) INTEGER */
/* The leading dimension of the array VT. */
/* LDVT >= MAX(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. */
/* U (input/output) REAL array, dimension (LDU, N) */
/* On entry, an NRU-by-N matrix U. */
/* On exit, U is overwritten by U * Q. */
/* Not referenced if NRU = 0. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= MAX(1,NRU). */
/* C (input/output) REAL array, dimension (LDC, NCC) */
/* On entry, an N-by-NCC matrix C. */
/* On exit, C is overwritten by Q**T * C. */
/* Not referenced if NCC = 0. */
/* LDC (input) INTEGER */
/* The leading dimension of the array C. */
/* LDC >= MAX(1,N) if NCC > 0; LDC >=1 if NCC = 0. */
/* WORK (workspace) REAL array, dimension (4*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: If INFO = -i, the i-th argument had an illegal value */
/* > 0: */
/* if NCVT = NRU = NCC = 0, */
/* = 1, a split was marked by a positive value in E */
/* = 2, current block of Z not diagonalized after 30*N */
/* iterations (in inner while loop) */
/* = 3, termination criterion of outer while loop not met */
/* (program created more than N unreduced blocks) */
/* else NCVT = NRU = NCC = 0, */
/* the algorithm did not converge; D and E contain the */
/* elements of a bidiagonal matrix which is orthogonally */
/* similar to the input matrix B; if INFO = i, i */
/* elements of E have not converged to zero. */
/* Internal Parameters */
/* =================== */
/* TOLMUL REAL, default = MAX(10,MIN(100,EPS**(-1/8))) */
/* TOLMUL controls the convergence criterion of the QR loop. */
/* If it is positive, TOLMUL*EPS is the desired relative */
/* precision in the computed singular values. */
/* If it is negative, ABS(TOLMUL*EPS*sigma_max) is the */
/* desired absolute accuracy in the computed singular */
/* values (corresponds to relative accuracy */
/* ABS(TOLMUL*EPS) in the largest singular value. */
/* ABS(TOLMUL) should be between 1 and 1/EPS, and preferably */
/* between 10 (for fast convergence) and .1/EPS */
/* (for there to be some accuracy in the results). */
/* Default is to lose at either one eighth or 2 of the */
/* available decimal digits in each computed singular value */
/* (whichever is smaller). */
/* MAXITR INTEGER, default = 6 */
/* MAXITR controls the maximum number of passes of the */
/* algorithm through its inner loop. The algorithms stops */
/* (and so fails to converge) if the number of passes */
/* through the inner loop exceeds MAXITR*N**2. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1;
c__ -= c_offset;
--work;
/* Function Body */
*info = 0;
lower = lsame_(uplo, "L");
if (! lsame_(uplo, "U") && ! lower) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*ncvt < 0) {
*info = -3;
} else if (*nru < 0) {
*info = -4;
} else if (*ncc < 0) {
*info = -5;
} else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < MAX(1,*n)) {
*info = -9;
} else if (*ldu < MAX(1,*nru)) {
*info = -11;
} else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < MAX(1,*n)) {
*info = -13;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSQR", &i__1);
return 0;
}
if (*n == 0) {
return 0;
}
if (*n == 1) {
goto L160;
}
/* ROTATE is true if any singular vectors desired, false otherwise */
rotate = *ncvt > 0 || *nru > 0 || *ncc > 0;
/* If no singular vectors desired, use qd algorithm */
if (! rotate) {
slasq1_(n, &d__[1], &e[1], &work[1], info);
return 0;
}
nm1 = *n - 1;
nm12 = nm1 + nm1;
nm13 = nm12 + nm1;
idir = 0;
/* Get machine constants */
eps = slamch_("Epsilon");
unfl = slamch_("Safe minimum");
/* If matrix lower bidiagonal, rotate to be upper bidiagonal */
/* by applying Givens rotations on the left */
if (lower) {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
work[i__] = cs;
work[nm1 + i__] = sn;
/* L10: */
}
/* Update singular vectors if desired */
if (*nru > 0) {
slasr_("R", "V", "F", nru, n, &work[1], &work[*n], &u[u_offset],
ldu);
}
if (*ncc > 0) {
slasr_("L", "V", "F", n, ncc, &work[1], &work[*n], &c__[c_offset],
ldc);
}
}
/* Compute singular values to relative accuracy TOL */
/* (By setting TOL to be negative, algorithm will compute */
/* singular values to absolute accuracy ABS(TOL)*norm(input matrix)) */
/* Computing MAX */
/* Computing MIN */
d__1 = (double) eps;
r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b15);
r__1 = 10.f, r__2 = MIN(r__3,r__4);
tolmul = MAX(r__1,r__2);
tol = tolmul * eps;
/* Compute approximate maximum, minimum singular values */
smax = 0.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__2 = smax, r__3 = (r__1 = d__[i__], ABS(r__1));
smax = MAX(r__2,r__3);
/* L20: */
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__2 = smax, r__3 = (r__1 = e[i__], ABS(r__1));
smax = MAX(r__2,r__3);
/* L30: */
}
sminl = 0.f;
if (tol >= 0.f) {
/* Relative accuracy desired */
sminoa = ABS(d__[1]);
if (sminoa == 0.f) {
goto L50;
}
mu = sminoa;
i__1 = *n;
for (i__ = 2; i__ <= i__1; ++i__) {
mu = (r__2 = d__[i__], ABS(r__2)) * (mu / (mu + (r__1 = e[i__ -
1], ABS(r__1))));
sminoa = MIN(sminoa,mu);
if (sminoa == 0.f) {
goto L50;
}
/* L40: */
}
L50:
sminoa /= sqrt((float) (*n));
/* Computing MAX */
r__1 = tol * sminoa, r__2 = *n * 6 * *n * unfl;
thresh = MAX(r__1,r__2);
} else {
/* Absolute accuracy desired */
/* Computing MAX */
r__1 = ABS(tol) * smax, r__2 = *n * 6 * *n * unfl;
thresh = MAX(r__1,r__2);
}
/* Prepare for main iteration loop for the singular values */
/* (MAXIT is the maximum number of passes through the inner */
/* loop permitted before nonconvergence signalled.) */
maxit = *n * 6 * *n;
iter = 0;
oldll = -1;
oldm = -1;
/* M points to last element of unconverged part of matrix */
m = *n;
/* Begin main iteration loop */
L60:
/* Check for convergence or exceeding iteration count */
if (m <= 1) {
goto L160;
}
if (iter > maxit) {
goto L200;
}
/* Find diagonal block of matrix to work on */
if (tol < 0.f && (r__1 = d__[m], ABS(r__1)) <= thresh) {
d__[m] = 0.f;
}
smax = (r__1 = d__[m], ABS(r__1));
smin = smax;
i__1 = m - 1;
for (lll = 1; lll <= i__1; ++lll) {
ll = m - lll;
abss = (r__1 = d__[ll], ABS(r__1));
abse = (r__1 = e[ll], ABS(r__1));
if (tol < 0.f && abss <= thresh) {
d__[ll] = 0.f;
}
if (abse <= thresh) {
goto L80;
}
smin = MIN(smin,abss);
/* Computing MAX */
r__1 = MAX(smax,abss);
smax = MAX(r__1,abse);
/* L70: */
}
ll = 0;
goto L90;
L80:
e[ll] = 0.f;
/* Matrix splits since E(LL) = 0 */
if (ll == m - 1) {
/* Convergence of bottom singular value, return to top of loop */
--m;
goto L60;
}
L90:
++ll;
/* E(LL) through E(M-1) are nonzero, E(LL-1) is zero */
if (ll == m - 1) {
/* 2 by 2 block, handle separately */
slasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr,
&sinl, &cosl);
d__[m - 1] = sigmx;
e[m - 1] = 0.f;
d__[m] = sigmn;
/* Compute singular vectors, if desired */
if (*ncvt > 0) {
srot_(ncvt, &vt[m - 1 + vt_dim1], ldvt, &vt[m + vt_dim1], ldvt, &
cosr, &sinr);
}
if (*nru > 0) {
srot_(nru, &u[(m - 1) * u_dim1 + 1], &c__1, &u[m * u_dim1 + 1], &
c__1, &cosl, &sinl);
}
if (*ncc > 0) {
srot_(ncc, &c__[m - 1 + c_dim1], ldc, &c__[m + c_dim1], ldc, &
cosl, &sinl);
}
m += -2;
goto L60;
}
/* If working on new submatrix, choose shift direction */
/* (from larger end diagonal element towards smaller) */
if (ll > oldm || m < oldll) {
if ((r__1 = d__[ll], ABS(r__1)) >= (r__2 = d__[m], ABS(r__2))) {
/* Chase bulge from top (big end) to bottom (small end) */
idir = 1;
} else {
/* Chase bulge from bottom (big end) to top (small end) */
idir = 2;
}
}
/* Apply convergence tests */
if (idir == 1) {
/* Run convergence test in forward direction */
/* First apply standard test to bottom of matrix */
if ((r__2 = e[m - 1], ABS(r__2)) <= ABS(tol) * (r__1 = d__[m], ABS(
r__1)) || tol < 0.f && (r__3 = e[m - 1], ABS(r__3)) <=
thresh) {
e[m - 1] = 0.f;
goto L60;
}
if (tol >= 0.f) {
/* If relative accuracy desired, */
/* apply convergence criterion forward */
mu = (r__1 = d__[ll], ABS(r__1));
sminl = mu;
i__1 = m - 1;
for (lll = ll; lll <= i__1; ++lll) {
if ((r__1 = e[lll], ABS(r__1)) <= tol * mu) {
e[lll] = 0.f;
goto L60;
}
mu = (r__2 = d__[lll + 1], ABS(r__2)) * (mu / (mu + (r__1 =
e[lll], ABS(r__1))));
sminl = MIN(sminl,mu);
/* L100: */
}
}
} else {
/* Run convergence test in backward direction */
/* First apply standard test to top of matrix */
if ((r__2 = e[ll], ABS(r__2)) <= ABS(tol) * (r__1 = d__[ll], ABS(
r__1)) || tol < 0.f && (r__3 = e[ll], ABS(r__3)) <= thresh) {
e[ll] = 0.f;
goto L60;
}
if (tol >= 0.f) {
/* If relative accuracy desired, */
/* apply convergence criterion backward */
mu = (r__1 = d__[m], ABS(r__1));
sminl = mu;
i__1 = ll;
for (lll = m - 1; lll >= i__1; --lll) {
if ((r__1 = e[lll], ABS(r__1)) <= tol * mu) {
e[lll] = 0.f;
goto L60;
}
mu = (r__2 = d__[lll], ABS(r__2)) * (mu / (mu + (r__1 = e[
lll], ABS(r__1))));
sminl = MIN(sminl,mu);
/* L110: */
}
}
}
oldll = ll;
oldm = m;
/* Compute shift. First, test if shifting would ruin relative */
/* accuracy, and if so set the shift to zero. */
/* Computing MAX */
r__1 = eps, r__2 = tol * .01f;
if (tol >= 0.f && *n * tol * (sminl / smax) <= MAX(r__1,r__2)) {
/* Use a zero shift to avoid loss of relative accuracy */
shift = 0.f;
} else {
/* Compute the shift from 2-by-2 block at end of matrix */
if (idir == 1) {
sll = (r__1 = d__[ll], ABS(r__1));
slas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__);
} else {
sll = (r__1 = d__[m], ABS(r__1));
slas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__);
}
/* Test if shift negligible, and if so set to zero */
if (sll > 0.f) {
/* Computing 2nd power */
r__1 = shift / sll;
if (r__1 * r__1 < eps) {
shift = 0.f;
}
}
}
/* Increment iteration count */
iter = iter + m - ll;
/* If SHIFT = 0, do simplified QR iteration */
if (shift == 0.f) {
if (idir == 1) {
/* Chase bulge from top to bottom */
/* Save cosines and sines for later singular vector updates */
cs = 1.f;
oldcs = 1.f;
i__1 = m - 1;
for (i__ = ll; i__ <= i__1; ++i__) {
r__1 = d__[i__] * cs;
slartg_(&r__1, &e[i__], &cs, &sn, &r__);
if (i__ > ll) {
e[i__ - 1] = oldsn * r__;
}
r__1 = oldcs * r__;
r__2 = d__[i__ + 1] * sn;
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
work[i__ - ll + 1] = cs;
work[i__ - ll + 1 + nm1] = sn;
work[i__ - ll + 1 + nm12] = oldcs;
work[i__ - ll + 1 + nm13] = oldsn;
/* L120: */
}
h__ = d__[m] * cs;
d__[m] = h__ * oldcs;
e[m - 1] = h__ * oldsn;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &vt[
ll + vt_dim1], ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
+ 1], &u[ll * u_dim1 + 1], ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
+ 1], &c__[ll + c_dim1], ldc);
}
/* Test convergence */
if ((r__1 = e[m - 1], ABS(r__1)) <= thresh) {
e[m - 1] = 0.f;
}
} else {
/* Chase bulge from bottom to top */
/* Save cosines and sines for later singular vector updates */
cs = 1.f;
oldcs = 1.f;
i__1 = ll + 1;
for (i__ = m; i__ >= i__1; --i__) {
r__1 = d__[i__] * cs;
slartg_(&r__1, &e[i__ - 1], &cs, &sn, &r__);
if (i__ < m) {
e[i__] = oldsn * r__;
}
r__1 = oldcs * r__;
r__2 = d__[i__ - 1] * sn;
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
work[i__ - ll] = cs;
work[i__ - ll + nm1] = -sn;
work[i__ - ll + nm12] = oldcs;
work[i__ - ll + nm13] = -oldsn;
/* L130: */
}
h__ = d__[ll] * cs;
d__[ll] = h__ * oldcs;
e[ll] = h__ * oldsn;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
nm13 + 1], &vt[ll + vt_dim1], ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u[ll *
u_dim1 + 1], ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &c__[
ll + c_dim1], ldc);
}
/* Test convergence */
if ((r__1 = e[ll], ABS(r__1)) <= thresh) {
e[ll] = 0.f;
}
}
} else {
/* Use nonzero shift */
if (idir == 1) {
/* Chase bulge from top to bottom */
/* Save cosines and sines for later singular vector updates */
f = ((r__1 = d__[ll], ABS(r__1)) - shift) * (r_sign(&c_b49, &d__[
ll]) + shift / d__[ll]);
g = e[ll];
i__1 = m - 1;
for (i__ = ll; i__ <= i__1; ++i__) {
slartg_(&f, &g, &cosr, &sinr, &r__);
if (i__ > ll) {
e[i__ - 1] = r__;
}
f = cosr * d__[i__] + sinr * e[i__];
e[i__] = cosr * e[i__] - sinr * d__[i__];
g = sinr * d__[i__ + 1];
d__[i__ + 1] = cosr * d__[i__ + 1];
slartg_(&f, &g, &cosl, &sinl, &r__);
d__[i__] = r__;
f = cosl * e[i__] + sinl * d__[i__ + 1];
d__[i__ + 1] = cosl * d__[i__ + 1] - sinl * e[i__];
if (i__ < m - 1) {
g = sinl * e[i__ + 1];
e[i__ + 1] = cosl * e[i__ + 1];
}
work[i__ - ll + 1] = cosr;
work[i__ - ll + 1 + nm1] = sinr;
work[i__ - ll + 1 + nm12] = cosl;
work[i__ - ll + 1 + nm13] = sinl;
/* L140: */
}
e[m - 1] = f;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &vt[
ll + vt_dim1], ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
+ 1], &u[ll * u_dim1 + 1], ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
+ 1], &c__[ll + c_dim1], ldc);
}
/* Test convergence */
if ((r__1 = e[m - 1], ABS(r__1)) <= thresh) {
e[m - 1] = 0.f;
}
} else {
/* Chase bulge from bottom to top */
/* Save cosines and sines for later singular vector updates */
f = ((r__1 = d__[m], ABS(r__1)) - shift) * (r_sign(&c_b49, &d__[
m]) + shift / d__[m]);
g = e[m - 1];
i__1 = ll + 1;
for (i__ = m; i__ >= i__1; --i__) {
slartg_(&f, &g, &cosr, &sinr, &r__);
if (i__ < m) {
e[i__] = r__;
}
f = cosr * d__[i__] + sinr * e[i__ - 1];
e[i__ - 1] = cosr * e[i__ - 1] - sinr * d__[i__];
g = sinr * d__[i__ - 1];
d__[i__ - 1] = cosr * d__[i__ - 1];
slartg_(&f, &g, &cosl, &sinl, &r__);
d__[i__] = r__;
f = cosl * e[i__ - 1] + sinl * d__[i__ - 1];
d__[i__ - 1] = cosl * d__[i__ - 1] - sinl * e[i__ - 1];
if (i__ > ll + 1) {
g = sinl * e[i__ - 2];
e[i__ - 2] = cosl * e[i__ - 2];
}
work[i__ - ll] = cosr;
work[i__ - ll + nm1] = -sinr;
work[i__ - ll + nm12] = cosl;
work[i__ - ll + nm13] = -sinl;
/* L150: */
}
e[ll] = f;
/* Test convergence */
if ((r__1 = e[ll], ABS(r__1)) <= thresh) {
e[ll] = 0.f;
}
/* Update singular vectors if desired */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
nm13 + 1], &vt[ll + vt_dim1], ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u[ll *
u_dim1 + 1], ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &c__[
ll + c_dim1], ldc);
}
}
}
/* QR iteration finished, go back and check convergence */
goto L60;
/* All singular values converged, so make them positive */
L160:
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] < 0.f) {
d__[i__] = -d__[i__];
/* Change sign of singular vectors, if desired */
if (*ncvt > 0) {
sscal_(ncvt, &c_b72, &vt[i__ + vt_dim1], ldvt);
}
}
/* L170: */
}
/* Sort the singular values into decreasing order (insertion sort on */
/* singular values, but only one transposition per singular vector) */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Scan for smallest D(I) */
isub = 1;
smin = d__[1];
i__2 = *n + 1 - i__;
for (j = 2; j <= i__2; ++j) {
if (d__[j] <= smin) {
isub = j;
smin = d__[j];
}
/* L180: */
}
if (isub != *n + 1 - i__) {
/* Swap singular values and vectors */
d__[isub] = d__[*n + 1 - i__];
d__[*n + 1 - i__] = smin;
if (*ncvt > 0) {
sswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[*n + 1 - i__ +
vt_dim1], ldvt);
}
if (*nru > 0) {
sswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[(*n + 1 - i__) *
u_dim1 + 1], &c__1);
}
if (*ncc > 0) {
sswap_(ncc, &c__[isub + c_dim1], ldc, &c__[*n + 1 - i__ +
c_dim1], ldc);
}
}
/* L190: */
}
goto L220;
/* Maximum number of iterations exceeded, failure to converge */
L200:
*info = 0;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (e[i__] != 0.f) {
++(*info);
}
/* L210: */
}
L220:
return 0;
/* End of SBDSQR */
} /* sbdsqr_ */
/* Subroutine */ int clalsd_(char *uplo, integer *smlsiz, integer *n, integer
*nrhs, real *d__, real *e, complex *b, integer *ldb, real *rcond,
integer *rank, complex *work, real *rwork, integer *iwork, integer *
info)
{
/* System generated locals */
integer b_dim1, b_offset, i__1, i__2, i__3, i__4, i__5, i__6;
real r__1;
complex q__1;
/* Builtin functions */
double r_imag(complex *), log(doublereal), r_sign(real *, real *);
/* Local variables */
integer c__, i__, j, k;
real r__;
integer s, u, z__;
real cs;
integer bx;
real sn;
integer st, vt, nm1, st1;
real eps;
integer iwk;
real tol;
integer difl, difr;
real rcnd;
integer jcol, irwb, perm, nsub, nlvl, sqre, bxst, jrow, irwu, jimag,
jreal;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
integer irwib;
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
integer poles, sizei, irwrb, nsize;
extern /* Subroutine */ int csrot_(integer *, complex *, integer *,
complex *, integer *, real *, real *);
integer irwvt, icmpq1, icmpq2;
extern /* Subroutine */ int clalsa_(integer *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, real *,
integer *, real *, integer *, real *, real *, real *, real *,
integer *, integer *, integer *, integer *, real *, real *, real *
, real *, integer *, integer *), clascl_(char *, integer *,
integer *, real *, real *, integer *, integer *, complex *,
integer *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *),
clacpy_(char *, integer *, integer *, complex *, integer *,
complex *, integer *), claset_(char *, integer *, integer
*, complex *, complex *, complex *, integer *), xerbla_(
char *, integer *), slascl_(char *, integer *, integer *,
real *, real *, integer *, integer *, real *, integer *, integer *
);
extern integer isamax_(integer *, real *, integer *);
integer givcol;
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *),
slaset_(char *, integer *, integer *, real *, real *, real *,
integer *), slartg_(real *, real *, real *, real *, real *
);
real orgnrm;
integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
extern /* Subroutine */ int slasrt_(char *, integer *, real *, integer *);
integer givptr, nrwork, irwwrk, smlszp;
/* -- LAPACK routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CLALSD uses the singular value decomposition of A to solve the least */
/* squares problem of finding X to minimize the Euclidean norm of each */
/* column of A*X-B, where A is N-by-N upper bidiagonal, and X and B */
/* are N-by-NRHS. The solution X overwrites B. */
/* The singular values of A smaller than RCOND times the largest */
/* singular value are treated as zero in solving the least squares */
/* problem; in this case a minimum norm solution is returned. */
/* The actual singular values are returned in D in ascending order. */
/* This code makes very mild assumptions about floating point */
/* arithmetic. It will work on machines with a guard digit in */
/* add/subtract, or on those binary machines without guard digits */
/* which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. */
/* It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': D and E define an upper bidiagonal matrix. */
/* = 'L': D and E define a lower bidiagonal matrix. */
/* SMLSIZ (input) INTEGER */
/* The maximum size of the subproblems at the bottom of the */
/* computation tree. */
/* N (input) INTEGER */
/* The dimension of the bidiagonal matrix. N >= 0. */
/* NRHS (input) INTEGER */
/* The number of columns of B. NRHS must be at least 1. */
/* D (input/output) REAL array, dimension (N) */
/* On entry D contains the main diagonal of the bidiagonal */
/* matrix. On exit, if INFO = 0, D contains its singular values. */
/* E (input/output) REAL array, dimension (N-1) */
/* Contains the super-diagonal entries of the bidiagonal matrix. */
/* On exit, E has been destroyed. */
/* B (input/output) COMPLEX array, dimension (LDB,NRHS) */
/* On input, B contains the right hand sides of the least */
/* squares problem. On output, B contains the solution X. */
/* LDB (input) INTEGER */
/* The leading dimension of B in the calling subprogram. */
/* LDB must be at least max(1,N). */
/* RCOND (input) REAL */
/* The singular values of A less than or equal to RCOND times */
/* the largest singular value are treated as zero in solving */
/* the least squares problem. If RCOND is negative, */
/* machine precision is used instead. */
/* For example, if diag(S)*X=B were the least squares problem, */
/* where diag(S) is a diagonal matrix of singular values, the */
/* solution would be X(i) = B(i) / S(i) if S(i) is greater than */
/* RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to */
/* RCOND*max(S). */
/* RANK (output) INTEGER */
/* The number of singular values of A greater than RCOND times */
/* the largest singular value. */
/* WORK (workspace) COMPLEX array, dimension (N * NRHS). */
/* RWORK (workspace) REAL array, dimension at least */
/* (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2), */
/* where */
/* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) */
/* IWORK (workspace) INTEGER array, dimension (3*N*NLVL + 11*N). */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: The algorithm failed to compute an singular value while */
/* working on the submatrix lying in rows and columns */
/* INFO/(N+1) through MOD(INFO,N+1). */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Ren-Cang Li, Computer Science Division, University of */
/* California at Berkeley, USA */
/* Osni Marques, LBNL/NERSC, USA */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--work;
--rwork;
--iwork;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -3;
} else if (*nrhs < 1) {
*info = -4;
} else if (*ldb < 1 || *ldb < *n) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CLALSD", &i__1);
return 0;
}
eps = slamch_("Epsilon");
/* Set up the tolerance. */
if (*rcond <= 0.f || *rcond >= 1.f) {
rcnd = eps;
} else {
rcnd = *rcond;
}
*rank = 0;
/* Quick return if possible. */
if (*n == 0) {
return 0;
} else if (*n == 1) {
if (d__[1] == 0.f) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
} else {
*rank = 1;
clascl_("G", &c__0, &c__0, &d__[1], &c_b10, &c__1, nrhs, &b[
b_offset], ldb, info);
d__[1] = dabs(d__[1]);
}
return 0;
}
/* Rotate the matrix if it is lower bidiagonal. */
if (*(unsigned char *)uplo == 'L') {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (*nrhs == 1) {
csrot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], &
c__1, &cs, &sn);
} else {
rwork[(i__ << 1) - 1] = cs;
rwork[i__ * 2] = sn;
}
/* L10: */
}
if (*nrhs > 1) {
i__1 = *nrhs;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n - 1;
for (j = 1; j <= i__2; ++j) {
cs = rwork[(j << 1) - 1];
sn = rwork[j * 2];
csrot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__
* b_dim1], &c__1, &cs, &sn);
/* L20: */
}
/* L30: */
}
}
}
/* Scale. */
nm1 = *n - 1;
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
claset_("A", n, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
return 0;
}
slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, &c__1, &d__[1], n, info);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b10, &nm1, &c__1, &e[1], &nm1,
info);
/* If N is smaller than the minimum divide size SMLSIZ, then solve */
/* the problem with another solver. */
if (*n <= *smlsiz) {
irwu = 1;
irwvt = irwu + *n * *n;
irwwrk = irwvt + *n * *n;
irwrb = irwwrk;
irwib = irwrb + *n * *nrhs;
irwb = irwib + *n * *nrhs;
slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwu], n);
slaset_("A", n, n, &c_b35, &c_b10, &rwork[irwvt], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &rwork[irwvt], n,
&rwork[irwu], n, &rwork[irwwrk], &c__1, &rwork[irwwrk], info);
if (*info != 0) {
return 0;
}
/* In the real version, B is passed to SLASDQ and multiplied */
/* internally by Q'. Here B is complex and that product is */
/* computed below in two steps (real and imaginary parts). */
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
i__3 = jrow + jcol * b_dim1;
rwork[j] = b[i__3].r;
/* L40: */
}
/* L50: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
&c_b35, &rwork[irwrb], n);
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L60: */
}
/* L70: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
&c_b35, &rwork[irwib], n);
jreal = irwrb - 1;
jimag = irwib - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++jreal;
++jimag;
i__3 = jrow + jcol * b_dim1;
i__4 = jreal;
i__5 = jimag;
q__1.r = rwork[i__4], q__1.i = rwork[i__5];
b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L80: */
}
/* L90: */
}
tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] <= tol) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[i__ + b_dim1], ldb);
} else {
clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &b[
i__ + b_dim1], ldb, info);
++(*rank);
}
/* L100: */
}
/* Since B is complex, the following call to SGEMM is performed */
/* in two steps (real and imaginary parts). That is for V * B */
/* (in the real version of the code V' is stored in WORK). */
/* CALL SGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO, */
/* $ WORK( NWORK ), N ) */
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
i__3 = jrow + jcol * b_dim1;
rwork[j] = b[i__3].r;
/* L110: */
}
/* L120: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb],
n, &c_b35, &rwork[irwrb], n);
j = irwb - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L130: */
}
/* L140: */
}
sgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb],
n, &c_b35, &rwork[irwib], n);
jreal = irwrb - 1;
jimag = irwib - 1;
i__1 = *nrhs;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = 1; jrow <= i__2; ++jrow) {
++jreal;
++jimag;
i__3 = jrow + jcol * b_dim1;
i__4 = jreal;
i__5 = jimag;
q__1.r = rwork[i__4], q__1.i = rwork[i__5];
b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L150: */
}
/* L160: */
}
/* Unscale. */
slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n,
info);
slasrt_("D", n, &d__[1], info);
clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset],
ldb, info);
return 0;
}
/* Book-keeping and setting up some constants. */
nlvl = (integer) (log((real) (*n) / (real) (*smlsiz + 1)) / log(2.f)) + 1;
smlszp = *smlsiz + 1;
u = 1;
vt = *smlsiz * *n + 1;
difl = vt + smlszp * *n;
difr = difl + nlvl * *n;
z__ = difr + (nlvl * *n << 1);
c__ = z__ + nlvl * *n;
s = c__ + *n;
poles = s + *n;
givnum = poles + (nlvl << 1) * *n;
nrwork = givnum + (nlvl << 1) * *n;
bx = 1;
irwrb = nrwork;
irwib = irwrb + *smlsiz * *nrhs;
irwb = irwib + *smlsiz * *nrhs;
sizei = *n + 1;
k = sizei + *n;
givptr = k + *n;
perm = givptr + *n;
givcol = perm + nlvl * *n;
iwk = givcol + (nlvl * *n << 1);
st = 1;
sqre = 0;
icmpq1 = 1;
icmpq2 = 0;
nsub = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L170: */
}
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
++nsub;
iwork[nsub] = st;
/* Subproblem found. First determine its size and then */
/* apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - st + 1;
iwork[sizei + nsub - 1] = nsize;
} else {
/* A subproblem with E(NM1) small. This implies an */
/* 1-by-1 subproblem at D(N), which is not solved */
/* explicitly. */
nsize = i__ - st + 1;
iwork[sizei + nsub - 1] = nsize;
++nsub;
iwork[nsub] = *n;
iwork[sizei + nsub - 1] = 1;
ccopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n);
}
st1 = st - 1;
if (nsize == 1) {
/* This is a 1-by-1 subproblem and is not solved */
/* explicitly. */
ccopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
} else if (nsize <= *smlsiz) {
/* This is a small subproblem and is solved by SLASDQ. */
slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[vt + st1],
n);
slaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[u + st1],
n);
slasdq_("U", &c__0, &nsize, &nsize, &nsize, &c__0, &d__[st], &
e[st], &rwork[vt + st1], n, &rwork[u + st1], n, &
rwork[nrwork], &c__1, &rwork[nrwork], info)
;
if (*info != 0) {
return 0;
}
/* In the real version, B is passed to SLASDQ and multiplied */
/* internally by Q'. Here B is complex and that product is */
/* computed below in two steps (real and imaginary parts). */
j = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++j;
i__4 = jrow + jcol * b_dim1;
rwork[j] = b[i__4].r;
/* L180: */
}
/* L190: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &
nsize);
j = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++j;
rwork[j] = r_imag(&b[jrow + jcol * b_dim1]);
/* L200: */
}
/* L210: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &
nsize);
jreal = irwrb - 1;
jimag = irwib - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = jrow + jcol * b_dim1;
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
b[i__4].r = q__1.r, b[i__4].i = q__1.i;
/* L220: */
}
/* L230: */
}
clacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx +
st1], n);
} else {
/* A large problem. Solve it using divide and conquer. */
slasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
rwork[u + st1], n, &rwork[vt + st1], &iwork[k + st1],
&rwork[difl + st1], &rwork[difr + st1], &rwork[z__ +
st1], &rwork[poles + st1], &iwork[givptr + st1], &
iwork[givcol + st1], n, &iwork[perm + st1], &rwork[
givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &
rwork[nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
bxst = bx + st1;
clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, &
work[bxst], n, &rwork[u + st1], n, &rwork[vt + st1], &
iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1]
, &rwork[z__ + st1], &rwork[poles + st1], &iwork[
givptr + st1], &iwork[givcol + st1], n, &iwork[perm +
st1], &rwork[givnum + st1], &rwork[c__ + st1], &rwork[
s + st1], &rwork[nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
st = i__ + 1;
}
/* L240: */
}
/* Apply the singular values and treat the tiny ones as zero. */
tol = rcnd * (r__1 = d__[isamax_(n, &d__[1], &c__1)], dabs(r__1));
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Some of the elements in D can be negative because 1-by-1 */
/* subproblems were not solved explicitly. */
if ((r__1 = d__[i__], dabs(r__1)) <= tol) {
claset_("A", &c__1, nrhs, &c_b1, &c_b1, &work[bx + i__ - 1], n);
} else {
++(*rank);
clascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &work[
bx + i__ - 1], n, info);
}
d__[i__] = (r__1 = d__[i__], dabs(r__1));
/* L250: */
}
/* Now apply back the right singular vectors. */
icmpq2 = 1;
i__1 = nsub;
for (i__ = 1; i__ <= i__1; ++i__) {
st = iwork[i__];
st1 = st - 1;
nsize = iwork[sizei + i__ - 1];
bxst = bx + st1;
if (nsize == 1) {
ccopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb);
} else if (nsize <= *smlsiz) {
/* Since B and BX are complex, the following call to SGEMM */
/* is performed in two steps (real and imaginary parts). */
/* CALL SGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE, */
/* $ RWORK( VT+ST1 ), N, RWORK( BXST ), N, ZERO, */
/* $ B( ST, 1 ), LDB ) */
j = bxst - *n - 1;
jreal = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
j += *n;
i__3 = nsize;
for (jrow = 1; jrow <= i__3; ++jrow) {
++jreal;
i__4 = j + jrow;
rwork[jreal] = work[i__4].r;
/* L260: */
}
/* L270: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1],
n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &nsize);
j = bxst - *n - 1;
jimag = irwb - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
j += *n;
i__3 = nsize;
for (jrow = 1; jrow <= i__3; ++jrow) {
++jimag;
rwork[jimag] = r_imag(&work[j + jrow]);
/* L280: */
}
/* L290: */
}
sgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1],
n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &nsize);
jreal = irwrb - 1;
jimag = irwib - 1;
i__2 = *nrhs;
for (jcol = 1; jcol <= i__2; ++jcol) {
i__3 = st + nsize - 1;
for (jrow = st; jrow <= i__3; ++jrow) {
++jreal;
++jimag;
i__4 = jrow + jcol * b_dim1;
i__5 = jreal;
i__6 = jimag;
q__1.r = rwork[i__5], q__1.i = rwork[i__6];
b[i__4].r = q__1.r, b[i__4].i = q__1.i;
/* L300: */
}
/* L310: */
}
} else {
clalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st +
b_dim1], ldb, &rwork[u + st1], n, &rwork[vt + st1], &
iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1], &
rwork[z__ + st1], &rwork[poles + st1], &iwork[givptr +
st1], &iwork[givcol + st1], n, &iwork[perm + st1], &rwork[
givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &rwork[
nrwork], &iwork[iwk], info);
if (*info != 0) {
return 0;
}
}
/* L320: */
}
/* Unscale and sort the singular values. */
slascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, info);
slasrt_("D", n, &d__[1], info);
clascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], ldb,
info);
return 0;
/* End of CLALSD */
} /* clalsd_ */
/* Subroutine */ int ssteqr_(char *compz, integer *n, real *d__, real *e,
real *z__, integer *ldz, real *work, integer *info)
{
/* System generated locals */
integer z_dim1, z_offset, i__1, i__2;
real r__1, r__2;
/* Local variables */
real b, c__, f, g;
integer i__, j, k, l, m;
real p, r__, s;
integer l1, ii, mm, lm1, mm1, nm1;
real rt1, rt2, eps;
integer lsv;
real tst, eps2;
integer lend, jtot;
real anorm;
integer lendm1, lendp1;
integer iscale;
real safmin;
real safmax;
integer lendsv;
real ssfmin;
integer nmaxit, icompz;
real ssfmax;
/* -- LAPACK routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* SSTEQR computes all eigenvalues and, optionally, eigenvectors of a */
/* symmetric tridiagonal matrix using the implicit QL or QR method. */
/* The eigenvectors of a full or band symmetric matrix can also be found */
/* if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to */
/* tridiagonal form. */
/* Arguments */
/* ========= */
/* COMPZ (input) CHARACTER*1 */
/* = 'N': Compute eigenvalues only. */
/* = 'V': Compute eigenvalues and eigenvectors of the original */
/* symmetric matrix. On entry, Z must contain the */
/* orthogonal matrix used to reduce the original matrix */
/* to tridiagonal form. */
/* = 'I': Compute eigenvalues and eigenvectors of the */
/* tridiagonal matrix. Z is initialized to the identity */
/* matrix. */
/* N (input) INTEGER */
/* The order of the matrix. N >= 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, the diagonal elements of the tridiagonal matrix. */
/* On exit, if INFO = 0, the eigenvalues in ascending order. */
/* E (input/output) REAL array, dimension (N-1) */
/* On entry, the (n-1) subdiagonal elements of the tridiagonal */
/* matrix. */
/* On exit, E has been destroyed. */
/* Z (input/output) REAL array, dimension (LDZ, N) */
/* On entry, if COMPZ = 'V', then Z contains the orthogonal */
/* matrix used in the reduction to tridiagonal form. */
/* On exit, if INFO = 0, then if COMPZ = 'V', Z contains the */
/* orthonormal eigenvectors of the original symmetric matrix, */
/* and if COMPZ = 'I', Z contains the orthonormal eigenvectors */
/* of the symmetric tridiagonal matrix. */
/* If COMPZ = 'N', then Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1, and if */
/* eigenvectors are desired, then LDZ >= max(1,N). */
/* WORK (workspace) REAL array, dimension (max(1,2*N-2)) */
/* If COMPZ = 'N', then WORK is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* > 0: the algorithm has failed to find all the eigenvalues in */
/* a total of 30*N iterations; if INFO = i, then i */
/* elements of E have not converged to zero; on exit, D */
/* and E contain the elements of a symmetric tridiagonal */
/* matrix which is orthogonally similar to the original */
/* matrix. */
/* ===================================================================== */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
if (lsame_(compz, "N")) {
icompz = 0;
} else if (lsame_(compz, "V")) {
icompz = 1;
} else if (lsame_(compz, "I")) {
icompz = 2;
} else {
icompz = -1;
}
if (icompz < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEQR", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (icompz == 2) {
z__[z_dim1 + 1] = 1.f;
}
return 0;
}
/* Determine the unit roundoff and over/underflow thresholds. */
eps = slamch_("E");
/* Computing 2nd power */
r__1 = eps;
eps2 = r__1 * r__1;
safmin = slamch_("S");
safmax = 1.f / safmin;
ssfmax = sqrt(safmax) / 3.f;
ssfmin = sqrt(safmin) / eps2;
/* Compute the eigenvalues and eigenvectors of the tridiagonal */
/* matrix. */
if (icompz == 2) {
slaset_("Full", n, n, &c_b9, &c_b10, &z__[z_offset], ldz);
}
nmaxit = *n * 30;
jtot = 0;
/* Determine where the matrix splits and choose QL or QR iteration */
/* for each block, according to whether top or bottom diagonal */
/* element is smaller. */
l1 = 1;
nm1 = *n - 1;
L10:
if (l1 > *n) {
goto L160;
}
if (l1 > 1) {
e[l1 - 1] = 0.f;
}
if (l1 <= nm1) {
i__1 = nm1;
for (m = l1; m <= i__1; ++m) {
tst = (r__1 = e[m], dabs(r__1));
if (tst == 0.f) {
goto L30;
}
if (tst <= sqrt((r__1 = d__[m], dabs(r__1))) * sqrt((r__2 = d__[m
+ 1], dabs(r__2))) * eps) {
e[m] = 0.f;
goto L30;
}
}
}
m = *n;
L30:
l = l1;
lsv = l;
lend = m;
lendsv = lend;
l1 = m + 1;
if (lend == l) {
goto L10;
}
/* Scale submatrix in rows and columns L to LEND */
i__1 = lend - l + 1;
anorm = slanst_("I", &i__1, &d__[l], &e[l]);
iscale = 0;
if (anorm == 0.f) {
goto L10;
}
if (anorm > ssfmax) {
iscale = 1;
i__1 = lend - l + 1;
slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &d__[l], n,
info);
i__1 = lend - l;
slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &e[l], n,
info);
} else if (anorm < ssfmin) {
iscale = 2;
i__1 = lend - l + 1;
slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &d__[l], n,
info);
i__1 = lend - l;
slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &e[l], n,
info);
}
/* Choose between QL and QR iteration */
if ((r__1 = d__[lend], dabs(r__1)) < (r__2 = d__[l], dabs(r__2))) {
lend = lsv;
l = lendsv;
}
if (lend > l) {
/* QL Iteration */
/* Look for small subdiagonal element. */
L40:
if (l != lend) {
lendm1 = lend - 1;
i__1 = lendm1;
for (m = l; m <= i__1; ++m) {
/* Computing 2nd power */
r__2 = (r__1 = e[m], dabs(r__1));
tst = r__2 * r__2;
if (tst <= eps2 * (r__1 = d__[m], dabs(r__1)) * (r__2 = d__[m
+ 1], dabs(r__2)) + safmin) {
goto L60;
}
}
}
m = lend;
L60:
if (m < lend) {
e[m] = 0.f;
}
p = d__[l];
if (m == l) {
goto L80;
}
/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2 */
/* to compute its eigensystem. */
if (m == l + 1) {
if (icompz > 0) {
slaev2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2, &c__, &s);
work[l] = c__;
work[*n - 1 + l] = s;
slasr_("R", "V", "B", n, &c__2, &work[l], &work[*n - 1 + l], &
z__[l * z_dim1 + 1], ldz);
} else {
slae2_(&d__[l], &e[l], &d__[l + 1], &rt1, &rt2);
}
d__[l] = rt1;
d__[l + 1] = rt2;
e[l] = 0.f;
l += 2;
if (l <= lend) {
goto L40;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (d__[l + 1] - p) / (e[l] * 2.f);
r__ = slapy2_(&g, &c_b10);
g = d__[m] - p + e[l] / (g + r_sign(&r__, &g));
s = 1.f;
c__ = 1.f;
p = 0.f;
/* Inner loop */
mm1 = m - 1;
i__1 = l;
for (i__ = mm1; i__ >= i__1; --i__) {
f = s * e[i__];
b = c__ * e[i__];
slartg_(&g, &f, &c__, &s, &r__);
if (i__ != m - 1) {
e[i__ + 1] = r__;
}
g = d__[i__ + 1] - p;
r__ = (d__[i__] - g) * s + c__ * 2.f * b;
p = s * r__;
d__[i__ + 1] = g + p;
g = c__ * r__ - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = -s;
}
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = m - l + 1;
slasr_("R", "V", "B", n, &mm, &work[l], &work[*n - 1 + l], &z__[l
* z_dim1 + 1], ldz);
}
d__[l] -= p;
e[l] = g;
goto L40;
/* Eigenvalue found. */
L80:
d__[l] = p;
++l;
if (l <= lend) {
goto L40;
}
goto L140;
} else {
/* QR Iteration */
/* Look for small superdiagonal element. */
L90:
if (l != lend) {
lendp1 = lend + 1;
i__1 = lendp1;
for (m = l; m >= i__1; --m) {
/* Computing 2nd power */
r__2 = (r__1 = e[m - 1], dabs(r__1));
tst = r__2 * r__2;
if (tst <= eps2 * (r__1 = d__[m], dabs(r__1)) * (r__2 = d__[m
- 1], dabs(r__2)) + safmin) {
goto L110;
}
}
}
m = lend;
L110:
if (m > lend) {
e[m - 1] = 0.f;
}
p = d__[l];
if (m == l) {
goto L130;
}
/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2 */
/* to compute its eigensystem. */
if (m == l - 1) {
if (icompz > 0) {
slaev2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2, &c__, &s)
;
work[m] = c__;
work[*n - 1 + m] = s;
slasr_("R", "V", "F", n, &c__2, &work[m], &work[*n - 1 + m], &
z__[(l - 1) * z_dim1 + 1], ldz);
} else {
slae2_(&d__[l - 1], &e[l - 1], &d__[l], &rt1, &rt2);
}
d__[l - 1] = rt1;
d__[l] = rt2;
e[l - 1] = 0.f;
l += -2;
if (l >= lend) {
goto L90;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (d__[l - 1] - p) / (e[l - 1] * 2.f);
r__ = slapy2_(&g, &c_b10);
g = d__[m] - p + e[l - 1] / (g + r_sign(&r__, &g));
s = 1.f;
c__ = 1.f;
p = 0.f;
/* Inner loop */
lm1 = l - 1;
i__1 = lm1;
for (i__ = m; i__ <= i__1; ++i__) {
f = s * e[i__];
b = c__ * e[i__];
slartg_(&g, &f, &c__, &s, &r__);
if (i__ != m) {
e[i__ - 1] = r__;
}
g = d__[i__] - p;
r__ = (d__[i__ + 1] - g) * s + c__ * 2.f * b;
p = s * r__;
d__[i__] = g + p;
g = c__ * r__ - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
work[i__] = c__;
work[*n - 1 + i__] = s;
}
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = l - m + 1;
slasr_("R", "V", "F", n, &mm, &work[m], &work[*n - 1 + m], &z__[m
* z_dim1 + 1], ldz);
}
d__[l] -= p;
e[lm1] = g;
goto L90;
/* Eigenvalue found. */
L130:
d__[l] = p;
--l;
if (l >= lend) {
goto L90;
}
goto L140;
}
/* Undo scaling if necessary */
L140:
if (iscale == 1) {
i__1 = lendsv - lsv + 1;
slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &d__[lsv],
n, info);
i__1 = lendsv - lsv;
slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &e[lsv], n,
info);
} else if (iscale == 2) {
i__1 = lendsv - lsv + 1;
slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &d__[lsv],
n, info);
i__1 = lendsv - lsv;
slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &e[lsv], n,
info);
}
/* Check for no convergence to an eigenvalue after a total */
/* of N*MAXIT iterations. */
if (jtot < nmaxit) {
goto L10;
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (e[i__] != 0.f) {
++(*info);
}
}
goto L190;
/* Order eigenvalues and eigenvectors. */
L160:
if (icompz == 0) {
/* Use Quick Sort */
slasrt_("I", n, &d__[1], info);
} else {
/* Use Selection Sort to minimize swaps of eigenvectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
k = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] < p) {
k = j;
p = d__[j];
}
}
if (k != i__) {
d__[k] = d__[i__];
d__[i__] = p;
sswap_(n, &z__[i__ * z_dim1 + 1], &c__1, &z__[k * z_dim1 + 1],
&c__1);
}
}
}
L190:
return 0;
/* End of SSTEQR */
} /* ssteqr_ */
/* Subroutine */ int sgbbrd_(char *vect, integer *m, integer *n, integer *ncc,
integer *kl, integer *ku, real *ab, integer *ldab, real *d__, real *
e, real *q, integer *ldq, real *pt, integer *ldpt, real *c__, integer
*ldc, real *work, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SGBBRD reduces a real general m-by-n band matrix A to upper
bidiagonal form B by an orthogonal transformation: Q' * A * P = B.
The routine computes B, and optionally forms Q or P', or computes
Q'*C for a given matrix C.
Arguments
=========
VECT (input) CHARACTER*1
Specifies whether or not the matrices Q and P' are to be
formed.
= 'N': do not form Q or P';
= 'Q': form Q only;
= 'P': form P' only;
= 'B': form both.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NCC (input) INTEGER
The number of columns of the matrix C. NCC >= 0.
KL (input) INTEGER
The number of subdiagonals of the matrix A. KL >= 0.
KU (input) INTEGER
The number of superdiagonals of the matrix A. KU >= 0.
AB (input/output) REAL array, dimension (LDAB,N)
On entry, the m-by-n band matrix A, stored in rows 1 to
KL+KU+1. The j-th column of A is stored in the j-th column of
the array AB as follows:
AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
On exit, A is overwritten by values generated during the
reduction.
LDAB (input) INTEGER
The leading dimension of the array A. LDAB >= KL+KU+1.
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B.
E (output) REAL array, dimension (min(M,N)-1)
The superdiagonal elements of the bidiagonal matrix B.
Q (output) REAL array, dimension (LDQ,M)
If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
If VECT = 'N' or 'P', the array Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q.
LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
PT (output) REAL array, dimension (LDPT,N)
If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
If VECT = 'N' or 'Q', the array PT is not referenced.
LDPT (input) INTEGER
The leading dimension of the array PT.
LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
C (input/output) REAL array, dimension (LDC,NCC)
On entry, an m-by-ncc matrix C.
On exit, C is overwritten by Q'*C.
C is not referenced if NCC = 0.
LDC (input) INTEGER
The leading dimension of the array C.
LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
WORK (workspace) REAL array, dimension (2*max(M,N))
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static real c_b8 = 0.f;
static real c_b9 = 1.f;
static integer c__1 = 1;
/* System generated locals */
integer ab_dim1, ab_offset, c_dim1, c_offset, pt_dim1, pt_offset, q_dim1,
q_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
/* Local variables */
static integer inca;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static integer i__, j, l;
extern logical lsame_(char *, char *);
static logical wantb, wantc;
static integer minmn;
static logical wantq;
static integer j1, j2, kb;
static real ra, rb, rc;
static integer kk, ml, mn, nr, mu;
static real rs;
extern /* Subroutine */ int xerbla_(char *, integer *), slaset_(
char *, integer *, integer *, real *, real *, real *, integer *), slartg_(real *, real *, real *, real *, real *);
static integer kb1;
extern /* Subroutine */ int slargv_(integer *, real *, integer *, real *,
integer *, real *, integer *);
static integer ml0;
extern /* Subroutine */ int slartv_(integer *, real *, integer *, real *,
integer *, real *, real *, integer *);
static logical wantpt;
static integer mu0, klm, kun, nrt, klu1;
#define c___ref(a_1,a_2) c__[(a_2)*c_dim1 + a_1]
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1]
#define pt_ref(a_1,a_2) pt[(a_2)*pt_dim1 + a_1]
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1 * 1;
ab -= ab_offset;
--d__;
--e;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
pt_dim1 = *ldpt;
pt_offset = 1 + pt_dim1 * 1;
pt -= pt_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1 * 1;
c__ -= c_offset;
--work;
/* Function Body */
wantb = lsame_(vect, "B");
wantq = lsame_(vect, "Q") || wantb;
wantpt = lsame_(vect, "P") || wantb;
wantc = *ncc > 0;
klu1 = *kl + *ku + 1;
*info = 0;
if (! wantq && ! wantpt && ! lsame_(vect, "N")) {
*info = -1;
} else if (*m < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ncc < 0) {
*info = -4;
} else if (*kl < 0) {
*info = -5;
} else if (*ku < 0) {
*info = -6;
} else if (*ldab < klu1) {
*info = -8;
} else if (*ldq < 1 || wantq && *ldq < max(1,*m)) {
*info = -12;
} else if (*ldpt < 1 || wantpt && *ldpt < max(1,*n)) {
*info = -14;
} else if (*ldc < 1 || wantc && *ldc < max(1,*m)) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGBBRD", &i__1);
return 0;
}
/* Initialize Q and P' to the unit matrix, if needed */
if (wantq) {
slaset_("Full", m, m, &c_b8, &c_b9, &q[q_offset], ldq);
}
if (wantpt) {
slaset_("Full", n, n, &c_b8, &c_b9, &pt[pt_offset], ldpt);
}
/* Quick return if possible. */
if (*m == 0 || *n == 0) {
return 0;
}
minmn = min(*m,*n);
if (*kl + *ku > 1) {
/* Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
first to lower bidiagonal form and then transform to upper
bidiagonal */
if (*ku > 0) {
ml0 = 1;
mu0 = 2;
} else {
ml0 = 2;
mu0 = 1;
}
/* Wherever possible, plane rotations are generated and applied in
vector operations of length NR over the index set J1:J2:KLU1.
The sines of the plane rotations are stored in WORK(1:max(m,n))
and the cosines in WORK(max(m,n)+1:2*max(m,n)). */
mn = max(*m,*n);
/* Computing MIN */
i__1 = *m - 1;
klm = min(i__1,*kl);
/* Computing MIN */
i__1 = *n - 1;
kun = min(i__1,*ku);
kb = klm + kun;
kb1 = kb + 1;
inca = kb1 * *ldab;
nr = 0;
j1 = klm + 2;
j2 = 1 - kun;
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th column and i-th row of matrix to bidiagonal form */
ml = klm + 1;
mu = kun + 1;
i__2 = kb;
for (kk = 1; kk <= i__2; ++kk) {
j1 += kb;
j2 += kb;
/* generate plane rotations to annihilate nonzero elements
which have been created below the band */
if (nr > 0) {
slargv_(&nr, &ab_ref(klu1, j1 - klm - 1), &inca, &work[j1]
, &kb1, &work[mn + j1], &kb1);
}
/* apply plane rotations from the left */
i__3 = kb;
for (l = 1; l <= i__3; ++l) {
if (j2 - klm + l - 1 > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab_ref(klu1 - l, j1 - klm + l - 1), &
inca, &ab_ref(klu1 - l + 1, j1 - klm + l - 1),
&inca, &work[mn + j1], &work[j1], &kb1);
}
/* L10: */
}
if (ml > ml0) {
if (ml <= *m - i__ + 1) {
/* generate plane rotation to annihilate a(i+ml-1,i)
within the band, and apply rotation from the left */
slartg_(&ab_ref(*ku + ml - 1, i__), &ab_ref(*ku + ml,
i__), &work[mn + i__ + ml - 1], &work[i__ +
ml - 1], &ra);
ab_ref(*ku + ml - 1, i__) = ra;
if (i__ < *n) {
/* Computing MIN */
i__4 = *ku + ml - 2, i__5 = *n - i__;
i__3 = min(i__4,i__5);
i__6 = *ldab - 1;
i__7 = *ldab - 1;
srot_(&i__3, &ab_ref(*ku + ml - 2, i__ + 1), &
i__6, &ab_ref(*ku + ml - 1, i__ + 1), &
i__7, &work[mn + i__ + ml - 1], &work[i__
+ ml - 1]);
}
}
++nr;
j1 -= kb1;
}
if (wantq) {
/* accumulate product of plane rotations in Q */
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4)
{
srot_(m, &q_ref(1, j - 1), &c__1, &q_ref(1, j), &c__1,
&work[mn + j], &work[j]);
/* L20: */
}
}
if (wantc) {
/* apply plane rotations to C */
i__4 = j2;
i__3 = kb1;
for (j = j1; i__3 < 0 ? j >= i__4 : j <= i__4; j += i__3)
{
srot_(ncc, &c___ref(j - 1, 1), ldc, &c___ref(j, 1),
ldc, &work[mn + j], &work[j]);
/* L30: */
}
}
if (j2 + kun > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 -= kb1;
}
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4) {
/* create nonzero element a(j-1,j+ku) above the band
and store it in WORK(n+1:2*n) */
work[j + kun] = work[j] * ab_ref(1, j + kun);
ab_ref(1, j + kun) = work[mn + j] * ab_ref(1, j + kun);
/* L40: */
}
/* generate plane rotations to annihilate nonzero elements
which have been generated above the band */
if (nr > 0) {
slargv_(&nr, &ab_ref(1, j1 + kun - 1), &inca, &work[j1 +
kun], &kb1, &work[mn + j1 + kun], &kb1);
}
/* apply plane rotations from the right */
i__4 = kb;
for (l = 1; l <= i__4; ++l) {
if (j2 + l - 1 > *m) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab_ref(l + 1, j1 + kun - 1), &inca, &
ab_ref(l, j1 + kun), &inca, &work[mn + j1 +
kun], &work[j1 + kun], &kb1);
}
/* L50: */
}
if (ml == ml0 && mu > mu0) {
if (mu <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i,i+mu-1)
within the band, and apply rotation from the right */
slartg_(&ab_ref(*ku - mu + 3, i__ + mu - 2), &ab_ref(*
ku - mu + 2, i__ + mu - 1), &work[mn + i__ +
mu - 1], &work[i__ + mu - 1], &ra);
ab_ref(*ku - mu + 3, i__ + mu - 2) = ra;
/* Computing MIN */
i__3 = *kl + mu - 2, i__5 = *m - i__;
i__4 = min(i__3,i__5);
srot_(&i__4, &ab_ref(*ku - mu + 4, i__ + mu - 2), &
c__1, &ab_ref(*ku - mu + 3, i__ + mu - 1), &
c__1, &work[mn + i__ + mu - 1], &work[i__ +
mu - 1]);
}
++nr;
j1 -= kb1;
}
if (wantpt) {
/* accumulate product of plane rotations in P' */
i__4 = j2;
i__3 = kb1;
for (j = j1; i__3 < 0 ? j >= i__4 : j <= i__4; j += i__3)
{
srot_(n, &pt_ref(j + kun - 1, 1), ldpt, &pt_ref(j +
kun, 1), ldpt, &work[mn + j + kun], &work[j +
kun]);
/* L60: */
}
}
if (j2 + kb > *m) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 -= kb1;
}
i__3 = j2;
i__4 = kb1;
for (j = j1; i__4 < 0 ? j >= i__3 : j <= i__3; j += i__4) {
/* create nonzero element a(j+kl+ku,j+ku-1) below the
band and store it in WORK(1:n) */
work[j + kb] = work[j + kun] * ab_ref(klu1, j + kun);
ab_ref(klu1, j + kun) = work[mn + j + kun] * ab_ref(klu1,
j + kun);
/* L70: */
}
if (ml > ml0) {
--ml;
} else {
--mu;
}
/* L80: */
}
/* L90: */
}
}
if (*ku == 0 && *kl > 0) {
/* A has been reduced to lower bidiagonal form
Transform lower bidiagonal form to upper bidiagonal by applying
plane rotations from the left, storing diagonal elements in D
and off-diagonal elements in E
Computing MIN */
i__2 = *m - 1;
i__1 = min(i__2,*n);
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&ab_ref(1, i__), &ab_ref(2, i__), &rc, &rs, &ra);
d__[i__] = ra;
if (i__ < *n) {
e[i__] = rs * ab_ref(1, i__ + 1);
ab_ref(1, i__ + 1) = rc * ab_ref(1, i__ + 1);
}
if (wantq) {
srot_(m, &q_ref(1, i__), &c__1, &q_ref(1, i__ + 1), &c__1, &
rc, &rs);
}
if (wantc) {
srot_(ncc, &c___ref(i__, 1), ldc, &c___ref(i__ + 1, 1), ldc, &
rc, &rs);
}
/* L100: */
}
if (*m <= *n) {
d__[*m] = ab_ref(1, *m);
}
} else if (*ku > 0) {
/* A has been reduced to upper bidiagonal form */
if (*m < *n) {
/* Annihilate a(m,m+1) by applying plane rotations from the
right, storing diagonal elements in D and off-diagonal
elements in E */
rb = ab_ref(*ku, *m + 1);
for (i__ = *m; i__ >= 1; --i__) {
slartg_(&ab_ref(*ku + 1, i__), &rb, &rc, &rs, &ra);
d__[i__] = ra;
if (i__ > 1) {
rb = -rs * ab_ref(*ku, i__);
e[i__ - 1] = rc * ab_ref(*ku, i__);
}
if (wantpt) {
srot_(n, &pt_ref(i__, 1), ldpt, &pt_ref(*m + 1, 1), ldpt,
&rc, &rs);
}
/* L110: */
}
} else {
/* Copy off-diagonal elements to E and diagonal elements to D */
i__1 = minmn - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab_ref(*ku, i__ + 1);
/* L120: */
}
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab_ref(*ku + 1, i__);
/* L130: */
}
}
} else {
/* A is diagonal. Set elements of E to zero and copy diagonal
elements to D. */
i__1 = minmn - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.f;
/* L140: */
}
i__1 = minmn;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab_ref(1, i__);
/* L150: */
}
}
return 0;
/* End of SGBBRD */
} /* sgbbrd_ */
/* Subroutine */ int slags2_(logical *upper, real *a1, real *a2, real *a3,
real *b1, real *b2, real *b3, real *csu, real *snu, real *csv, real *
snv, real *csq, real *snq)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
that if ( UPPER ) then
U'*A*Q = U'*( A1 A2 )*Q = ( x 0 )
( 0 A3 ) ( x x )
and
V'*B*Q = V'*( B1 B2 )*Q = ( x 0 )
( 0 B3 ) ( x x )
or if ( .NOT.UPPER ) then
U'*A*Q = U'*( A1 0 )*Q = ( x x )
( A2 A3 ) ( 0 x )
and
V'*B*Q = V'*( B1 0 )*Q = ( x x )
( B2 B3 ) ( 0 x )
The rows of the transformed A and B are parallel, where
U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ )
( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ )
Z' denotes the transpose of Z.
Arguments
=========
UPPER (input) LOGICAL
= .TRUE.: the input matrices A and B are upper triangular.
= .FALSE.: the input matrices A and B are lower triangular.
A1 (input) REAL
A2 (input) REAL
A3 (input) REAL
On entry, A1, A2 and A3 are elements of the input 2-by-2
upper (lower) triangular matrix A.
B1 (input) REAL
B2 (input) REAL
B3 (input) REAL
On entry, B1, B2 and B3 are elements of the input 2-by-2
upper (lower) triangular matrix B.
CSU (output) REAL
SNU (output) REAL
The desired orthogonal matrix U.
CSV (output) REAL
SNV (output) REAL
The desired orthogonal matrix V.
CSQ (output) REAL
SNQ (output) REAL
The desired orthogonal matrix Q.
===================================================================== */
/* System generated locals */
real r__1;
/* Local variables */
static real aua11, aua12, aua21, aua22, avb11, avb12, avb21, avb22, ua11r,
ua22r, vb11r, vb22r, a, b, c__, d__, r__, s1, s2;
extern /* Subroutine */ int slasv2_(real *, real *, real *, real *, real *
, real *, real *, real *, real *), slartg_(real *, real *, real *,
real *, real *);
static real ua11, ua12, ua21, ua22, vb11, vb12, vb21, vb22, csl, csr, snl,
snr;
if (*upper) {
/* Input matrices A and B are upper triangular matrices
Form matrix C = A*adj(B) = ( a b )
( 0 d ) */
a = *a1 * *b3;
d__ = *a3 * *b1;
b = *a2 * *b1 - *a1 * *b2;
/* The SVD of real 2-by-2 triangular C
( CSL -SNL )*( A B )*( CSR SNR ) = ( R 0 )
( SNL CSL ) ( 0 D ) ( -SNR CSR ) ( 0 T ) */
slasv2_(&a, &b, &d__, &s1, &s2, &snr, &csr, &snl, &csl);
if (dabs(csl) >= dabs(snl) || dabs(csr) >= dabs(snr)) {
/* Compute the (1,1) and (1,2) elements of U'*A and V'*B,
and (1,2) element of |U|'*|A| and |V|'*|B|. */
ua11r = csl * *a1;
ua12 = csl * *a2 + snl * *a3;
vb11r = csr * *b1;
vb12 = csr * *b2 + snr * *b3;
aua12 = dabs(csl) * dabs(*a2) + dabs(snl) * dabs(*a3);
avb12 = dabs(csr) * dabs(*b2) + dabs(snr) * dabs(*b3);
/* zero (1,2) elements of U'*A and V'*B */
if (dabs(ua11r) + dabs(ua12) != 0.f) {
if (aua12 / (dabs(ua11r) + dabs(ua12)) <= avb12 / (dabs(vb11r)
+ dabs(vb12))) {
r__1 = -ua11r;
slartg_(&r__1, &ua12, csq, snq, &r__);
} else {
r__1 = -vb11r;
slartg_(&r__1, &vb12, csq, snq, &r__);
}
} else {
r__1 = -vb11r;
slartg_(&r__1, &vb12, csq, snq, &r__);
}
*csu = csl;
*snu = -snl;
*csv = csr;
*snv = -snr;
} else {
/* Compute the (2,1) and (2,2) elements of U'*A and V'*B,
and (2,2) element of |U|'*|A| and |V|'*|B|. */
ua21 = -snl * *a1;
ua22 = -snl * *a2 + csl * *a3;
vb21 = -snr * *b1;
vb22 = -snr * *b2 + csr * *b3;
aua22 = dabs(snl) * dabs(*a2) + dabs(csl) * dabs(*a3);
avb22 = dabs(snr) * dabs(*b2) + dabs(csr) * dabs(*b3);
/* zero (2,2) elements of U'*A and V'*B, and then swap. */
if (dabs(ua21) + dabs(ua22) != 0.f) {
if (aua22 / (dabs(ua21) + dabs(ua22)) <= avb22 / (dabs(vb21)
+ dabs(vb22))) {
r__1 = -ua21;
slartg_(&r__1, &ua22, csq, snq, &r__);
} else {
r__1 = -vb21;
slartg_(&r__1, &vb22, csq, snq, &r__);
}
} else {
r__1 = -vb21;
slartg_(&r__1, &vb22, csq, snq, &r__);
}
*csu = snl;
*snu = csl;
*csv = snr;
*snv = csr;
}
} else {
/* Input matrices A and B are lower triangular matrices
Form matrix C = A*adj(B) = ( a 0 )
( c d ) */
a = *a1 * *b3;
d__ = *a3 * *b1;
c__ = *a2 * *b3 - *a3 * *b2;
/* The SVD of real 2-by-2 triangular C
( CSL -SNL )*( A 0 )*( CSR SNR ) = ( R 0 )
( SNL CSL ) ( C D ) ( -SNR CSR ) ( 0 T ) */
slasv2_(&a, &c__, &d__, &s1, &s2, &snr, &csr, &snl, &csl);
if (dabs(csr) >= dabs(snr) || dabs(csl) >= dabs(snl)) {
/* Compute the (2,1) and (2,2) elements of U'*A and V'*B,
and (2,1) element of |U|'*|A| and |V|'*|B|. */
ua21 = -snr * *a1 + csr * *a2;
ua22r = csr * *a3;
vb21 = -snl * *b1 + csl * *b2;
vb22r = csl * *b3;
aua21 = dabs(snr) * dabs(*a1) + dabs(csr) * dabs(*a2);
avb21 = dabs(snl) * dabs(*b1) + dabs(csl) * dabs(*b2);
/* zero (2,1) elements of U'*A and V'*B. */
if (dabs(ua21) + dabs(ua22r) != 0.f) {
if (aua21 / (dabs(ua21) + dabs(ua22r)) <= avb21 / (dabs(vb21)
+ dabs(vb22r))) {
slartg_(&ua22r, &ua21, csq, snq, &r__);
} else {
slartg_(&vb22r, &vb21, csq, snq, &r__);
}
} else {
slartg_(&vb22r, &vb21, csq, snq, &r__);
}
*csu = csr;
*snu = -snr;
*csv = csl;
*snv = -snl;
} else {
/* Compute the (1,1) and (1,2) elements of U'*A and V'*B,
and (1,1) element of |U|'*|A| and |V|'*|B|. */
ua11 = csr * *a1 + snr * *a2;
ua12 = snr * *a3;
vb11 = csl * *b1 + snl * *b2;
vb12 = snl * *b3;
aua11 = dabs(csr) * dabs(*a1) + dabs(snr) * dabs(*a2);
avb11 = dabs(csl) * dabs(*b1) + dabs(snl) * dabs(*b2);
/* zero (1,1) elements of U'*A and V'*B, and then swap. */
if (dabs(ua11) + dabs(ua12) != 0.f) {
if (aua11 / (dabs(ua11) + dabs(ua12)) <= avb11 / (dabs(vb11)
+ dabs(vb12))) {
slartg_(&ua12, &ua11, csq, snq, &r__);
} else {
slartg_(&vb12, &vb11, csq, snq, &r__);
}
} else {
slartg_(&vb12, &vb11, csq, snq, &r__);
}
*csu = snr;
*snu = csr;
*csv = snl;
*snv = csl;
}
}
return 0;
/* End of SLAGS2 */
} /* slags2_ */
/* Subroutine */ int sgeev_(char *jobvl, char *jobvr, integer *n, real *a,
integer *lda, real *wr, real *wi, real *vl, integer *ldvl, real *vr,
integer *ldvr, real *work, integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1,
i__2, i__3;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, k;
real r__, cs, sn;
integer ihi;
real scl;
integer ilo;
real dum[1], eps;
integer ibal;
char side[1];
real anrm;
integer ierr, itau, iwrk, nout;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
extern doublereal snrm2_(integer *, real *, integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
extern doublereal slapy2_(real *, real *);
extern /* Subroutine */ int slabad_(real *, real *);
logical scalea;
real cscale;
extern /* Subroutine */ int sgebak_(char *, char *, integer *, integer *,
integer *, real *, integer *, real *, integer *, integer *), sgebal_(char *, integer *, real *, integer *,
integer *, integer *, real *, integer *);
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int sgehrd_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *, integer *), xerbla_(char
*, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
logical select[1];
real bignum;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), slartg_(real *, real *,
real *, real *, real *), sorghr_(integer *, integer *, integer *,
real *, integer *, real *, real *, integer *, integer *), shseqr_(
char *, char *, integer *, integer *, integer *, real *, integer *
, real *, real *, real *, integer *, real *, integer *, integer *), strevc_(char *, char *, logical *, integer *,
real *, integer *, real *, integer *, real *, integer *, integer *
, integer *, real *, integer *);
integer minwrk, maxwrk;
logical wantvl;
real smlnum;
integer hswork;
logical lquery, wantvr;
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGEEV computes for an N-by-N real nonsymmetric matrix A, the */
/* eigenvalues and, optionally, the left and/or right eigenvectors. */
/* The right eigenvector v(j) of A satisfies */
/* A * v(j) = lambda(j) * v(j) */
/* where lambda(j) is its eigenvalue. */
/* The left eigenvector u(j) of A satisfies */
/* u(j)**H * A = lambda(j) * u(j)**H */
/* where u(j)**H denotes the conjugate transpose of u(j). */
/* The computed eigenvectors are normalized to have Euclidean norm */
/* equal to 1 and largest component real. */
/* Arguments */
/* ========= */
/* JOBVL (input) CHARACTER*1 */
/* = 'N': left eigenvectors of A are not computed; */
/* = 'V': left eigenvectors of A are computed. */
/* JOBVR (input) CHARACTER*1 */
/* = 'N': right eigenvectors of A are not computed; */
/* = 'V': right eigenvectors of A are computed. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* On exit, A has been overwritten. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* WR (output) REAL array, dimension (N) */
/* WI (output) REAL array, dimension (N) */
/* WR and WI contain the real and imaginary parts, */
/* respectively, of the computed eigenvalues. Complex */
/* conjugate pairs of eigenvalues appear consecutively */
/* with the eigenvalue having the positive imaginary part */
/* first. */
/* VL (output) REAL array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left eigenvectors u(j) are stored one */
/* after another in the columns of VL, in the same order */
/* as their eigenvalues. */
/* If JOBVL = 'N', VL is not referenced. */
/* If the j-th eigenvalue is real, then u(j) = VL(:,j), */
/* the j-th column of VL. */
/* If the j-th and (j+1)-st eigenvalues form a complex */
/* conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and */
/* u(j+1) = VL(:,j) - i*VL(:,j+1). */
/* LDVL (input) INTEGER */
/* The leading dimension of the array VL. LDVL >= 1; if */
/* JOBVL = 'V', LDVL >= N. */
/* VR (output) REAL array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right eigenvectors v(j) are stored one */
/* after another in the columns of VR, in the same order */
/* as their eigenvalues. */
/* If JOBVR = 'N', VR is not referenced. */
/* If the j-th eigenvalue is real, then v(j) = VR(:,j), */
/* the j-th column of VR. */
/* If the j-th and (j+1)-st eigenvalues form a complex */
/* conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and */
/* v(j+1) = VR(:,j) - i*VR(:,j+1). */
/* LDVR (input) INTEGER */
/* The leading dimension of the array VR. LDVR >= 1; if */
/* JOBVR = 'V', LDVR >= N. */
/* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= max(1,3*N), and */
/* if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good */
/* performance, LWORK must generally be larger. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = i, the QR algorithm failed to compute all the */
/* eigenvalues, and no eigenvectors have been computed; */
/* elements i+1:N of WR and WI contain eigenvalues which */
/* have converged. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--wr;
--wi;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
wantvl = lsame_(jobvl, "V");
wantvr = lsame_(jobvr, "V");
if (! wantvl && ! lsame_(jobvl, "N")) {
*info = -1;
} else if (! wantvr && ! lsame_(jobvr, "N")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldvl < 1 || wantvl && *ldvl < *n) {
*info = -9;
} else if (*ldvr < 1 || wantvr && *ldvr < *n) {
*info = -11;
}
/* Compute workspace */
/* (Note: Comments in the code beginning "Workspace:" describe the */
/* minimal amount of workspace needed at that point in the code, */
/* as well as the preferred amount for good performance. */
/* NB refers to the optimal block size for the immediately */
/* following subroutine, as returned by ILAENV. */
/* HSWORK refers to the workspace preferred by SHSEQR, as */
/* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, */
/* the worst case.) */
if (*info == 0) {
if (*n == 0) {
minwrk = 1;
maxwrk = 1;
} else {
maxwrk = (*n << 1) + *n * ilaenv_(&c__1, "SGEHRD", " ", n, &c__1,
n, &c__0);
if (wantvl) {
minwrk = *n << 2;
/* Computing MAX */
i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1,
"SORGHR", " ", n, &c__1, n, &c_n1);
maxwrk = max(i__1,i__2);
shseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[
1], &vl[vl_offset], ldvl, &work[1], &c_n1, info);
hswork = work[1];
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = *
n + hswork;
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk, i__2 = *n << 2;
maxwrk = max(i__1,i__2);
} else if (wantvr) {
minwrk = *n << 2;
/* Computing MAX */
i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1,
"SORGHR", " ", n, &c__1, n, &c_n1);
maxwrk = max(i__1,i__2);
shseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[
1], &vr[vr_offset], ldvr, &work[1], &c_n1, info);
hswork = work[1];
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = *
n + hswork;
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk, i__2 = *n << 2;
maxwrk = max(i__1,i__2);
} else {
minwrk = *n * 3;
shseqr_("E", "N", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[
1], &vr[vr_offset], ldvr, &work[1], &c_n1, info);
hswork = work[1];
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = *
n + hswork;
maxwrk = max(i__1,i__2);
}
maxwrk = max(maxwrk,minwrk);
}
work[1] = (real) maxwrk;
if (*lwork < minwrk && ! lquery) {
*info = -13;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGEEV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1.f / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = slange_("M", n, n, &a[a_offset], lda, dum);
scalea = FALSE_;
if (anrm > 0.f && anrm < smlnum) {
scalea = TRUE_;
cscale = smlnum;
} else if (anrm > bignum) {
scalea = TRUE_;
cscale = bignum;
}
if (scalea) {
slascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, &
ierr);
}
/* Balance the matrix */
/* (Workspace: need N) */
ibal = 1;
sgebal_("B", n, &a[a_offset], lda, &ilo, &ihi, &work[ibal], &ierr);
/* Reduce to upper Hessenberg form */
/* (Workspace: need 3*N, prefer 2*N+N*NB) */
itau = ibal + *n;
iwrk = itau + *n;
i__1 = *lwork - iwrk + 1;
sgehrd_(n, &ilo, &ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1,
&ierr);
if (wantvl) {
/* Want left eigenvectors */
/* Copy Householder vectors to VL */
*(unsigned char *)side = 'L';
slacpy_("L", n, n, &a[a_offset], lda, &vl[vl_offset], ldvl)
;
/* Generate orthogonal matrix in VL */
/* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) */
i__1 = *lwork - iwrk + 1;
sorghr_(n, &ilo, &ihi, &vl[vl_offset], ldvl, &work[itau], &work[iwrk],
&i__1, &ierr);
/* Perform QR iteration, accumulating Schur vectors in VL */
/* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_("S", "V", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], &
vl[vl_offset], ldvl, &work[iwrk], &i__1, info);
if (wantvr) {
/* Want left and right eigenvectors */
/* Copy Schur vectors to VR */
*(unsigned char *)side = 'B';
slacpy_("F", n, n, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr);
}
} else if (wantvr) {
/* Want right eigenvectors */
/* Copy Householder vectors to VR */
*(unsigned char *)side = 'R';
slacpy_("L", n, n, &a[a_offset], lda, &vr[vr_offset], ldvr)
;
/* Generate orthogonal matrix in VR */
/* (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) */
i__1 = *lwork - iwrk + 1;
sorghr_(n, &ilo, &ihi, &vr[vr_offset], ldvr, &work[itau], &work[iwrk],
&i__1, &ierr);
/* Perform QR iteration, accumulating Schur vectors in VR */
/* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_("S", "V", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], &
vr[vr_offset], ldvr, &work[iwrk], &i__1, info);
} else {
/* Compute eigenvalues only */
/* (Workspace: need N+1, prefer N+HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_("E", "N", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], &
vr[vr_offset], ldvr, &work[iwrk], &i__1, info);
}
/* If INFO > 0 from SHSEQR, then quit */
if (*info > 0) {
goto L50;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors */
/* (Workspace: need 4*N) */
strevc_(side, "B", select, n, &a[a_offset], lda, &vl[vl_offset], ldvl,
&vr[vr_offset], ldvr, n, &nout, &work[iwrk], &ierr);
}
if (wantvl) {
/* Undo balancing of left eigenvectors */
/* (Workspace: need N) */
sgebak_("B", "L", n, &ilo, &ihi, &work[ibal], n, &vl[vl_offset], ldvl,
&ierr);
/* Normalize left eigenvectors and make largest component real */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (wi[i__] == 0.f) {
scl = 1.f / snrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1);
sscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1);
} else if (wi[i__] > 0.f) {
r__1 = snrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1);
r__2 = snrm2_(n, &vl[(i__ + 1) * vl_dim1 + 1], &c__1);
scl = 1.f / slapy2_(&r__1, &r__2);
sscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1);
sscal_(n, &scl, &vl[(i__ + 1) * vl_dim1 + 1], &c__1);
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
r__1 = vl[k + i__ * vl_dim1];
/* Computing 2nd power */
r__2 = vl[k + (i__ + 1) * vl_dim1];
work[iwrk + k - 1] = r__1 * r__1 + r__2 * r__2;
/* L10: */
}
k = isamax_(n, &work[iwrk], &c__1);
slartg_(&vl[k + i__ * vl_dim1], &vl[k + (i__ + 1) * vl_dim1],
&cs, &sn, &r__);
srot_(n, &vl[i__ * vl_dim1 + 1], &c__1, &vl[(i__ + 1) *
vl_dim1 + 1], &c__1, &cs, &sn);
vl[k + (i__ + 1) * vl_dim1] = 0.f;
}
/* L20: */
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors */
/* (Workspace: need N) */
sgebak_("B", "R", n, &ilo, &ihi, &work[ibal], n, &vr[vr_offset], ldvr,
&ierr);
/* Normalize right eigenvectors and make largest component real */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (wi[i__] == 0.f) {
scl = 1.f / snrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1);
sscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1);
} else if (wi[i__] > 0.f) {
r__1 = snrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1);
r__2 = snrm2_(n, &vr[(i__ + 1) * vr_dim1 + 1], &c__1);
scl = 1.f / slapy2_(&r__1, &r__2);
sscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1);
sscal_(n, &scl, &vr[(i__ + 1) * vr_dim1 + 1], &c__1);
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
r__1 = vr[k + i__ * vr_dim1];
/* Computing 2nd power */
r__2 = vr[k + (i__ + 1) * vr_dim1];
work[iwrk + k - 1] = r__1 * r__1 + r__2 * r__2;
/* L30: */
}
k = isamax_(n, &work[iwrk], &c__1);
slartg_(&vr[k + i__ * vr_dim1], &vr[k + (i__ + 1) * vr_dim1],
&cs, &sn, &r__);
srot_(n, &vr[i__ * vr_dim1 + 1], &c__1, &vr[(i__ + 1) *
vr_dim1 + 1], &c__1, &cs, &sn);
vr[k + (i__ + 1) * vr_dim1] = 0.f;
}
/* L40: */
}
}
/* Undo scaling if necessary */
L50:
if (scalea) {
i__1 = *n - *info;
/* Computing MAX */
i__3 = *n - *info;
i__2 = max(i__3,1);
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[*info +
1], &i__2, &ierr);
i__1 = *n - *info;
/* Computing MAX */
i__3 = *n - *info;
i__2 = max(i__3,1);
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[*info +
1], &i__2, &ierr);
if (*info > 0) {
i__1 = ilo - 1;
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[1],
n, &ierr);
i__1 = ilo - 1;
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[1],
n, &ierr);
}
}
work[1] = (real) maxwrk;
return 0;
/* End of SGEEV */
} /* sgeev_ */
/* Subroutine */ int ssteqr_(char *compz, int *n, real *d, real *e, real *
z, int *ldz, real *work, int *info)
{
/* -- LAPACK routine (version 2.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
SSTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric tridiagonal matrix using the implicit QL or QR method.
The eigenvectors of a full or band symmetric matrix can also be found
if SSYTRD or SSPTRD or SSBTRD has been used to reduce this matrix to
tridiagonal form.
Arguments
=========
COMPZ (input) CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors of the original
symmetric matrix. On entry, Z must contain the
orthogonal matrix used to reduce the original matrix
to tridiagonal form.
= 'I': Compute eigenvalues and eigenvectors of the
tridiagonal matrix. Z is initialized to the identity
matrix.
N (input) INTEGER
The order of the matrix. N >= 0.
D (input/output) REAL array, dimension (N)
On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E (input/output) REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix.
On exit, E has been destroyed.
Z (input/output) REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', then Z contains the orthogonal
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
orthonormal eigenvectors of the original symmetric matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
eigenvectors are desired, then LDZ >= max(1,N).
WORK (workspace) REAL array, dimension (max(1,2*N-2))
If COMPZ = 'N', then WORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm has failed to find all the eigenvalues in
a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero; on exit, D
and E contain the elements of a symmetric tridiagonal
matrix which is orthogonally similar to the original
matrix.
=====================================================================
Test the input parameters.
Parameter adjustments
Function Body */
/* Table of constant values */
static real c_b9 = 0.f;
static real c_b10 = 1.f;
static int c__0 = 0;
static int c__1 = 1;
static int c__2 = 2;
/* System generated locals */
/* Unused variables commented out by MDG on 03-09-05
int z_dim1, z_offset;
*/
int i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal), r_sign(real *, real *);
/* Local variables */
static int lend, jtot;
extern /* Subroutine */ int slae2_(real *, real *, real *, real *, real *)
;
static real b, c, f, g;
static int i, j, k, l, m;
static real p, r, s;
extern logical lsame_(char *, char *);
static real anorm;
extern /* Subroutine */ int slasr_(char *, char *, char *, int *,
int *, real *, real *, real *, int *);
static int l1;
extern /* Subroutine */ int sswap_(int *, real *, int *, real *,
int *);
static int lendm1, lendp1;
extern /* Subroutine */ int slaev2_(real *, real *, real *, real *, real *
, real *, real *);
extern doublereal slapy2_(real *, real *);
static int ii, mm, iscale;
extern doublereal slamch_(char *);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, int *);
static real safmax;
extern /* Subroutine */ int slascl_(char *, int *, int *, real *,
real *, int *, int *, real *, int *, int *);
static int lendsv;
extern /* Subroutine */ int slartg_(real *, real *, real *, real *, real *
), slaset_(char *, int *, int *, real *, real *, real *,
int *);
static real ssfmin;
static int nmaxit, icompz;
static real ssfmax;
extern doublereal slanst_(char *, int *, real *, real *);
extern /* Subroutine */ int slasrt_(char *, int *, real *, int *);
static int lm1, mm1, nm1;
static real rt1, rt2, eps;
static int lsv;
static real tst, eps2;
#define D(I) d[(I)-1]
#define E(I) e[(I)-1]
#define WORK(I) work[(I)-1]
#define Z(I,J) z[(I)-1 + ((J)-1)* ( *ldz)]
*info = 0;
if (lsame_(compz, "N")) {
icompz = 0;
} else if (lsame_(compz, "V")) {
icompz = 1;
} else if (lsame_(compz, "I")) {
icompz = 2;
} else {
icompz = -1;
}
if (icompz < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
/*
} else if (*ldz < 1 || icompz > 0 && *ldz < max(1,*n)) {
*/
/* Paretheses added by MDG on 03-09-05 */
} else if ((*ldz < 1 || icompz > 0) && (*ldz < max(1,*n))) {
*info = -6;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSTEQR", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (*n == 1) {
if (icompz == 2) {
Z(1,1) = 1.f;
}
return 0;
}
/* Determine the unit roundoff and over/underflow thresholds. */
eps = slamch_("E");
/* Computing 2nd power */
r__1 = eps;
eps2 = r__1 * r__1;
safmin = slamch_("S");
safmax = 1.f / safmin;
ssfmax = sqrt(safmax) / 3.f;
ssfmin = sqrt(safmin) / eps2;
/* Compute the eigenvalues and eigenvectors of the tridiagonal
matrix. */
if (icompz == 2) {
slaset_("Full", n, n, &c_b9, &c_b10, &Z(1,1), ldz);
}
nmaxit = *n * 30;
jtot = 0;
/* Determine where the matrix splits and choose QL or QR iteration
for each block, according to whether top or bottom diagonal
element is smaller. */
l1 = 1;
nm1 = *n - 1;
L10:
if (l1 > *n) {
goto L160;
}
if (l1 > 1) {
E(l1 - 1) = 0.f;
}
if (l1 <= nm1) {
i__1 = nm1;
for (m = l1; m <= nm1; ++m) {
tst = (r__1 = E(m), dabs(r__1));
if (tst == 0.f) {
goto L30;
}
if (tst <= sqrt((r__1 = D(m), dabs(r__1))) * sqrt((r__2 = D(m + 1)
, dabs(r__2))) * eps) {
E(m) = 0.f;
goto L30;
}
/* L20: */
}
}
m = *n;
L30:
l = l1;
lsv = l;
lend = m;
lendsv = lend;
l1 = m + 1;
if (lend == l) {
goto L10;
}
/* Scale submatrix in rows and columns L to LEND */
i__1 = lend - l + 1;
anorm = slanst_("I", &i__1, &D(l), &E(l));
iscale = 0;
if (anorm == 0.f) {
goto L10;
}
if (anorm > ssfmax) {
iscale = 1;
i__1 = lend - l + 1;
slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &D(l), n,
info);
i__1 = lend - l;
slascl_("G", &c__0, &c__0, &anorm, &ssfmax, &i__1, &c__1, &E(l), n,
info);
} else if (anorm < ssfmin) {
iscale = 2;
i__1 = lend - l + 1;
slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &D(l), n,
info);
i__1 = lend - l;
slascl_("G", &c__0, &c__0, &anorm, &ssfmin, &i__1, &c__1, &E(l), n,
info);
}
/* Choose between QL and QR iteration */
if ((r__1 = D(lend), dabs(r__1)) < (r__2 = D(l), dabs(r__2))) {
lend = lsv;
l = lendsv;
}
if (lend > l) {
/* QL Iteration
Look for small subdiagonal element. */
L40:
if (l != lend) {
lendm1 = lend - 1;
i__1 = lendm1;
for (m = l; m <= lendm1; ++m) {
/* Computing 2nd power */
r__2 = (r__1 = E(m), dabs(r__1));
tst = r__2 * r__2;
if (tst <= eps2 * (r__1 = D(m), dabs(r__1)) * (r__2 = D(m + 1)
, dabs(r__2)) + safmin) {
goto L60;
}
/* L50: */
}
}
m = lend;
L60:
if (m < lend) {
E(m) = 0.f;
}
p = D(l);
if (m == l) {
goto L80;
}
/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
to compute its eigensystem. */
if (m == l + 1) {
if (icompz > 0) {
slaev2_(&D(l), &E(l), &D(l + 1), &rt1, &rt2, &c, &s);
WORK(l) = c;
WORK(*n - 1 + l) = s;
slasr_("R", "V", "B", n, &c__2, &WORK(l), &WORK(*n - 1 + l), &
Z(1,l), ldz);
} else {
slae2_(&D(l), &E(l), &D(l + 1), &rt1, &rt2);
}
D(l) = rt1;
D(l + 1) = rt2;
E(l) = 0.f;
l += 2;
if (l <= lend) {
goto L40;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (D(l + 1) - p) / (E(l) * 2.f);
r = slapy2_(&g, &c_b10);
g = D(m) - p + E(l) / (g + r_sign(&r, &g));
s = 1.f;
c = 1.f;
p = 0.f;
/* Inner loop */
mm1 = m - 1;
i__1 = l;
for (i = mm1; i >= l; --i) {
f = s * E(i);
b = c * E(i);
slartg_(&g, &f, &c, &s, &r);
if (i != m - 1) {
E(i + 1) = r;
}
g = D(i + 1) - p;
r = (D(i) - g) * s + c * 2.f * b;
p = s * r;
D(i + 1) = g + p;
g = c * r - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
WORK(i) = c;
WORK(*n - 1 + i) = -(doublereal)s;
}
/* L70: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = m - l + 1;
slasr_("R", "V", "B", n, &mm, &WORK(l), &WORK(*n - 1 + l), &Z(1,l), ldz);
}
D(l) -= p;
E(l) = g;
goto L40;
/* Eigenvalue found. */
L80:
D(l) = p;
++l;
if (l <= lend) {
goto L40;
}
goto L140;
} else {
/* QR Iteration
Look for small superdiagonal element. */
L90:
if (l != lend) {
lendp1 = lend + 1;
i__1 = lendp1;
for (m = l; m >= lendp1; --m) {
/* Computing 2nd power */
r__2 = (r__1 = E(m - 1), dabs(r__1));
tst = r__2 * r__2;
if (tst <= eps2 * (r__1 = D(m), dabs(r__1)) * (r__2 = D(m - 1)
, dabs(r__2)) + safmin) {
goto L110;
}
/* L100: */
}
}
m = lend;
L110:
if (m > lend) {
E(m - 1) = 0.f;
}
p = D(l);
if (m == l) {
goto L130;
}
/* If remaining matrix is 2-by-2, use SLAE2 or SLAEV2
to compute its eigensystem. */
if (m == l - 1) {
if (icompz > 0) {
slaev2_(&D(l - 1), &E(l - 1), &D(l), &rt1, &rt2, &c, &s);
WORK(m) = c;
WORK(*n - 1 + m) = s;
slasr_("R", "V", "F", n, &c__2, &WORK(m), &WORK(*n - 1 + m), &
Z(1,l-1), ldz);
} else {
slae2_(&D(l - 1), &E(l - 1), &D(l), &rt1, &rt2);
}
D(l - 1) = rt1;
D(l) = rt2;
E(l - 1) = 0.f;
l += -2;
if (l >= lend) {
goto L90;
}
goto L140;
}
if (jtot == nmaxit) {
goto L140;
}
++jtot;
/* Form shift. */
g = (D(l - 1) - p) / (E(l - 1) * 2.f);
r = slapy2_(&g, &c_b10);
g = D(m) - p + E(l - 1) / (g + r_sign(&r, &g));
s = 1.f;
c = 1.f;
p = 0.f;
/* Inner loop */
lm1 = l - 1;
i__1 = lm1;
for (i = m; i <= lm1; ++i) {
f = s * E(i);
b = c * E(i);
slartg_(&g, &f, &c, &s, &r);
if (i != m) {
E(i - 1) = r;
}
g = D(i) - p;
r = (D(i + 1) - g) * s + c * 2.f * b;
p = s * r;
D(i) = g + p;
g = c * r - b;
/* If eigenvectors are desired, then save rotations. */
if (icompz > 0) {
WORK(i) = c;
WORK(*n - 1 + i) = s;
}
/* L120: */
}
/* If eigenvectors are desired, then apply saved rotations. */
if (icompz > 0) {
mm = l - m + 1;
slasr_("R", "V", "F", n, &mm, &WORK(m), &WORK(*n - 1 + m), &Z(1,m), ldz);
}
D(l) -= p;
E(lm1) = g;
goto L90;
/* Eigenvalue found. */
L130:
D(l) = p;
--l;
if (l >= lend) {
goto L90;
}
goto L140;
}
/* Undo scaling if necessary */
L140:
if (iscale == 1) {
i__1 = lendsv - lsv + 1;
slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &D(lsv), n,
info);
i__1 = lendsv - lsv;
slascl_("G", &c__0, &c__0, &ssfmax, &anorm, &i__1, &c__1, &E(lsv), n,
info);
} else if (iscale == 2) {
i__1 = lendsv - lsv + 1;
slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &D(lsv), n,
info);
i__1 = lendsv - lsv;
slascl_("G", &c__0, &c__0, &ssfmin, &anorm, &i__1, &c__1, &E(lsv), n,
info);
}
/* Check for no convergence to an eigenvalue after a total
of N*MAXIT iterations. */
if (jtot < nmaxit) {
goto L10;
}
i__1 = *n - 1;
for (i = 1; i <= *n-1; ++i) {
if (E(i) != 0.f) {
++(*info);
}
/* L150: */
}
goto L190;
/* Order eigenvalues and eigenvectors. */
L160:
if (icompz == 0) {
/* Use Quick Sort */
slasrt_("I", n, &D(1), info);
} else {
/* Use Selection Sort to minimize swaps of eigenvectors */
i__1 = *n;
for (ii = 2; ii <= *n; ++ii) {
i = ii - 1;
k = i;
p = D(i);
i__2 = *n;
for (j = ii; j <= *n; ++j) {
if (D(j) < p) {
k = j;
p = D(j);
}
/* L170: */
}
if (k != i) {
D(k) = D(i);
D(i) = p;
sswap_(n, &Z(1,i), &c__1, &Z(1,k), &
c__1);
}
/* L180: */
}
}
L190:
return 0;
/* End of SSTEQR */
} /* ssteqr_ */
/* Subroutine */ int slagv2_(real *a, integer *lda, real *b, integer *ldb,
real *alphar, real *alphai, real *beta, real *csl, real *snl, real *
csr, real *snr)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
SLAGV2 computes the Generalized Schur factorization of a real 2-by-2
matrix pencil (A,B) where B is upper triangular. This routine
computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
SNR such that
1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
types), then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
where b11 >= b22 > 0.
Arguments
=========
A (input/output) REAL array, dimension (LDA, 2)
On entry, the 2 x 2 matrix A.
On exit, A is overwritten by the ``A-part'' of the
generalized Schur form.
LDA (input) INTEGER
THe leading dimension of the array A. LDA >= 2.
B (input/output) REAL array, dimension (LDB, 2)
On entry, the upper triangular 2 x 2 matrix B.
On exit, B is overwritten by the ``B-part'' of the
generalized Schur form.
LDB (input) INTEGER
THe leading dimension of the array B. LDB >= 2.
ALPHAR (output) REAL array, dimension (2)
ALPHAI (output) REAL array, dimension (2)
BETA (output) REAL array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
be zero.
CSL (output) REAL
The cosine of the left rotation matrix.
SNL (output) REAL
The sine of the left rotation matrix.
CSR (output) REAL
The cosine of the right rotation matrix.
SNR (output) REAL
The sine of the right rotation matrix.
Further Details
===============
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__2 = 2;
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset;
real r__1, r__2, r__3, r__4, r__5, r__6;
/* Local variables */
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *), slag2_(real *, integer *, real *,
integer *, real *, real *, real *, real *, real *, real *);
static real r__, t, anorm, bnorm, h1, h2, h3, scale1, scale2;
extern /* Subroutine */ int slasv2_(real *, real *, real *, real *, real *
, real *, real *, real *, real *);
extern doublereal slapy2_(real *, real *);
static real ascale, bscale, wi, qq, rr;
extern doublereal slamch_(char *);
static real safmin;
extern /* Subroutine */ int slartg_(real *, real *, real *, real *, real *
);
static real wr1, wr2, ulp;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--alphar;
--alphai;
--beta;
/* Function Body */
safmin = slamch_("S");
ulp = slamch_("P");
/* Scale A
Computing MAX */
r__5 = (r__1 = a_ref(1, 1), dabs(r__1)) + (r__2 = a_ref(2, 1), dabs(r__2))
, r__6 = (r__3 = a_ref(1, 2), dabs(r__3)) + (r__4 = a_ref(2, 2),
dabs(r__4)), r__5 = max(r__5,r__6);
anorm = dmax(r__5,safmin);
ascale = 1.f / anorm;
a_ref(1, 1) = ascale * a_ref(1, 1);
a_ref(1, 2) = ascale * a_ref(1, 2);
a_ref(2, 1) = ascale * a_ref(2, 1);
a_ref(2, 2) = ascale * a_ref(2, 2);
/* Scale B
Computing MAX */
r__4 = (r__3 = b_ref(1, 1), dabs(r__3)), r__5 = (r__1 = b_ref(1, 2), dabs(
r__1)) + (r__2 = b_ref(2, 2), dabs(r__2)), r__4 = max(r__4,r__5);
bnorm = dmax(r__4,safmin);
bscale = 1.f / bnorm;
b_ref(1, 1) = bscale * b_ref(1, 1);
b_ref(1, 2) = bscale * b_ref(1, 2);
b_ref(2, 2) = bscale * b_ref(2, 2);
/* Check if A can be deflated */
if ((r__1 = a_ref(2, 1), dabs(r__1)) <= ulp) {
*csl = 1.f;
*snl = 0.f;
*csr = 1.f;
*snr = 0.f;
a_ref(2, 1) = 0.f;
b_ref(2, 1) = 0.f;
/* Check if B is singular */
} else if ((r__1 = b_ref(1, 1), dabs(r__1)) <= ulp) {
slartg_(&a_ref(1, 1), &a_ref(2, 1), csl, snl, &r__);
*csr = 1.f;
*snr = 0.f;
srot_(&c__2, &a_ref(1, 1), lda, &a_ref(2, 1), lda, csl, snl);
srot_(&c__2, &b_ref(1, 1), ldb, &b_ref(2, 1), ldb, csl, snl);
a_ref(2, 1) = 0.f;
b_ref(1, 1) = 0.f;
b_ref(2, 1) = 0.f;
} else if ((r__1 = b_ref(2, 2), dabs(r__1)) <= ulp) {
slartg_(&a_ref(2, 2), &a_ref(2, 1), csr, snr, &t);
*snr = -(*snr);
srot_(&c__2, &a_ref(1, 1), &c__1, &a_ref(1, 2), &c__1, csr, snr);
srot_(&c__2, &b_ref(1, 1), &c__1, &b_ref(1, 2), &c__1, csr, snr);
*csl = 1.f;
*snl = 0.f;
a_ref(2, 1) = 0.f;
b_ref(2, 1) = 0.f;
b_ref(2, 2) = 0.f;
} else {
/* B is nonsingular, first compute the eigenvalues of (A,B) */
slag2_(&a[a_offset], lda, &b[b_offset], ldb, &safmin, &scale1, &
scale2, &wr1, &wr2, &wi);
if (wi == 0.f) {
/* two real eigenvalues, compute s*A-w*B */
h1 = scale1 * a_ref(1, 1) - wr1 * b_ref(1, 1);
h2 = scale1 * a_ref(1, 2) - wr1 * b_ref(1, 2);
h3 = scale1 * a_ref(2, 2) - wr1 * b_ref(2, 2);
rr = slapy2_(&h1, &h2);
r__1 = scale1 * a_ref(2, 1);
qq = slapy2_(&r__1, &h3);
if (rr > qq) {
/* find right rotation matrix to zero 1,1 element of
(sA - wB) */
slartg_(&h2, &h1, csr, snr, &t);
} else {
/* find right rotation matrix to zero 2,1 element of
(sA - wB) */
r__1 = scale1 * a_ref(2, 1);
slartg_(&h3, &r__1, csr, snr, &t);
}
*snr = -(*snr);
srot_(&c__2, &a_ref(1, 1), &c__1, &a_ref(1, 2), &c__1, csr, snr);
srot_(&c__2, &b_ref(1, 1), &c__1, &b_ref(1, 2), &c__1, csr, snr);
/* compute inf norms of A and B
Computing MAX */
r__5 = (r__1 = a_ref(1, 1), dabs(r__1)) + (r__2 = a_ref(1, 2),
dabs(r__2)), r__6 = (r__3 = a_ref(2, 1), dabs(r__3)) + (
r__4 = a_ref(2, 2), dabs(r__4));
h1 = dmax(r__5,r__6);
/* Computing MAX */
r__5 = (r__1 = b_ref(1, 1), dabs(r__1)) + (r__2 = b_ref(1, 2),
dabs(r__2)), r__6 = (r__3 = b_ref(2, 1), dabs(r__3)) + (
r__4 = b_ref(2, 2), dabs(r__4));
h2 = dmax(r__5,r__6);
if (scale1 * h1 >= dabs(wr1) * h2) {
/* find left rotation matrix Q to zero out B(2,1) */
slartg_(&b_ref(1, 1), &b_ref(2, 1), csl, snl, &r__);
} else {
/* find left rotation matrix Q to zero out A(2,1) */
slartg_(&a_ref(1, 1), &a_ref(2, 1), csl, snl, &r__);
}
srot_(&c__2, &a_ref(1, 1), lda, &a_ref(2, 1), lda, csl, snl);
srot_(&c__2, &b_ref(1, 1), ldb, &b_ref(2, 1), ldb, csl, snl);
a_ref(2, 1) = 0.f;
b_ref(2, 1) = 0.f;
} else {
/* a pair of complex conjugate eigenvalues
first compute the SVD of the matrix B */
slasv2_(&b_ref(1, 1), &b_ref(1, 2), &b_ref(2, 2), &r__, &t, snr,
csr, snl, csl);
/* Form (A,B) := Q(A,B)Z' where Q is left rotation matrix and
Z is right rotation matrix computed from SLASV2 */
srot_(&c__2, &a_ref(1, 1), lda, &a_ref(2, 1), lda, csl, snl);
srot_(&c__2, &b_ref(1, 1), ldb, &b_ref(2, 1), ldb, csl, snl);
srot_(&c__2, &a_ref(1, 1), &c__1, &a_ref(1, 2), &c__1, csr, snr);
srot_(&c__2, &b_ref(1, 1), &c__1, &b_ref(1, 2), &c__1, csr, snr);
b_ref(2, 1) = 0.f;
b_ref(1, 2) = 0.f;
}
}
/* Unscaling */
a_ref(1, 1) = anorm * a_ref(1, 1);
a_ref(2, 1) = anorm * a_ref(2, 1);
a_ref(1, 2) = anorm * a_ref(1, 2);
a_ref(2, 2) = anorm * a_ref(2, 2);
b_ref(1, 1) = bnorm * b_ref(1, 1);
b_ref(2, 1) = bnorm * b_ref(2, 1);
b_ref(1, 2) = bnorm * b_ref(1, 2);
b_ref(2, 2) = bnorm * b_ref(2, 2);
if (wi == 0.f) {
alphar[1] = a_ref(1, 1);
alphar[2] = a_ref(2, 2);
alphai[1] = 0.f;
alphai[2] = 0.f;
beta[1] = b_ref(1, 1);
beta[2] = b_ref(2, 2);
} else {
alphar[1] = anorm * wr1 / scale1 / bnorm;
alphai[1] = anorm * wi / scale1 / bnorm;
alphar[2] = alphar[1];
alphai[2] = -alphai[1];
beta[1] = 1.f;
beta[2] = 1.f;
}
/* L10: */
return 0;
/* End of SLAGV2 */
} /* slagv2_ */
/* Subroutine */ int snapps_(integer *n, integer *kev, integer *np, real *
shiftr, real *shifti, real *v, integer *ldv, real *h__, integer *ldh,
real *resid, real *q, integer *ldq, real *workl, real *workd)
{
/* Initialized data */
static logical first = TRUE_;
/* System generated locals */
integer h_dim1, h_offset, v_dim1, v_offset, q_dim1, q_offset, i__1, i__2,
i__3, i__4;
real r__1, r__2;
/* Local variables */
static real c__, f, g;
static integer i__, j;
static real r__, s, t, u[3], t0, t1, h11, h12, h21, h22, h32;
static integer jj, ir, nr;
static real tau, ulp, tst1;
static integer iend;
static real unfl, ovfl;
static logical cconj;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
slarf_(char *, integer *, integer *, real *, integer *, real *,
real *, integer *, real *, ftnlen), sgemv_(char *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *, ftnlen), scopy_(integer *, real *, integer *,
real *, integer *), saxpy_(integer *, real *, real *, integer *,
real *, integer *), ivout_(integer *, integer *, integer *,
integer *, char *, ftnlen), smout_(integer *, integer *, integer *
, real *, integer *, integer *, char *, ftnlen), svout_(integer *,
integer *, real *, integer *, char *, ftnlen);
extern doublereal slapy2_(real *, real *);
extern /* Subroutine */ int slabad_(real *, real *);
extern doublereal slamch_(char *, ftnlen);
static real sigmai;
extern /* Subroutine */ int second_(real *);
static real sigmar;
static integer istart, kplusp, msglvl;
static real smlnum;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *, ftnlen), slarfg_(integer *, real *,
real *, integer *, real *), slaset_(char *, integer *, integer *,
real *, real *, real *, integer *, ftnlen), slartg_(real *, real *
, real *, real *, real *);
extern doublereal slanhs_(char *, integer *, real *, integer *, real *,
ftnlen);
/* %----------------------------------------------------% */
/* | Include files for debugging and timing information | */
/* %----------------------------------------------------% */
/* \SCCS Information: @(#) */
/* FILE: debug.h SID: 2.3 DATE OF SID: 11/16/95 RELEASE: 2 */
/* %---------------------------------% */
/* | See debug.doc for documentation | */
/* %---------------------------------% */
/* %------------------% */
/* | Scalar Arguments | */
/* %------------------% */
/* %--------------------------------% */
/* | See stat.doc for documentation | */
/* %--------------------------------% */
/* \SCCS Information: @(#) */
/* FILE: stat.h SID: 2.2 DATE OF SID: 11/16/95 RELEASE: 2 */
/* %-----------------% */
/* | Array Arguments | */
/* %-----------------% */
/* %------------% */
/* | Parameters | */
/* %------------% */
/* %------------------------% */
/* | Local Scalars & Arrays | */
/* %------------------------% */
/* %----------------------% */
/* | External Subroutines | */
/* %----------------------% */
/* %--------------------% */
/* | External Functions | */
/* %--------------------% */
/* %----------------------% */
/* | Intrinsics Functions | */
/* %----------------------% */
/* %----------------% */
/* | Data statments | */
/* %----------------% */
/* Parameter adjustments */
--workd;
--resid;
--workl;
--shifti;
--shiftr;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
/* Function Body */
/* %-----------------------% */
/* | Executable Statements | */
/* %-----------------------% */
if (first) {
/* %-----------------------------------------------% */
/* | Set machine-dependent constants for the | */
/* | stopping criterion. If norm(H) <= sqrt(OVFL), | */
/* | overflow should not occur. | */
/* | REFERENCE: LAPACK subroutine slahqr | */
/* %-----------------------------------------------% */
unfl = slamch_("safe minimum", (ftnlen)12);
ovfl = 1.f / unfl;
slabad_(&unfl, &ovfl);
ulp = slamch_("precision", (ftnlen)9);
smlnum = unfl * (*n / ulp);
first = FALSE_;
}
/* %-------------------------------% */
/* | Initialize timing statistics | */
/* | & message level for debugging | */
/* %-------------------------------% */
second_(&t0);
msglvl = debug_1.mnapps;
kplusp = *kev + *np;
/* %--------------------------------------------% */
/* | Initialize Q to the identity to accumulate | */
/* | the rotations and reflections | */
/* %--------------------------------------------% */
slaset_("All", &kplusp, &kplusp, &c_b5, &c_b6, &q[q_offset], ldq, (ftnlen)
3);
/* %----------------------------------------------% */
/* | Quick return if there are no shifts to apply | */
/* %----------------------------------------------% */
if (*np == 0) {
goto L9000;
}
/* %----------------------------------------------% */
/* | Chase the bulge with the application of each | */
/* | implicit shift. Each shift is applied to the | */
/* | whole matrix including each block. | */
/* %----------------------------------------------% */
cconj = FALSE_;
i__1 = *np;
for (jj = 1; jj <= i__1; ++jj) {
sigmar = shiftr[jj];
sigmai = shifti[jj];
if (msglvl > 2) {
ivout_(&debug_1.logfil, &c__1, &jj, &debug_1.ndigit, "_napps: sh"
"ift number.", (ftnlen)21);
svout_(&debug_1.logfil, &c__1, &sigmar, &debug_1.ndigit, "_napps"
": The real part of the shift ", (ftnlen)35);
svout_(&debug_1.logfil, &c__1, &sigmai, &debug_1.ndigit, "_napps"
": The imaginary part of the shift ", (ftnlen)40);
}
/* %-------------------------------------------------% */
/* | The following set of conditionals is necessary | */
/* | in order that complex conjugate pairs of shifts | */
/* | are applied together or not at all. | */
/* %-------------------------------------------------% */
if (cconj) {
/* %-----------------------------------------% */
/* | cconj = .true. means the previous shift | */
/* | had non-zero imaginary part. | */
/* %-----------------------------------------% */
cconj = FALSE_;
goto L110;
} else if (jj < *np && dabs(sigmai) > 0.f) {
/* %------------------------------------% */
/* | Start of a complex conjugate pair. | */
/* %------------------------------------% */
cconj = TRUE_;
} else if (jj == *np && dabs(sigmai) > 0.f) {
/* %----------------------------------------------% */
/* | The last shift has a nonzero imaginary part. | */
/* | Don't apply it; thus the order of the | */
/* | compressed H is order KEV+1 since only np-1 | */
/* | were applied. | */
/* %----------------------------------------------% */
++(*kev);
goto L110;
}
istart = 1;
L20:
/* %--------------------------------------------------% */
/* | if sigmai = 0 then | */
/* | Apply the jj-th shift ... | */
/* | else | */
/* | Apply the jj-th and (jj+1)-th together ... | */
/* | (Note that jj < np at this point in the code) | */
/* | end | */
/* | to the current block of H. The next do loop | */
/* | determines the current block ; | */
/* %--------------------------------------------------% */
i__2 = kplusp - 1;
for (i__ = istart; i__ <= i__2; ++i__) {
/* %----------------------------------------% */
/* | Check for splitting and deflation. Use | */
/* | a standard test as in the QR algorithm | */
/* | REFERENCE: LAPACK subroutine slahqr | */
/* %----------------------------------------% */
tst1 = (r__1 = h__[i__ + i__ * h_dim1], dabs(r__1)) + (r__2 = h__[
i__ + 1 + (i__ + 1) * h_dim1], dabs(r__2));
if (tst1 == 0.f) {
i__3 = kplusp - jj + 1;
tst1 = slanhs_("1", &i__3, &h__[h_offset], ldh, &workl[1], (
ftnlen)1);
}
/* Computing MAX */
r__2 = ulp * tst1;
if ((r__1 = h__[i__ + 1 + i__ * h_dim1], dabs(r__1)) <= dmax(r__2,
smlnum)) {
if (msglvl > 0) {
ivout_(&debug_1.logfil, &c__1, &i__, &debug_1.ndigit,
"_napps: matrix splitting at row/column no.", (
ftnlen)42);
ivout_(&debug_1.logfil, &c__1, &jj, &debug_1.ndigit,
"_napps: matrix splitting with shift number.", (
ftnlen)43);
svout_(&debug_1.logfil, &c__1, &h__[i__ + 1 + i__ *
h_dim1], &debug_1.ndigit, "_napps: off diagonal "
"element.", (ftnlen)29);
}
iend = i__;
h__[i__ + 1 + i__ * h_dim1] = 0.f;
goto L40;
}
/* L30: */
}
iend = kplusp;
L40:
if (msglvl > 2) {
ivout_(&debug_1.logfil, &c__1, &istart, &debug_1.ndigit, "_napps"
": Start of current block ", (ftnlen)31);
ivout_(&debug_1.logfil, &c__1, &iend, &debug_1.ndigit, "_napps: "
"End of current block ", (ftnlen)29);
}
/* %------------------------------------------------% */
/* | No reason to apply a shift to block of order 1 | */
/* %------------------------------------------------% */
if (istart == iend) {
goto L100;
}
/* %------------------------------------------------------% */
/* | If istart + 1 = iend then no reason to apply a | */
/* | complex conjugate pair of shifts on a 2 by 2 matrix. | */
/* %------------------------------------------------------% */
if (istart + 1 == iend && dabs(sigmai) > 0.f) {
goto L100;
}
h11 = h__[istart + istart * h_dim1];
h21 = h__[istart + 1 + istart * h_dim1];
if (dabs(sigmai) <= 0.f) {
/* %---------------------------------------------% */
/* | Real-valued shift ==> apply single shift QR | */
/* %---------------------------------------------% */
f = h11 - sigmar;
g = h21;
i__2 = iend - 1;
for (i__ = istart; i__ <= i__2; ++i__) {
/* %-----------------------------------------------------% */
/* | Contruct the plane rotation G to zero out the bulge | */
/* %-----------------------------------------------------% */
slartg_(&f, &g, &c__, &s, &r__);
if (i__ > istart) {
/* %-------------------------------------------% */
/* | The following ensures that h(1:iend-1,1), | */
/* | the first iend-2 off diagonal of elements | */
/* | H, remain non negative. | */
/* %-------------------------------------------% */
if (r__ < 0.f) {
r__ = -r__;
c__ = -c__;
s = -s;
}
h__[i__ + (i__ - 1) * h_dim1] = r__;
h__[i__ + 1 + (i__ - 1) * h_dim1] = 0.f;
}
/* %---------------------------------------------% */
/* | Apply rotation to the left of H; H <- G'*H | */
/* %---------------------------------------------% */
i__3 = kplusp;
for (j = i__; j <= i__3; ++j) {
t = c__ * h__[i__ + j * h_dim1] + s * h__[i__ + 1 + j *
h_dim1];
h__[i__ + 1 + j * h_dim1] = -s * h__[i__ + j * h_dim1] +
c__ * h__[i__ + 1 + j * h_dim1];
h__[i__ + j * h_dim1] = t;
/* L50: */
}
/* %---------------------------------------------% */
/* | Apply rotation to the right of H; H <- H*G | */
/* %---------------------------------------------% */
/* Computing MIN */
i__4 = i__ + 2;
i__3 = min(i__4,iend);
for (j = 1; j <= i__3; ++j) {
t = c__ * h__[j + i__ * h_dim1] + s * h__[j + (i__ + 1) *
h_dim1];
h__[j + (i__ + 1) * h_dim1] = -s * h__[j + i__ * h_dim1]
+ c__ * h__[j + (i__ + 1) * h_dim1];
h__[j + i__ * h_dim1] = t;
/* L60: */
}
/* %----------------------------------------------------% */
/* | Accumulate the rotation in the matrix Q; Q <- Q*G | */
/* %----------------------------------------------------% */
/* Computing MIN */
i__4 = i__ + jj;
i__3 = min(i__4,kplusp);
for (j = 1; j <= i__3; ++j) {
t = c__ * q[j + i__ * q_dim1] + s * q[j + (i__ + 1) *
q_dim1];
q[j + (i__ + 1) * q_dim1] = -s * q[j + i__ * q_dim1] +
c__ * q[j + (i__ + 1) * q_dim1];
q[j + i__ * q_dim1] = t;
/* L70: */
}
/* %---------------------------% */
/* | Prepare for next rotation | */
/* %---------------------------% */
if (i__ < iend - 1) {
f = h__[i__ + 1 + i__ * h_dim1];
g = h__[i__ + 2 + i__ * h_dim1];
}
/* L80: */
}
/* %-----------------------------------% */
/* | Finished applying the real shift. | */
/* %-----------------------------------% */
} else {
/* %----------------------------------------------------% */
/* | Complex conjugate shifts ==> apply double shift QR | */
/* %----------------------------------------------------% */
h12 = h__[istart + (istart + 1) * h_dim1];
h22 = h__[istart + 1 + (istart + 1) * h_dim1];
h32 = h__[istart + 2 + (istart + 1) * h_dim1];
/* %---------------------------------------------------------% */
/* | Compute 1st column of (H - shift*I)*(H - conj(shift)*I) | */
/* %---------------------------------------------------------% */
s = sigmar * 2.f;
t = slapy2_(&sigmar, &sigmai);
u[0] = (h11 * (h11 - s) + t * t) / h21 + h12;
u[1] = h11 + h22 - s;
u[2] = h32;
i__2 = iend - 1;
for (i__ = istart; i__ <= i__2; ++i__) {
/* Computing MIN */
i__3 = 3, i__4 = iend - i__ + 1;
nr = min(i__3,i__4);
/* %-----------------------------------------------------% */
/* | Construct Householder reflector G to zero out u(1). | */
/* | G is of the form I - tau*( 1 u )' * ( 1 u' ). | */
/* %-----------------------------------------------------% */
slarfg_(&nr, u, &u[1], &c__1, &tau);
if (i__ > istart) {
h__[i__ + (i__ - 1) * h_dim1] = u[0];
h__[i__ + 1 + (i__ - 1) * h_dim1] = 0.f;
if (i__ < iend - 1) {
h__[i__ + 2 + (i__ - 1) * h_dim1] = 0.f;
}
}
u[0] = 1.f;
/* %--------------------------------------% */
/* | Apply the reflector to the left of H | */
/* %--------------------------------------% */
i__3 = kplusp - i__ + 1;
slarf_("Left", &nr, &i__3, u, &c__1, &tau, &h__[i__ + i__ *
h_dim1], ldh, &workl[1], (ftnlen)4);
/* %---------------------------------------% */
/* | Apply the reflector to the right of H | */
/* %---------------------------------------% */
/* Computing MIN */
i__3 = i__ + 3;
ir = min(i__3,iend);
slarf_("Right", &ir, &nr, u, &c__1, &tau, &h__[i__ * h_dim1 +
1], ldh, &workl[1], (ftnlen)5);
/* %-----------------------------------------------------% */
/* | Accumulate the reflector in the matrix Q; Q <- Q*G | */
/* %-----------------------------------------------------% */
slarf_("Right", &kplusp, &nr, u, &c__1, &tau, &q[i__ * q_dim1
+ 1], ldq, &workl[1], (ftnlen)5);
/* %----------------------------% */
/* | Prepare for next reflector | */
/* %----------------------------% */
if (i__ < iend - 1) {
u[0] = h__[i__ + 1 + i__ * h_dim1];
u[1] = h__[i__ + 2 + i__ * h_dim1];
if (i__ < iend - 2) {
u[2] = h__[i__ + 3 + i__ * h_dim1];
}
}
/* L90: */
}
/* %--------------------------------------------% */
/* | Finished applying a complex pair of shifts | */
/* | to the current block | */
/* %--------------------------------------------% */
}
L100:
/* %---------------------------------------------------------% */
/* | Apply the same shift to the next block if there is any. | */
/* %---------------------------------------------------------% */
istart = iend + 1;
if (iend < kplusp) {
goto L20;
}
/* %---------------------------------------------% */
/* | Loop back to the top to get the next shift. | */
/* %---------------------------------------------% */
L110:
;
}
/* %--------------------------------------------------% */
/* | Perform a similarity transformation that makes | */
/* | sure that H will have non negative sub diagonals | */
/* %--------------------------------------------------% */
i__1 = *kev;
for (j = 1; j <= i__1; ++j) {
if (h__[j + 1 + j * h_dim1] < 0.f) {
i__2 = kplusp - j + 1;
sscal_(&i__2, &c_b43, &h__[j + 1 + j * h_dim1], ldh);
/* Computing MIN */
i__3 = j + 2;
i__2 = min(i__3,kplusp);
sscal_(&i__2, &c_b43, &h__[(j + 1) * h_dim1 + 1], &c__1);
/* Computing MIN */
i__3 = j + *np + 1;
i__2 = min(i__3,kplusp);
sscal_(&i__2, &c_b43, &q[(j + 1) * q_dim1 + 1], &c__1);
}
/* L120: */
}
i__1 = *kev;
for (i__ = 1; i__ <= i__1; ++i__) {
/* %--------------------------------------------% */
/* | Final check for splitting and deflation. | */
/* | Use a standard test as in the QR algorithm | */
/* | REFERENCE: LAPACK subroutine slahqr | */
/* %--------------------------------------------% */
tst1 = (r__1 = h__[i__ + i__ * h_dim1], dabs(r__1)) + (r__2 = h__[i__
+ 1 + (i__ + 1) * h_dim1], dabs(r__2));
if (tst1 == 0.f) {
tst1 = slanhs_("1", kev, &h__[h_offset], ldh, &workl[1], (ftnlen)
1);
}
/* Computing MAX */
r__1 = ulp * tst1;
if (h__[i__ + 1 + i__ * h_dim1] <= dmax(r__1,smlnum)) {
h__[i__ + 1 + i__ * h_dim1] = 0.f;
}
/* L130: */
}
/* %-------------------------------------------------% */
/* | Compute the (kev+1)-st column of (V*Q) and | */
/* | temporarily store the result in WORKD(N+1:2*N). | */
/* | This is needed in the residual update since we | */
/* | cannot GUARANTEE that the corresponding entry | */
/* | of H would be zero as in exact arithmetic. | */
/* %-------------------------------------------------% */
if (h__[*kev + 1 + *kev * h_dim1] > 0.f) {
sgemv_("N", n, &kplusp, &c_b6, &v[v_offset], ldv, &q[(*kev + 1) *
q_dim1 + 1], &c__1, &c_b5, &workd[*n + 1], &c__1, (ftnlen)1);
}
/* %----------------------------------------------------------% */
/* | Compute column 1 to kev of (V*Q) in backward order | */
/* | taking advantage of the upper Hessenberg structure of Q. | */
/* %----------------------------------------------------------% */
i__1 = *kev;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = kplusp - i__ + 1;
sgemv_("N", n, &i__2, &c_b6, &v[v_offset], ldv, &q[(*kev - i__ + 1) *
q_dim1 + 1], &c__1, &c_b5, &workd[1], &c__1, (ftnlen)1);
scopy_(n, &workd[1], &c__1, &v[(kplusp - i__ + 1) * v_dim1 + 1], &
c__1);
/* L140: */
}
/* %-------------------------------------------------% */
/* | Move v(:,kplusp-kev+1:kplusp) into v(:,1:kev). | */
/* %-------------------------------------------------% */
slacpy_("A", n, kev, &v[(kplusp - *kev + 1) * v_dim1 + 1], ldv, &v[
v_offset], ldv, (ftnlen)1);
/* %--------------------------------------------------------------% */
/* | Copy the (kev+1)-st column of (V*Q) in the appropriate place | */
/* %--------------------------------------------------------------% */
if (h__[*kev + 1 + *kev * h_dim1] > 0.f) {
scopy_(n, &workd[*n + 1], &c__1, &v[(*kev + 1) * v_dim1 + 1], &c__1);
}
/* %-------------------------------------% */
/* | Update the residual vector: | */
/* | r <- sigmak*r + betak*v(:,kev+1) | */
/* | where | */
/* | sigmak = (e_{kplusp}'*Q)*e_{kev} | */
/* | betak = e_{kev+1}'*H*e_{kev} | */
/* %-------------------------------------% */
sscal_(n, &q[kplusp + *kev * q_dim1], &resid[1], &c__1);
if (h__[*kev + 1 + *kev * h_dim1] > 0.f) {
saxpy_(n, &h__[*kev + 1 + *kev * h_dim1], &v[(*kev + 1) * v_dim1 + 1],
&c__1, &resid[1], &c__1);
}
if (msglvl > 1) {
svout_(&debug_1.logfil, &c__1, &q[kplusp + *kev * q_dim1], &
debug_1.ndigit, "_napps: sigmak = (e_{kev+p}^T*Q)*e_{kev}", (
ftnlen)40);
svout_(&debug_1.logfil, &c__1, &h__[*kev + 1 + *kev * h_dim1], &
debug_1.ndigit, "_napps: betak = e_{kev+1}^T*H*e_{kev}", (
ftnlen)37);
ivout_(&debug_1.logfil, &c__1, kev, &debug_1.ndigit, "_napps: Order "
"of the final Hessenberg matrix ", (ftnlen)45);
if (msglvl > 2) {
smout_(&debug_1.logfil, kev, kev, &h__[h_offset], ldh, &
debug_1.ndigit, "_napps: updated Hessenberg matrix H for"
" next iteration", (ftnlen)54);
}
}
L9000:
second_(&t1);
timing_1.tnapps += t1 - t0;
return 0;
/* %---------------% */
/* | End of snapps | */
/* %---------------% */
} /* snapps_ */
/*< >*/
/* Subroutine */ int stgsja_(const char *jobu, const char *jobv, const char *jobq, integer *m,
integer *p, integer *n, integer *k, integer *l, real *a, integer *lda,
real *b, integer *ldb, real *tola, real *tolb, real *alpha, real *
beta, real *u, integer *ldu, real *v, integer *ldv, real *q, integer *
ldq, real *work, integer *ncycle, integer *info, ftnlen jobu_len,
ftnlen jobv_len, ftnlen jobq_len)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1,
u_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4;
real r__1;
/* Local variables */
integer i__, j;
real a1, a2, a3, b1, b2, b3, csq, csu, csv, snq, rwk, snu, snv;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
real gamma;
extern logical lsame_(const char *, const char *, ftnlen, ftnlen);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
logical initq, initu, initv, wantq, upper;
real error, ssmin;
logical wantu, wantv;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slags2_(logical *, real *, real *, real *, real *,
real *, real *, real *, real *, real *, real *, real *, real *);
integer kcycle;
extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen), slapll_(
integer *, real *, integer *, real *, integer *, real *), slartg_(
real *, real *, real *, real *, real *), slaset_(char *, integer *
, integer *, real *, real *, real *, integer *, ftnlen);
(void)jobu_len;
(void)jobv_len;
(void)jobq_len;
/* -- LAPACK routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* June 30, 1999 */
/* .. Scalar Arguments .. */
/*< CHARACTER JOBQ, JOBU, JOBV >*/
/*< >*/
/*< REAL TOLA, TOLB >*/
/* .. */
/* .. Array Arguments .. */
/*< >*/
/* .. */
/* Purpose */
/* ======= */
/* STGSJA computes the generalized singular value decomposition (GSVD) */
/* of two real upper triangular (or trapezoidal) matrices A and B. */
/* On entry, it is assumed that matrices A and B have the following */
/* forms, which may be obtained by the preprocessing subroutine SGGSVP */
/* from a general M-by-N matrix A and P-by-N matrix B: */
/* N-K-L K L */
/* A = K ( 0 A12 A13 ) if M-K-L >= 0; */
/* L ( 0 0 A23 ) */
/* M-K-L ( 0 0 0 ) */
/* N-K-L K L */
/* A = K ( 0 A12 A13 ) if M-K-L < 0; */
/* M-K ( 0 0 A23 ) */
/* N-K-L K L */
/* B = L ( 0 0 B13 ) */
/* P-L ( 0 0 0 ) */
/* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular */
/* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, */
/* otherwise A23 is (M-K)-by-L upper trapezoidal. */
/* On exit, */
/* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), */
/* where U, V and Q are orthogonal matrices, Z' denotes the transpose */
/* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are */
/* ``diagonal'' matrices, which are of the following structures: */
/* If M-K-L >= 0, */
/* K L */
/* D1 = K ( I 0 ) */
/* L ( 0 C ) */
/* M-K-L ( 0 0 ) */
/* K L */
/* D2 = L ( 0 S ) */
/* P-L ( 0 0 ) */
/* N-K-L K L */
/* ( 0 R ) = K ( 0 R11 R12 ) K */
/* L ( 0 0 R22 ) L */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */
/* S = diag( BETA(K+1), ... , BETA(K+L) ), */
/* C**2 + S**2 = I. */
/* R is stored in A(1:K+L,N-K-L+1:N) on exit. */
/* If M-K-L < 0, */
/* K M-K K+L-M */
/* D1 = K ( I 0 0 ) */
/* M-K ( 0 C 0 ) */
/* K M-K K+L-M */
/* D2 = M-K ( 0 S 0 ) */
/* K+L-M ( 0 0 I ) */
/* P-L ( 0 0 0 ) */
/* N-K-L K M-K K+L-M */
/* ( 0 R ) = K ( 0 R11 R12 R13 ) */
/* M-K ( 0 0 R22 R23 ) */
/* K+L-M ( 0 0 0 R33 ) */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */
/* S = diag( BETA(K+1), ... , BETA(M) ), */
/* C**2 + S**2 = I. */
/* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored */
/* ( 0 R22 R23 ) */
/* in B(M-K+1:L,N+M-K-L+1:N) on exit. */
/* The computation of the orthogonal transformation matrices U, V or Q */
/* is optional. These matrices may either be formed explicitly, or they */
/* may be postmultiplied into input matrices U1, V1, or Q1. */
/* Arguments */
/* ========= */
/* JOBU (input) CHARACTER*1 */
/* = 'U': U must contain an orthogonal matrix U1 on entry, and */
/* the product U1*U is returned; */
/* = 'I': U is initialized to the unit matrix, and the */
/* orthogonal matrix U is returned; */
/* = 'N': U is not computed. */
/* JOBV (input) CHARACTER*1 */
/* = 'V': V must contain an orthogonal matrix V1 on entry, and */
/* the product V1*V is returned; */
/* = 'I': V is initialized to the unit matrix, and the */
/* orthogonal matrix V is returned; */
/* = 'N': V is not computed. */
/* JOBQ (input) CHARACTER*1 */
/* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and */
/* the product Q1*Q is returned; */
/* = 'I': Q is initialized to the unit matrix, and the */
/* orthogonal matrix Q is returned; */
/* = 'N': Q is not computed. */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* P (input) INTEGER */
/* The number of rows of the matrix B. P >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrices A and B. N >= 0. */
/* K (input) INTEGER */
/* L (input) INTEGER */
/* K and L specify the subblocks in the input matrices A and B: */
/* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) */
/* of A and B, whose GSVD is going to be computed by STGSJA. */
/* See Further details. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular */
/* matrix R or part of R. See Purpose for details. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* B (input/output) REAL array, dimension (LDB,N) */
/* On entry, the P-by-N matrix B. */
/* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains */
/* a part of R. See Purpose for details. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,P). */
/* TOLA (input) REAL */
/* TOLB (input) REAL */
/* TOLA and TOLB are the convergence criteria for the Jacobi- */
/* Kogbetliantz iteration procedure. Generally, they are the */
/* same as used in the preprocessing step, say */
/* TOLA = max(M,N)*norm(A)*MACHEPS, */
/* TOLB = max(P,N)*norm(B)*MACHEPS. */
/* ALPHA (output) REAL array, dimension (N) */
/* BETA (output) REAL array, dimension (N) */
/* On exit, ALPHA and BETA contain the generalized singular */
/* value pairs of A and B; */
/* ALPHA(1:K) = 1, */
/* BETA(1:K) = 0, */
/* and if M-K-L >= 0, */
/* ALPHA(K+1:K+L) = diag(C), */
/* BETA(K+1:K+L) = diag(S), */
/* or if M-K-L < 0, */
/* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 */
/* BETA(K+1:M) = S, BETA(M+1:K+L) = 1. */
/* Furthermore, if K+L < N, */
/* ALPHA(K+L+1:N) = 0 and */
/* BETA(K+L+1:N) = 0. */
/* U (input/output) REAL array, dimension (LDU,M) */
/* On entry, if JOBU = 'U', U must contain a matrix U1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBU = 'I', U contains the orthogonal matrix U; */
/* if JOBU = 'U', U contains the product U1*U. */
/* If JOBU = 'N', U is not referenced. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= max(1,M) if */
/* JOBU = 'U'; LDU >= 1 otherwise. */
/* V (input/output) REAL array, dimension (LDV,P) */
/* On entry, if JOBV = 'V', V must contain a matrix V1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBV = 'I', V contains the orthogonal matrix V; */
/* if JOBV = 'V', V contains the product V1*V. */
/* If JOBV = 'N', V is not referenced. */
/* LDV (input) INTEGER */
/* The leading dimension of the array V. LDV >= max(1,P) if */
/* JOBV = 'V'; LDV >= 1 otherwise. */
/* Q (input/output) REAL array, dimension (LDQ,N) */
/* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBQ = 'I', Q contains the orthogonal matrix Q; */
/* if JOBQ = 'Q', Q contains the product Q1*Q. */
/* If JOBQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= max(1,N) if */
/* JOBQ = 'Q'; LDQ >= 1 otherwise. */
/* WORK (workspace) REAL array, dimension (2*N) */
/* NCYCLE (output) INTEGER */
/* The number of cycles required for convergence. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* = 1: the procedure does not converge after MAXIT cycles. */
/* Internal Parameters */
/* =================== */
/* MAXIT INTEGER */
/* MAXIT specifies the total loops that the iterative procedure */
/* may take. If after MAXIT cycles, the routine fails to */
/* converge, we return INFO = 1. */
/* Further Details */
/* =============== */
/* STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce */
/* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L */
/* matrix B13 to the form: */
/* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, */
/* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose */
/* of Z. C1 and S1 are diagonal matrices satisfying */
/* C1**2 + S1**2 = I, */
/* and R1 is an L-by-L nonsingular upper triangular matrix. */
/* ===================================================================== */
/* .. Parameters .. */
/*< INTEGER MAXIT >*/
/*< PARAMETER ( MAXIT = 40 ) >*/
/*< REAL ZERO, ONE >*/
/*< PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) >*/
/* .. */
/* .. Local Scalars .. */
/*< LOGICAL INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV >*/
/*< INTEGER I, J, KCYCLE >*/
/*< >*/
/* .. */
/* .. External Functions .. */
/*< LOGICAL LSAME >*/
/*< EXTERNAL LSAME >*/
/* .. */
/* .. External Subroutines .. */
/*< >*/
/* .. */
/* .. Intrinsic Functions .. */
/*< INTRINSIC ABS, MAX, MIN >*/
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/*< INITU = LSAME( JOBU, 'I' ) >*/
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--work;
/* Function Body */
initu = lsame_(jobu, "I", (ftnlen)1, (ftnlen)1);
/*< WANTU = INITU .OR. LSAME( JOBU, 'U' ) >*/
wantu = initu || lsame_(jobu, "U", (ftnlen)1, (ftnlen)1);
/*< INITV = LSAME( JOBV, 'I' ) >*/
initv = lsame_(jobv, "I", (ftnlen)1, (ftnlen)1);
/*< WANTV = INITV .OR. LSAME( JOBV, 'V' ) >*/
wantv = initv || lsame_(jobv, "V", (ftnlen)1, (ftnlen)1);
/*< INITQ = LSAME( JOBQ, 'I' ) >*/
initq = lsame_(jobq, "I", (ftnlen)1, (ftnlen)1);
/*< WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' ) >*/
wantq = initq || lsame_(jobq, "Q", (ftnlen)1, (ftnlen)1);
/*< INFO = 0 >*/
*info = 0;
/*< IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN >*/
if (! (initu || wantu || lsame_(jobu, "N", (ftnlen)1, (ftnlen)1))) {
/*< INFO = -1 >*/
*info = -1;
/*< ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN >*/
} else if (! (initv || wantv || lsame_(jobv, "N", (ftnlen)1, (ftnlen)1)))
{
/*< INFO = -2 >*/
*info = -2;
/*< ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN >*/
} else if (! (initq || wantq || lsame_(jobq, "N", (ftnlen)1, (ftnlen)1)))
{
/*< INFO = -3 >*/
*info = -3;
/*< ELSE IF( M.LT.0 ) THEN >*/
} else if (*m < 0) {
/*< INFO = -4 >*/
*info = -4;
/*< ELSE IF( P.LT.0 ) THEN >*/
} else if (*p < 0) {
/*< INFO = -5 >*/
*info = -5;
/*< ELSE IF( N.LT.0 ) THEN >*/
} else if (*n < 0) {
/*< INFO = -6 >*/
*info = -6;
/*< ELSE IF( LDA.LT.MAX( 1, M ) ) THEN >*/
} else if (*lda < max(1,*m)) {
/*< INFO = -10 >*/
*info = -10;
/*< ELSE IF( LDB.LT.MAX( 1, P ) ) THEN >*/
} else if (*ldb < max(1,*p)) {
/*< INFO = -12 >*/
*info = -12;
/*< ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN >*/
} else if (*ldu < 1 || (wantu && *ldu < *m)) {
/*< INFO = -18 >*/
*info = -18;
/*< ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN >*/
} else if (*ldv < 1 || (wantv && *ldv < *p)) {
/*< INFO = -20 >*/
*info = -20;
/*< ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN >*/
} else if (*ldq < 1 || (wantq && *ldq < *n)) {
/*< INFO = -22 >*/
*info = -22;
/*< END IF >*/
}
/*< IF( INFO.NE.0 ) THEN >*/
if (*info != 0) {
/*< CALL XERBLA( 'STGSJA', -INFO ) >*/
i__1 = -(*info);
xerbla_("STGSJA", &i__1, (ftnlen)6);
/*< RETURN >*/
return 0;
/*< END IF >*/
}
/* Initialize U, V and Q, if necessary */
/*< >*/
if (initu) {
slaset_("Full", m, m, &c_b13, &c_b14, &u[u_offset], ldu, (ftnlen)4);
}
/*< >*/
if (initv) {
slaset_("Full", p, p, &c_b13, &c_b14, &v[v_offset], ldv, (ftnlen)4);
}
/*< >*/
if (initq) {
slaset_("Full", n, n, &c_b13, &c_b14, &q[q_offset], ldq, (ftnlen)4);
}
/* Loop until convergence */
/*< UPPER = .FALSE. >*/
upper = FALSE_;
/*< DO 40 KCYCLE = 1, MAXIT >*/
for (kcycle = 1; kcycle <= 40; ++kcycle) {
/*< UPPER = .NOT.UPPER >*/
upper = ! upper;
/*< DO 20 I = 1, L - 1 >*/
i__1 = *l - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/*< DO 10 J = I + 1, L >*/
i__2 = *l;
for (j = i__ + 1; j <= i__2; ++j) {
/*< A1 = ZERO >*/
a1 = (float)0.;
/*< A2 = ZERO >*/
a2 = (float)0.;
/*< A3 = ZERO >*/
a3 = (float)0.;
/*< >*/
if (*k + i__ <= *m) {
a1 = a[*k + i__ + (*n - *l + i__) * a_dim1];
}
/*< >*/
if (*k + j <= *m) {
a3 = a[*k + j + (*n - *l + j) * a_dim1];
}
/*< B1 = B( I, N-L+I ) >*/
b1 = b[i__ + (*n - *l + i__) * b_dim1];
/*< B3 = B( J, N-L+J ) >*/
b3 = b[j + (*n - *l + j) * b_dim1];
/*< IF( UPPER ) THEN >*/
if (upper) {
/*< >*/
if (*k + i__ <= *m) {
a2 = a[*k + i__ + (*n - *l + j) * a_dim1];
}
/*< B2 = B( I, N-L+J ) >*/
b2 = b[i__ + (*n - *l + j) * b_dim1];
/*< ELSE >*/
} else {
/*< >*/
if (*k + j <= *m) {
a2 = a[*k + j + (*n - *l + i__) * a_dim1];
}
/*< B2 = B( J, N-L+I ) >*/
b2 = b[j + (*n - *l + i__) * b_dim1];
/*< END IF >*/
}
/*< >*/
slags2_(&upper, &a1, &a2, &a3, &b1, &b2, &b3, &csu, &snu, &
csv, &snv, &csq, &snq);
/* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A */
/*< >*/
if (*k + j <= *m) {
srot_(l, &a[*k + j + (*n - *l + 1) * a_dim1], lda, &a[*k
+ i__ + (*n - *l + 1) * a_dim1], lda, &csu, &snu);
}
/* Update I-th and J-th rows of matrix B: V'*B */
/*< >*/
srot_(l, &b[j + (*n - *l + 1) * b_dim1], ldb, &b[i__ + (*n - *
l + 1) * b_dim1], ldb, &csv, &snv);
/* Update (N-L+I)-th and (N-L+J)-th columns of matrices */
/* A and B: A*Q and B*Q */
/*< >*/
/* Computing MIN */
i__4 = *k + *l;
i__3 = min(i__4,*m);
srot_(&i__3, &a[(*n - *l + j) * a_dim1 + 1], &c__1, &a[(*n - *
l + i__) * a_dim1 + 1], &c__1, &csq, &snq);
/*< >*/
srot_(l, &b[(*n - *l + j) * b_dim1 + 1], &c__1, &b[(*n - *l +
i__) * b_dim1 + 1], &c__1, &csq, &snq);
/*< IF( UPPER ) THEN >*/
if (upper) {
/*< >*/
if (*k + i__ <= *m) {
a[*k + i__ + (*n - *l + j) * a_dim1] = (float)0.;
}
/*< B( I, N-L+J ) = ZERO >*/
b[i__ + (*n - *l + j) * b_dim1] = (float)0.;
/*< ELSE >*/
} else {
/*< >*/
if (*k + j <= *m) {
a[*k + j + (*n - *l + i__) * a_dim1] = (float)0.;
}
/*< B( J, N-L+I ) = ZERO >*/
b[j + (*n - *l + i__) * b_dim1] = (float)0.;
/*< END IF >*/
}
/* Update orthogonal matrices U, V, Q, if desired. */
/*< >*/
if (wantu && *k + j <= *m) {
srot_(m, &u[(*k + j) * u_dim1 + 1], &c__1, &u[(*k + i__) *
u_dim1 + 1], &c__1, &csu, &snu);
}
/*< >*/
if (wantv) {
srot_(p, &v[j * v_dim1 + 1], &c__1, &v[i__ * v_dim1 + 1],
&c__1, &csv, &snv);
}
/*< >*/
if (wantq) {
srot_(n, &q[(*n - *l + j) * q_dim1 + 1], &c__1, &q[(*n - *
l + i__) * q_dim1 + 1], &c__1, &csq, &snq);
}
/*< 10 CONTINUE >*/
/* L10: */
}
/*< 20 CONTINUE >*/
/* L20: */
}
/*< IF( .NOT.UPPER ) THEN >*/
if (! upper) {
/* The matrices A13 and B13 were lower triangular at the start */
/* of the cycle, and are now upper triangular. */
/* Convergence test: test the parallelism of the corresponding */
/* rows of A and B. */
/*< ERROR = ZERO >*/
error = (float)0.;
/*< DO 30 I = 1, MIN( L, M-K ) >*/
/* Computing MIN */
i__2 = *l, i__3 = *m - *k;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
/*< CALL SCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 ) >*/
i__2 = *l - i__ + 1;
scopy_(&i__2, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda, &
work[1], &c__1);
/*< CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 ) >*/
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &work[*
l + 1], &c__1);
/*< CALL SLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN ) >*/
i__2 = *l - i__ + 1;
slapll_(&i__2, &work[1], &c__1, &work[*l + 1], &c__1, &ssmin);
/*< ERROR = MAX( ERROR, SSMIN ) >*/
error = dmax(error,ssmin);
/*< 30 CONTINUE >*/
/* L30: */
}
/*< >*/
if (dabs(error) <= dmin(*tola,*tolb)) {
goto L50;
}
/*< END IF >*/
}
/* End of cycle loop */
/*< 40 CONTINUE >*/
/* L40: */
}
/* The algorithm has not converged after MAXIT cycles. */
/*< INFO = 1 >*/
*info = 1;
/*< GO TO 100 >*/
goto L100;
/*< 50 CONTINUE >*/
L50:
/* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged. */
/* Compute the generalized singular value pairs (ALPHA, BETA), and */
/* set the triangular matrix R to array A. */
/*< DO 60 I = 1, K >*/
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
/*< ALPHA( I ) = ONE >*/
alpha[i__] = (float)1.;
/*< BETA( I ) = ZERO >*/
beta[i__] = (float)0.;
/*< 60 CONTINUE >*/
/* L60: */
}
/*< DO 70 I = 1, MIN( L, M-K ) >*/
/* Computing MIN */
i__2 = *l, i__3 = *m - *k;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
/*< A1 = A( K+I, N-L+I ) >*/
a1 = a[*k + i__ + (*n - *l + i__) * a_dim1];
/*< B1 = B( I, N-L+I ) >*/
b1 = b[i__ + (*n - *l + i__) * b_dim1];
/*< IF( A1.NE.ZERO ) THEN >*/
if (a1 != (float)0.) {
/*< GAMMA = B1 / A1 >*/
gamma = b1 / a1;
/* change sign if necessary */
/*< IF( GAMMA.LT.ZERO ) THEN >*/
if (gamma < (float)0.) {
/*< CALL SSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB ) >*/
i__2 = *l - i__ + 1;
sscal_(&i__2, &c_b43, &b[i__ + (*n - *l + i__) * b_dim1], ldb)
;
/*< >*/
if (wantv) {
sscal_(p, &c_b43, &v[i__ * v_dim1 + 1], &c__1);
}
/*< END IF >*/
}
/*< >*/
r__1 = dabs(gamma);
slartg_(&r__1, &c_b14, &beta[*k + i__], &alpha[*k + i__], &rwk);
/*< IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN >*/
if (alpha[*k + i__] >= beta[*k + i__]) {
/*< >*/
i__2 = *l - i__ + 1;
r__1 = (float)1. / alpha[*k + i__];
sscal_(&i__2, &r__1, &a[*k + i__ + (*n - *l + i__) * a_dim1],
lda);
/*< ELSE >*/
} else {
/*< >*/
i__2 = *l - i__ + 1;
r__1 = (float)1. / beta[*k + i__];
sscal_(&i__2, &r__1, &b[i__ + (*n - *l + i__) * b_dim1], ldb);
/*< >*/
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k
+ i__ + (*n - *l + i__) * a_dim1], lda);
/*< END IF >*/
}
/*< ELSE >*/
} else {
/*< ALPHA( K+I ) = ZERO >*/
alpha[*k + i__] = (float)0.;
/*< BETA( K+I ) = ONE >*/
beta[*k + i__] = (float)1.;
/*< >*/
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k +
i__ + (*n - *l + i__) * a_dim1], lda);
/*< END IF >*/
}
/*< 70 CONTINUE >*/
/* L70: */
}
/* Post-assignment */
/*< DO 80 I = M + 1, K + L >*/
i__1 = *k + *l;
for (i__ = *m + 1; i__ <= i__1; ++i__) {
/*< ALPHA( I ) = ZERO >*/
alpha[i__] = (float)0.;
/*< BETA( I ) = ONE >*/
beta[i__] = (float)1.;
/*< 80 CONTINUE >*/
/* L80: */
}
/*< IF( K+L.LT.N ) THEN >*/
if (*k + *l < *n) {
/*< DO 90 I = K + L + 1, N >*/
i__1 = *n;
for (i__ = *k + *l + 1; i__ <= i__1; ++i__) {
/*< ALPHA( I ) = ZERO >*/
alpha[i__] = (float)0.;
/*< BETA( I ) = ZERO >*/
beta[i__] = (float)0.;
/*< 90 CONTINUE >*/
/* L90: */
}
/*< END IF >*/
}
/*< 100 CONTINUE >*/
L100:
/*< NCYCLE = KCYCLE >*/
*ncycle = kcycle;
/*< RETURN >*/
return 0;
/* End of STGSJA */
/*< END >*/
} /* stgsja_ */
/* Subroutine */ int sgghrd_(char *compq, char *compz, integer *n, integer *
ilo, integer *ihi, real *a, integer *lda, real *b, integer *ldb, real
*q, integer *ldq, real *z__, integer *ldz, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2, i__3;
/* Local variables */
static integer jcol;
static real temp;
static integer jrow;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static real c__, s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
static integer icompq;
extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
real *, real *, integer *), slartg_(real *, real *, real *
, real *, real *);
static integer icompz;
static logical ilq, ilz;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
/* -- LAPACK routine (instrumented to count operations, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
---------------------- Begin Timing Code -------------------------
Common block to return operation count and iteration count
ITCNT is initialized to 0, OPS is only incremented
OPST is used to accumulate small contributions to OPS
to avoid roundoff error
----------------------- End Timing Code --------------------------
Purpose
=======
SGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a
general matrix and B is upper triangular: Q' * A * Z = H and
Q' * B * Z = T, where H is upper Hessenberg, T is upper triangular,
and Q and Z are orthogonal, and ' means transpose.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'
Arguments
=========
COMPQ (input) CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'V': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ (input) CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'V': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
It is assumed that A is already upper triangular in rows and
columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set
by a previous call to SGGBAL; otherwise they should be set
to 1 and N respectively.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
rest is set to zero.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q' B Z. The
elements below the diagonal are set to zero.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) REAL array, dimension (LDQ, N)
If COMPQ='N': Q is not referenced.
If COMPQ='I': on entry, Q need not be set, and on exit it
contains the orthogonal matrix Q, where Q'
is the product of the Givens transformations
which are applied to A and B on the left.
If COMPQ='V': on entry, Q must contain an orthogonal matrix
Q1, and on exit this is overwritten by Q1*Q.
LDQ (input) INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
Z (input/output) REAL array, dimension (LDZ, N)
If COMPZ='N': Z is not referenced.
If COMPZ='I': on entry, Z need not be set, and on exit it
contains the orthogonal matrix Z, which is
the product of the Givens transformations
which are applied to A and B on the right.
If COMPZ='V': on entry, Z must contain an orthogonal matrix
Z1, and on exit this is overwritten by Z1*Z.
LDZ (input) INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
This routine reduces A to Hessenberg and B to triangular form by
an unblocked reduction, as described in _Matrix_Computations_,
by Golub and Van Loan (Johns Hopkins Press.)
=====================================================================
Decode COMPQ
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
/* Function Body */
if (lsame_(compq, "N")) {
ilq = FALSE_;
icompq = 1;
} else if (lsame_(compq, "V")) {
ilq = TRUE_;
icompq = 2;
} else if (lsame_(compq, "I")) {
ilq = TRUE_;
icompq = 3;
} else {
icompq = 0;
}
/* Decode COMPZ */
if (lsame_(compz, "N")) {
ilz = FALSE_;
icompz = 1;
} else if (lsame_(compz, "V")) {
ilz = TRUE_;
icompz = 2;
} else if (lsame_(compz, "I")) {
ilz = TRUE_;
icompz = 3;
} else {
icompz = 0;
}
/* Test the input parameters. */
*info = 0;
if (icompq <= 0) {
*info = -1;
} else if (icompz <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ilo < 1) {
*info = -4;
} else if (*ihi > *n || *ihi < *ilo - 1) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (ilq && *ldq < *n || *ldq < 1) {
*info = -11;
} else if (ilz && *ldz < *n || *ldz < 1) {
*info = -13;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGGHRD", &i__1);
return 0;
}
/* Initialize Q and Z if desired. */
if (icompq == 3) {
slaset_("Full", n, n, &c_b10, &c_b11, &q[q_offset], ldq);
}
if (icompz == 3) {
slaset_("Full", n, n, &c_b10, &c_b11, &z__[z_offset], ldz);
}
/* Quick return if possible */
if (*n <= 1) {
return 0;
}
/* Zero out lower triangle of B */
i__1 = *n - 1;
for (jcol = 1; jcol <= i__1; ++jcol) {
i__2 = *n;
for (jrow = jcol + 1; jrow <= i__2; ++jrow) {
b_ref(jrow, jcol) = 0.f;
/* L10: */
}
/* L20: */
}
/* Reduce A and B */
i__1 = *ihi - 2;
for (jcol = *ilo; jcol <= i__1; ++jcol) {
i__2 = jcol + 2;
for (jrow = *ihi; jrow >= i__2; --jrow) {
/* Step 1: rotate rows JROW-1, JROW to kill A(JROW,JCOL) */
temp = a_ref(jrow - 1, jcol);
slartg_(&temp, &a_ref(jrow, jcol), &c__, &s, &a_ref(jrow - 1,
jcol));
a_ref(jrow, jcol) = 0.f;
i__3 = *n - jcol;
srot_(&i__3, &a_ref(jrow - 1, jcol + 1), lda, &a_ref(jrow, jcol +
1), lda, &c__, &s);
i__3 = *n + 2 - jrow;
srot_(&i__3, &b_ref(jrow - 1, jrow - 1), ldb, &b_ref(jrow, jrow -
1), ldb, &c__, &s);
if (ilq) {
srot_(n, &q_ref(1, jrow - 1), &c__1, &q_ref(1, jrow), &c__1, &
c__, &s);
}
/* Step 2: rotate columns JROW, JROW-1 to kill B(JROW,JROW-1) */
temp = b_ref(jrow, jrow);
slartg_(&temp, &b_ref(jrow, jrow - 1), &c__, &s, &b_ref(jrow,
jrow));
b_ref(jrow, jrow - 1) = 0.f;
srot_(ihi, &a_ref(1, jrow), &c__1, &a_ref(1, jrow - 1), &c__1, &
c__, &s);
i__3 = jrow - 1;
srot_(&i__3, &b_ref(1, jrow), &c__1, &b_ref(1, jrow - 1), &c__1, &
c__, &s);
if (ilz) {
srot_(n, &z___ref(1, jrow), &c__1, &z___ref(1, jrow - 1), &
c__1, &c__, &s);
}
/* L30: */
}
/* L40: */
}
/* ---------------------- Begin Timing Code -------------------------
Operation count: factor
* number of calls to SLARTG TEMP *7
* total number of rows/cols
rotated in A and B TEMP*[6n + 2(ihi-ilo) + 5]/6 *6
* rows rotated in Q TEMP*n/2 *6
* rows rotated in Z TEMP*n/2 *6 */
temp = (real) (*ihi - *ilo) * (real) (*ihi - *ilo - 1);
jrow = *n * 6 + (*ihi - *ilo << 1) + 12;
if (ilq) {
jrow += *n * 3;
}
if (ilz) {
jrow += *n * 3;
}
latime_1.ops += (real) jrow * temp;
latime_1.itcnt = 0.f;
/* ----------------------- End Timing Code -------------------------- */
return 0;
/* End of SGGHRD */
} /* sgghrd_ */
/* Subroutine */ int stgex2_(logical *wantq, logical *wantz, integer *n, real
*a, integer *lda, real *b, integer *ldb, real *q, integer *ldq, real *
z__, integer *ldz, integer *j1, integer *n1, integer *n2, real *work,
integer *lwork, integer *info)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22)
of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair
(A, B) by an orthogonal equivalence transformation.
(A, B) must be in generalized real Schur canonical form (as returned
by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2
diagonal blocks. B is upper triangular.
Optionally, the matrices Q and Z of generalized Schur vectors are
updated.
Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)'
Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)'
Arguments
=========
WANTQ (input) LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ (input) LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) REAL arrays, dimensions (LDA,N)
On entry, the matrix A in the pair (A, B).
On exit, the updated matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) REAL arrays, dimensions (LDB,N)
On entry, the matrix B in the pair (A, B).
On exit, the updated matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) REAL array, dimension (LDZ,N)
On entry, if WANTQ = .TRUE., the orthogonal matrix Q.
On exit, the updated matrix Q.
Not referenced if WANTQ = .FALSE..
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N.
Z (input/output) REAL array, dimension (LDZ,N)
On entry, if WANTZ =.TRUE., the orthogonal matrix Z.
On exit, the updated matrix Z.
Not referenced if WANTZ = .FALSE..
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N.
J1 (input) INTEGER
The index to the first block (A11, B11). 1 <= J1 <= N.
N1 (input) INTEGER
The order of the first block (A11, B11). N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block (A22, B22). N2 = 0, 1 or 2.
WORK (workspace) REAL array, dimension (LWORK).
LWORK (input) INTEGER
The dimension of the array WORK.
LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 )
INFO (output) INTEGER
=0: Successful exit
>0: If INFO = 1, the transformed matrix (A, B) would be
too far from generalized Schur form; the blocks are
not swapped and (A, B) and (Q, Z) are unchanged.
The problem of swapping is too ill-conditioned.
<0: If INFO = -16: LWORK is too small. Appropriate value
for LWORK is returned in WORK(1).
Further Details
===============
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
In the current code both weak and strong stability tests are
performed. The user can omit the strong stability test by changing
the internal logical parameter WANDS to .FALSE.. See ref. [2] for
details.
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software,
Report UMINF - 94.04, Department of Computing Science, Umea
University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87. To appear in Numerical Algorithms, 1996.
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__16 = 16;
static real c_b3 = 0.f;
static integer c__0 = 0;
static integer c__1 = 1;
static integer c__4 = 4;
static integer c__2 = 2;
static real c_b38 = 1.f;
static real c_b44 = -1.f;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static logical weak;
static real ddum;
static integer idum;
static real taul[4], dsum, taur[4], scpy[16] /* was [4][4] */,
tcpy[16] /* was [4][4] */;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static real f, g;
static integer i__, m;
static real s[16] /* was [4][4] */, t[16] /* was [4][4] */, scale,
bqra21, brqa21;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static real licop[16] /* was [4][4] */;
static integer linfo;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
static real ircop[16] /* was [4][4] */, dnorm;
static integer iwork[4];
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slagv2_(real *, integer *, real *, integer *, real *,
real *, real *, real *, real *, real *, real *), sgeqr2_(integer *
, integer *, real *, integer *, real *, real *, integer *),
sgerq2_(integer *, integer *, real *, integer *, real *, real *,
integer *);
static real be[2], ai[2];
extern /* Subroutine */ int sorg2r_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *), sorgr2_(integer *,
integer *, integer *, real *, integer *, real *, real *, integer *
);
static real ar[2], sa, sb, li[16] /* was [4][4] */;
extern /* Subroutine */ int sorm2r_(char *, char *, integer *, integer *,
integer *, real *, integer *, real *, real *, integer *, real *,
integer *), sormr2_(char *, char *, integer *,
integer *, integer *, real *, integer *, real *, real *, integer *
, real *, integer *);
static real dscale, ir[16] /* was [4][4] */;
extern /* Subroutine */ int stgsy2_(char *, integer *, integer *, integer
*, real *, integer *, real *, integer *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *, real *, real *,
real *, integer *, integer *, integer *);
static real ss;
extern doublereal slamch_(char *);
static real ws;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), slartg_(real *, real *,
real *, real *, real *);
static real thresh;
extern /* Subroutine */ int slassq_(integer *, real *, integer *, real *,
real *);
static real smlnum;
static logical strong;
static real eps;
#define scpy_ref(a_1,a_2) scpy[(a_2)*4 + a_1 - 5]
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define s_ref(a_1,a_2) s[(a_2)*4 + a_1 - 5]
#define t_ref(a_1,a_2) t[(a_2)*4 + a_1 - 5]
#define z___ref(a_1,a_2) z__[(a_2)*z_dim1 + a_1]
#define li_ref(a_1,a_2) li[(a_2)*4 + a_1 - 5]
#define ir_ref(a_1,a_2) ir[(a_2)*4 + a_1 - 5]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
/* Quick return if possible */
if (*n <= 1 || *n1 <= 0 || *n2 <= 0) {
return 0;
}
if (*n1 > *n || *j1 + *n1 > *n) {
return 0;
}
m = *n1 + *n2;
/* Computing MAX */
i__1 = *n * m, i__2 = m * m << 1;
if (*lwork < max(i__1,i__2)) {
*info = -16;
/* Computing MAX */
i__1 = *n * m, i__2 = m * m << 1;
work[1] = (real) max(i__1,i__2);
return 0;
}
weak = FALSE_;
strong = FALSE_;
/* Make a local copy of selected block */
scopy_(&c__16, &c_b3, &c__0, li, &c__1);
scopy_(&c__16, &c_b3, &c__0, ir, &c__1);
slacpy_("Full", &m, &m, &a_ref(*j1, *j1), lda, s, &c__4);
slacpy_("Full", &m, &m, &b_ref(*j1, *j1), ldb, t, &c__4);
/* Compute threshold for testing acceptance of swapping. */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
dscale = 0.f;
dsum = 1.f;
slacpy_("Full", &m, &m, s, &c__4, &work[1], &m);
i__1 = m * m;
slassq_(&i__1, &work[1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, t, &c__4, &work[1], &m);
i__1 = m * m;
slassq_(&i__1, &work[1], &c__1, &dscale, &dsum);
dnorm = dscale * sqrt(dsum);
/* Computing MAX */
r__1 = eps * 10.f * dnorm;
thresh = dmax(r__1,smlnum);
if (m == 2) {
/* CASE 1: Swap 1-by-1 and 1-by-1 blocks.
Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks
using Givens rotations and perform the swap tentatively. */
f = s_ref(2, 2) * t_ref(1, 1) - t_ref(2, 2) * s_ref(1, 1);
g = s_ref(2, 2) * t_ref(1, 2) - t_ref(2, 2) * s_ref(1, 2);
sb = (r__1 = t_ref(2, 2), dabs(r__1));
sa = (r__1 = s_ref(2, 2), dabs(r__1));
slartg_(&f, &g, &ir_ref(1, 2), &ir_ref(1, 1), &ddum);
ir_ref(2, 1) = -ir_ref(1, 2);
ir_ref(2, 2) = ir_ref(1, 1);
srot_(&c__2, &s_ref(1, 1), &c__1, &s_ref(1, 2), &c__1, &ir_ref(1, 1),
&ir_ref(2, 1));
srot_(&c__2, &t_ref(1, 1), &c__1, &t_ref(1, 2), &c__1, &ir_ref(1, 1),
&ir_ref(2, 1));
if (sa >= sb) {
slartg_(&s_ref(1, 1), &s_ref(2, 1), &li_ref(1, 1), &li_ref(2, 1),
&ddum);
} else {
slartg_(&t_ref(1, 1), &t_ref(2, 1), &li_ref(1, 1), &li_ref(2, 1),
&ddum);
}
srot_(&c__2, &s_ref(1, 1), &c__4, &s_ref(2, 1), &c__4, &li_ref(1, 1),
&li_ref(2, 1));
srot_(&c__2, &t_ref(1, 1), &c__4, &t_ref(2, 1), &c__4, &li_ref(1, 1),
&li_ref(2, 1));
li_ref(2, 2) = li_ref(1, 1);
li_ref(1, 2) = -li_ref(2, 1);
/* Weak stability test:
|S21| + |T21| <= O(EPS * F-norm((S, T))) */
ws = (r__1 = s_ref(2, 1), dabs(r__1)) + (r__2 = t_ref(2, 1), dabs(
r__2));
weak = ws <= thresh;
if (! weak) {
goto L70;
}
if (TRUE_) {
/* Strong stability test:
F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B))) */
slacpy_("Full", &m, &m, &a_ref(*j1, *j1), lda, &work[m * m + 1], &
m);
sgemm_("N", "N", &m, &m, &m, &c_b38, li, &c__4, s, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b44, &work[1], &m, ir, &c__4, &
c_b38, &work[m * m + 1], &m);
dscale = 0.f;
dsum = 1.f;
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, &b_ref(*j1, *j1), ldb, &work[m * m + 1], &
m);
sgemm_("N", "N", &m, &m, &m, &c_b38, li, &c__4, t, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b44, &work[1], &m, ir, &c__4, &
c_b38, &work[m * m + 1], &m);
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
ss = dscale * sqrt(dsum);
strong = ss <= thresh;
if (! strong) {
goto L70;
}
}
/* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
(A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). */
i__1 = *j1 + 1;
srot_(&i__1, &a_ref(1, *j1), &c__1, &a_ref(1, *j1 + 1), &c__1, &
ir_ref(1, 1), &ir_ref(2, 1));
i__1 = *j1 + 1;
srot_(&i__1, &b_ref(1, *j1), &c__1, &b_ref(1, *j1 + 1), &c__1, &
ir_ref(1, 1), &ir_ref(2, 1));
i__1 = *n - *j1 + 1;
srot_(&i__1, &a_ref(*j1, *j1), lda, &a_ref(*j1 + 1, *j1), lda, &
li_ref(1, 1), &li_ref(2, 1));
i__1 = *n - *j1 + 1;
srot_(&i__1, &b_ref(*j1, *j1), ldb, &b_ref(*j1 + 1, *j1), ldb, &
li_ref(1, 1), &li_ref(2, 1));
/* Set N1-by-N2 (2,1) - blocks to ZERO. */
a_ref(*j1 + 1, *j1) = 0.f;
b_ref(*j1 + 1, *j1) = 0.f;
/* Accumulate transformations into Q and Z if requested. */
if (*wantz) {
srot_(n, &z___ref(1, *j1), &c__1, &z___ref(1, *j1 + 1), &c__1, &
ir_ref(1, 1), &ir_ref(2, 1));
}
if (*wantq) {
srot_(n, &q_ref(1, *j1), &c__1, &q_ref(1, *j1 + 1), &c__1, &
li_ref(1, 1), &li_ref(2, 1));
}
/* Exit with INFO = 0 if swap was successfully performed. */
return 0;
} else {
/* CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2
and 2-by-2 blocks.
Solve the generalized Sylvester equation
S11 * R - L * S22 = SCALE * S12
T11 * R - L * T22 = SCALE * T12
for R and L. Solutions in LI and IR. */
slacpy_("Full", n1, n2, &t_ref(1, *n1 + 1), &c__4, li, &c__4);
slacpy_("Full", n1, n2, &s_ref(1, *n1 + 1), &c__4, &ir_ref(*n2 + 1, *
n1 + 1), &c__4);
stgsy2_("N", &c__0, n1, n2, s, &c__4, &s_ref(*n1 + 1, *n1 + 1), &c__4,
&ir_ref(*n2 + 1, *n1 + 1), &c__4, t, &c__4, &t_ref(*n1 + 1, *
n1 + 1), &c__4, li, &c__4, &scale, &dsum, &dscale, iwork, &
idum, &linfo);
/* Compute orthogonal matrix QL:
QL' * LI = [ TL ]
[ 0 ]
where
LI = [ -L ]
[ SCALE * identity(N2) ] */
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
sscal_(n1, &c_b44, &li_ref(1, i__), &c__1);
li_ref(*n1 + i__, i__) = scale;
/* L10: */
}
sgeqr2_(&m, n2, li, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorg2r_(&m, &m, n2, li, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
/* Compute orthogonal matrix RQ:
IR * RQ' = [ 0 TR],
where IR = [ SCALE * identity(N1), R ] */
i__1 = *n1;
for (i__ = 1; i__ <= i__1; ++i__) {
ir_ref(*n2 + i__, i__) = scale;
/* L20: */
}
sgerq2_(n1, &m, &ir_ref(*n2 + 1, 1), &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorgr2_(&m, &m, n1, ir, &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
/* Perform the swapping tentatively: */
sgemm_("T", "N", &m, &m, &m, &c_b38, li, &c__4, s, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b38, &work[1], &m, ir, &c__4, &c_b3,
s, &c__4);
sgemm_("T", "N", &m, &m, &m, &c_b38, li, &c__4, t, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b38, &work[1], &m, ir, &c__4, &c_b3,
t, &c__4);
slacpy_("F", &m, &m, s, &c__4, scpy, &c__4);
slacpy_("F", &m, &m, t, &c__4, tcpy, &c__4);
slacpy_("F", &m, &m, ir, &c__4, ircop, &c__4);
slacpy_("F", &m, &m, li, &c__4, licop, &c__4);
/* Triangularize the B-part by an RQ factorization.
Apply transformation (from left) to A-part, giving S. */
sgerq2_(&m, &m, t, &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sormr2_("R", "T", &m, &m, &m, t, &c__4, taur, s, &c__4, &work[1], &
linfo);
if (linfo != 0) {
goto L70;
}
sormr2_("L", "N", &m, &m, &m, t, &c__4, taur, ir, &c__4, &work[1], &
linfo);
if (linfo != 0) {
goto L70;
}
/* Compute F-norm(S21) in BRQA21. (T21 is 0.) */
dscale = 0.f;
dsum = 1.f;
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
slassq_(n1, &s_ref(*n2 + 1, i__), &c__1, &dscale, &dsum);
/* L30: */
}
brqa21 = dscale * sqrt(dsum);
/* Triangularize the B-part by a QR factorization.
Apply transformation (from right) to A-part, giving S. */
sgeqr2_(&m, &m, tcpy, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorm2r_("L", "T", &m, &m, &m, tcpy, &c__4, taul, scpy, &c__4, &work[1]
, info);
sorm2r_("R", "N", &m, &m, &m, tcpy, &c__4, taul, licop, &c__4, &work[
1], info);
if (linfo != 0) {
goto L70;
}
/* Compute F-norm(S21) in BQRA21. (T21 is 0.) */
dscale = 0.f;
dsum = 1.f;
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
slassq_(n1, &scpy_ref(*n2 + 1, i__), &c__1, &dscale, &dsum);
/* L40: */
}
bqra21 = dscale * sqrt(dsum);
/* Decide which method to use.
Weak stability test:
F-norm(S21) <= O(EPS * F-norm((S, T))) */
if (bqra21 <= brqa21 && bqra21 <= thresh) {
slacpy_("F", &m, &m, scpy, &c__4, s, &c__4);
slacpy_("F", &m, &m, tcpy, &c__4, t, &c__4);
slacpy_("F", &m, &m, ircop, &c__4, ir, &c__4);
slacpy_("F", &m, &m, licop, &c__4, li, &c__4);
} else if (brqa21 >= thresh) {
goto L70;
}
/* Set lower triangle of B-part to zero */
i__1 = m;
for (i__ = 2; i__ <= i__1; ++i__) {
i__2 = m - i__ + 1;
scopy_(&i__2, &c_b3, &c__0, &t_ref(i__, i__ - 1), &c__1);
/* L50: */
}
if (TRUE_) {
/* Strong stability test:
F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B))) */
slacpy_("Full", &m, &m, &a_ref(*j1, *j1), lda, &work[m * m + 1], &
m);
sgemm_("N", "N", &m, &m, &m, &c_b38, li, &c__4, s, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b44, &work[1], &m, ir, &c__4, &
c_b38, &work[m * m + 1], &m);
dscale = 0.f;
dsum = 1.f;
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, &b_ref(*j1, *j1), ldb, &work[m * m + 1], &
m);
sgemm_("N", "N", &m, &m, &m, &c_b38, li, &c__4, t, &c__4, &c_b3, &
work[1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b44, &work[1], &m, ir, &c__4, &
c_b38, &work[m * m + 1], &m);
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
ss = dscale * sqrt(dsum);
strong = ss <= thresh;
if (! strong) {
goto L70;
}
}
/* If the swap is accepted ("weakly" and "strongly"), apply the
transformations and set N1-by-N2 (2,1)-block to zero. */
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
scopy_(n1, &c_b3, &c__0, &s_ref(*n2 + 1, i__), &c__1);
/* L60: */
}
/* copy back M-by-M diagonal block starting at index J1 of (A, B) */
slacpy_("F", &m, &m, s, &c__4, &a_ref(*j1, *j1), lda);
slacpy_("F", &m, &m, t, &c__4, &b_ref(*j1, *j1), ldb);
scopy_(&c__16, &c_b3, &c__0, t, &c__1);
/* Standardize existing 2-by-2 blocks. */
i__1 = m * m;
scopy_(&i__1, &c_b3, &c__0, &work[1], &c__1);
work[1] = 1.f;
t_ref(1, 1) = 1.f;
idum = *lwork - m * m - 2;
if (*n2 > 1) {
slagv2_(&a_ref(*j1, *j1), lda, &b_ref(*j1, *j1), ldb, ar, ai, be,
&work[1], &work[2], &t_ref(1, 1), &t_ref(2, 1));
work[m + 1] = -work[2];
work[m + 2] = work[1];
t_ref(*n2, *n2) = t_ref(1, 1);
t_ref(1, 2) = -t_ref(2, 1);
}
work[m * m] = 1.f;
t_ref(m, m) = 1.f;
if (*n1 > 1) {
slagv2_(&a_ref(*j1 + *n2, *j1 + *n2), lda, &b_ref(*j1 + *n2, *j1
+ *n2), ldb, taur, taul, &work[m * m + 1], &work[*n2 * m
+ *n2 + 1], &work[*n2 * m + *n2 + 2], &t_ref(*n2 + 1, *n2
+ 1), &t_ref(m, m - 1));
work[m * m] = work[*n2 * m + *n2 + 1];
work[m * m - 1] = -work[*n2 * m + *n2 + 2];
t_ref(m, m) = t_ref(*n2 + 1, *n2 + 1);
t_ref(m - 1, m) = -t_ref(m, m - 1);
}
sgemm_("T", "N", n2, n1, n2, &c_b38, &work[1], &m, &a_ref(*j1, *j1 + *
n2), lda, &c_b3, &work[m * m + 1], n2);
slacpy_("Full", n2, n1, &work[m * m + 1], n2, &a_ref(*j1, *j1 + *n2),
lda);
sgemm_("T", "N", n2, n1, n2, &c_b38, &work[1], &m, &b_ref(*j1, *j1 + *
n2), ldb, &c_b3, &work[m * m + 1], n2);
slacpy_("Full", n2, n1, &work[m * m + 1], n2, &b_ref(*j1, *j1 + *n2),
ldb);
sgemm_("N", "N", &m, &m, &m, &c_b38, li, &c__4, &work[1], &m, &c_b3, &
work[m * m + 1], &m);
slacpy_("Full", &m, &m, &work[m * m + 1], &m, li, &c__4);
sgemm_("N", "N", n2, n1, n1, &c_b38, &a_ref(*j1, *j1 + *n2), lda, &
t_ref(*n2 + 1, *n2 + 1), &c__4, &c_b3, &work[1], n2);
slacpy_("Full", n2, n1, &work[1], n2, &a_ref(*j1, *j1 + *n2), lda);
sgemm_("N", "N", n2, n1, n1, &c_b38, &b_ref(*j1, *j1 + *n2), lda, &
t_ref(*n2 + 1, *n2 + 1), &c__4, &c_b3, &work[1], n2);
slacpy_("Full", n2, n1, &work[1], n2, &b_ref(*j1, *j1 + *n2), ldb);
sgemm_("T", "N", &m, &m, &m, &c_b38, ir, &c__4, t, &c__4, &c_b3, &
work[1], &m);
slacpy_("Full", &m, &m, &work[1], &m, ir, &c__4);
/* Accumulate transformations into Q and Z if requested. */
if (*wantq) {
sgemm_("N", "N", n, &m, &m, &c_b38, &q_ref(1, *j1), ldq, li, &
c__4, &c_b3, &work[1], n);
slacpy_("Full", n, &m, &work[1], n, &q_ref(1, *j1), ldq);
}
if (*wantz) {
sgemm_("N", "N", n, &m, &m, &c_b38, &z___ref(1, *j1), ldz, ir, &
c__4, &c_b3, &work[1], n);
slacpy_("Full", n, &m, &work[1], n, &z___ref(1, *j1), ldz);
}
/* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and
(A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). */
i__ = *j1 + m;
if (i__ <= *n) {
i__1 = *n - i__ + 1;
sgemm_("T", "N", &m, &i__1, &m, &c_b38, li, &c__4, &a_ref(*j1,
i__), lda, &c_b3, &work[1], &m);
i__1 = *n - i__ + 1;
slacpy_("Full", &m, &i__1, &work[1], &m, &a_ref(*j1, i__), lda);
i__1 = *n - i__ + 1;
sgemm_("T", "N", &m, &i__1, &m, &c_b38, li, &c__4, &b_ref(*j1,
i__), lda, &c_b3, &work[1], &m);
i__1 = *n - i__ + 1;
slacpy_("Full", &m, &i__1, &work[1], &m, &b_ref(*j1, i__), lda);
}
i__ = *j1 - 1;
if (i__ > 0) {
sgemm_("N", "N", &i__, &m, &m, &c_b38, &a_ref(1, *j1), lda, ir, &
c__4, &c_b3, &work[1], &i__);
slacpy_("Full", &i__, &m, &work[1], &i__, &a_ref(1, *j1), lda);
sgemm_("N", "N", &i__, &m, &m, &c_b38, &b_ref(1, *j1), ldb, ir, &
c__4, &c_b3, &work[1], &i__);
slacpy_("Full", &i__, &m, &work[1], &i__, &b_ref(1, *j1), ldb);
}
/* Exit with INFO = 0 if swap was successfully performed. */
return 0;
}
/* Exit with INFO = 1 if swap was rejected. */
L70:
*info = 1;
return 0;
/* End of STGEX2 */
} /* stgex2_ */
/* Subroutine */ int slaexc_(logical *wantq, integer *n, real *t, integer *
ldt, real *q, integer *ldq, integer *j1, integer *n1, integer *n2,
real *work, integer *info)
{
/* -- LAPACK auxiliary routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
February 29, 1992
Purpose
=======
SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
an upper quasi-triangular matrix T by an orthogonal similarity
transformation.
T must be in Schur canonical form, that is, block upper triangular
with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
has its diagonal elemnts equal and its off-diagonal elements of
opposite sign.
Arguments
=========
WANTQ (input) LOGICAL
= .TRUE. : accumulate the transformation in the matrix Q;
= .FALSE.: do not accumulate the transformation.
N (input) INTEGER
The order of the matrix T. N >= 0.
T (input/output) REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, the updated matrix T, again in Schur canonical form.
LDT (input) INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q (input/output) REAL array, dimension (LDQ,N)
On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
On exit, if WANTQ is .TRUE., the updated matrix Q.
If WANTQ is .FALSE., Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
J1 (input) INTEGER
The index of the first row of the first block T11.
N1 (input) INTEGER
The order of the first block T11. N1 = 0, 1 or 2.
N2 (input) INTEGER
The order of the second block T22. N2 = 0, 1 or 2.
WORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
= 1: the transformed matrix T would be too far from Schur
form; the blocks are not swapped and T and Q are
unchanged.
=====================================================================
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c__4 = 4;
static logical c_false = FALSE_;
static integer c_n1 = -1;
static integer c__2 = 2;
static integer c__3 = 3;
/* System generated locals */
integer q_dim1, q_offset, t_dim1, t_offset, i__1;
real r__1, r__2, r__3, r__4, r__5, r__6;
/* Local variables */
static integer ierr;
static real temp;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
static real d__[16] /* was [4][4] */;
static integer k;
static real u[3], scale, x[4] /* was [2][2] */, dnorm;
static integer j2, j3, j4;
static real xnorm, u1[3], u2[3];
extern /* Subroutine */ int slanv2_(real *, real *, real *, real *, real *
, real *, real *, real *, real *, real *), slasy2_(logical *,
logical *, integer *, integer *, integer *, real *, integer *,
real *, integer *, real *, integer *, real *, real *, integer *,
real *, integer *);
static integer nd;
static real cs, t11, t22, t33, sn;
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int slarfg_(integer *, real *, real *, integer *,
real *), slacpy_(char *, integer *, integer *, real *, integer *,
real *, integer *), slartg_(real *, real *, real *, real *
, real *);
static real thresh;
extern /* Subroutine */ int slarfx_(char *, integer *, integer *, real *,
real *, real *, integer *, real *);
static real smlnum, wi1, wi2, wr1, wr2, eps, tau, tau1, tau2;
#define d___ref(a_1,a_2) d__[(a_2)*4 + a_1 - 5]
#define q_ref(a_1,a_2) q[(a_2)*q_dim1 + a_1]
#define t_ref(a_1,a_2) t[(a_2)*t_dim1 + a_1]
#define x_ref(a_1,a_2) x[(a_2)*2 + a_1 - 3]
t_dim1 = *ldt;
t_offset = 1 + t_dim1 * 1;
t -= t_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--work;
/* Function Body */
*info = 0;
/* Quick return if possible */
if (*n == 0 || *n1 == 0 || *n2 == 0) {
return 0;
}
if (*j1 + *n1 > *n) {
return 0;
}
j2 = *j1 + 1;
j3 = *j1 + 2;
j4 = *j1 + 3;
if (*n1 == 1 && *n2 == 1) {
/* Swap two 1-by-1 blocks. */
t11 = t_ref(*j1, *j1);
t22 = t_ref(j2, j2);
/* Determine the transformation to perform the interchange. */
r__1 = t22 - t11;
slartg_(&t_ref(*j1, j2), &r__1, &cs, &sn, &temp);
/* Apply transformation to the matrix T. */
if (j3 <= *n) {
i__1 = *n - *j1 - 1;
srot_(&i__1, &t_ref(*j1, j3), ldt, &t_ref(j2, j3), ldt, &cs, &sn);
}
i__1 = *j1 - 1;
srot_(&i__1, &t_ref(1, *j1), &c__1, &t_ref(1, j2), &c__1, &cs, &sn);
t_ref(*j1, *j1) = t22;
t_ref(j2, j2) = t11;
if (*wantq) {
/* Accumulate transformation in the matrix Q. */
srot_(n, &q_ref(1, *j1), &c__1, &q_ref(1, j2), &c__1, &cs, &sn);
}
} else {
/* Swapping involves at least one 2-by-2 block.
Copy the diagonal block of order N1+N2 to the local array D
and compute its norm. */
nd = *n1 + *n2;
slacpy_("Full", &nd, &nd, &t_ref(*j1, *j1), ldt, d__, &c__4);
dnorm = slange_("Max", &nd, &nd, d__, &c__4, &work[1]);
/* Compute machine-dependent threshold for test for accepting
swap. */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
/* Computing MAX */
r__1 = eps * 10.f * dnorm;
thresh = dmax(r__1,smlnum);
/* Solve T11*X - X*T22 = scale*T12 for X. */
slasy2_(&c_false, &c_false, &c_n1, n1, n2, d__, &c__4, &d___ref(*n1 +
1, *n1 + 1), &c__4, &d___ref(1, *n1 + 1), &c__4, &scale, x, &
c__2, &xnorm, &ierr);
/* Swap the adjacent diagonal blocks. */
k = *n1 + *n1 + *n2 - 3;
switch (k) {
case 1: goto L10;
case 2: goto L20;
case 3: goto L30;
}
L10:
/* N1 = 1, N2 = 2: generate elementary reflector H so that:
( scale, X11, X12 ) H = ( 0, 0, * ) */
u[0] = scale;
u[1] = x_ref(1, 1);
u[2] = x_ref(1, 2);
slarfg_(&c__3, &u[2], u, &c__1, &tau);
u[2] = 1.f;
t11 = t_ref(*j1, *j1);
/* Perform swap provisionally on diagonal block in D. */
slarfx_("L", &c__3, &c__3, u, &tau, d__, &c__4, &work[1]);
slarfx_("R", &c__3, &c__3, u, &tau, d__, &c__4, &work[1]);
/* Test whether to reject swap.
Computing MAX */
r__4 = (r__1 = d___ref(3, 1), dabs(r__1)), r__5 = (r__2 = d___ref(3,
2), dabs(r__2)), r__4 = max(r__4,r__5), r__5 = (r__3 =
d___ref(3, 3) - t11, dabs(r__3));
if (dmax(r__4,r__5) > thresh) {
goto L50;
}
/* Accept swap: apply transformation to the entire matrix T. */
i__1 = *n - *j1 + 1;
slarfx_("L", &c__3, &i__1, u, &tau, &t_ref(*j1, *j1), ldt, &work[1]);
slarfx_("R", &j2, &c__3, u, &tau, &t_ref(1, *j1), ldt, &work[1]);
t_ref(j3, *j1) = 0.f;
t_ref(j3, j2) = 0.f;
t_ref(j3, j3) = t11;
if (*wantq) {
/* Accumulate transformation in the matrix Q. */
slarfx_("R", n, &c__3, u, &tau, &q_ref(1, *j1), ldq, &work[1]);
}
goto L40;
L20:
/* N1 = 2, N2 = 1: generate elementary reflector H so that:
H ( -X11 ) = ( * )
( -X21 ) = ( 0 )
( scale ) = ( 0 ) */
u[0] = -x_ref(1, 1);
u[1] = -x_ref(2, 1);
u[2] = scale;
slarfg_(&c__3, u, &u[1], &c__1, &tau);
u[0] = 1.f;
t33 = t_ref(j3, j3);
/* Perform swap provisionally on diagonal block in D. */
slarfx_("L", &c__3, &c__3, u, &tau, d__, &c__4, &work[1]);
slarfx_("R", &c__3, &c__3, u, &tau, d__, &c__4, &work[1]);
/* Test whether to reject swap.
Computing MAX */
r__4 = (r__1 = d___ref(2, 1), dabs(r__1)), r__5 = (r__2 = d___ref(3,
1), dabs(r__2)), r__4 = max(r__4,r__5), r__5 = (r__3 =
d___ref(1, 1) - t33, dabs(r__3));
if (dmax(r__4,r__5) > thresh) {
goto L50;
}
/* Accept swap: apply transformation to the entire matrix T. */
slarfx_("R", &j3, &c__3, u, &tau, &t_ref(1, *j1), ldt, &work[1]);
i__1 = *n - *j1;
slarfx_("L", &c__3, &i__1, u, &tau, &t_ref(*j1, j2), ldt, &work[1]);
t_ref(*j1, *j1) = t33;
t_ref(j2, *j1) = 0.f;
t_ref(j3, *j1) = 0.f;
if (*wantq) {
/* Accumulate transformation in the matrix Q. */
slarfx_("R", n, &c__3, u, &tau, &q_ref(1, *j1), ldq, &work[1]);
}
goto L40;
L30:
/* N1 = 2, N2 = 2: generate elementary reflectors H(1) and H(2) so
that:
H(2) H(1) ( -X11 -X12 ) = ( * * )
( -X21 -X22 ) ( 0 * )
( scale 0 ) ( 0 0 )
( 0 scale ) ( 0 0 ) */
u1[0] = -x_ref(1, 1);
u1[1] = -x_ref(2, 1);
u1[2] = scale;
slarfg_(&c__3, u1, &u1[1], &c__1, &tau1);
u1[0] = 1.f;
temp = -tau1 * (x_ref(1, 2) + u1[1] * x_ref(2, 2));
u2[0] = -temp * u1[1] - x_ref(2, 2);
u2[1] = -temp * u1[2];
u2[2] = scale;
slarfg_(&c__3, u2, &u2[1], &c__1, &tau2);
u2[0] = 1.f;
/* Perform swap provisionally on diagonal block in D. */
slarfx_("L", &c__3, &c__4, u1, &tau1, d__, &c__4, &work[1])
;
slarfx_("R", &c__4, &c__3, u1, &tau1, d__, &c__4, &work[1])
;
slarfx_("L", &c__3, &c__4, u2, &tau2, &d___ref(2, 1), &c__4, &work[1]);
slarfx_("R", &c__4, &c__3, u2, &tau2, &d___ref(1, 2), &c__4, &work[1]);
/* Test whether to reject swap.
Computing MAX */
r__5 = (r__1 = d___ref(3, 1), dabs(r__1)), r__6 = (r__2 = d___ref(3,
2), dabs(r__2)), r__5 = max(r__5,r__6), r__6 = (r__3 =
d___ref(4, 1), dabs(r__3)), r__5 = max(r__5,r__6), r__6 = (
r__4 = d___ref(4, 2), dabs(r__4));
if (dmax(r__5,r__6) > thresh) {
goto L50;
}
/* Accept swap: apply transformation to the entire matrix T. */
i__1 = *n - *j1 + 1;
slarfx_("L", &c__3, &i__1, u1, &tau1, &t_ref(*j1, *j1), ldt, &work[1]);
slarfx_("R", &j4, &c__3, u1, &tau1, &t_ref(1, *j1), ldt, &work[1]);
i__1 = *n - *j1 + 1;
slarfx_("L", &c__3, &i__1, u2, &tau2, &t_ref(j2, *j1), ldt, &work[1]);
slarfx_("R", &j4, &c__3, u2, &tau2, &t_ref(1, j2), ldt, &work[1]);
t_ref(j3, *j1) = 0.f;
t_ref(j3, j2) = 0.f;
t_ref(j4, *j1) = 0.f;
t_ref(j4, j2) = 0.f;
if (*wantq) {
/* Accumulate transformation in the matrix Q. */
slarfx_("R", n, &c__3, u1, &tau1, &q_ref(1, *j1), ldq, &work[1]);
slarfx_("R", n, &c__3, u2, &tau2, &q_ref(1, j2), ldq, &work[1]);
}
L40:
if (*n2 == 2) {
/* Standardize new 2-by-2 block T11 */
slanv2_(&t_ref(*j1, *j1), &t_ref(*j1, j2), &t_ref(j2, *j1), &
t_ref(j2, j2), &wr1, &wi1, &wr2, &wi2, &cs, &sn);
i__1 = *n - *j1 - 1;
srot_(&i__1, &t_ref(*j1, *j1 + 2), ldt, &t_ref(j2, *j1 + 2), ldt,
&cs, &sn);
i__1 = *j1 - 1;
srot_(&i__1, &t_ref(1, *j1), &c__1, &t_ref(1, j2), &c__1, &cs, &
sn);
if (*wantq) {
srot_(n, &q_ref(1, *j1), &c__1, &q_ref(1, j2), &c__1, &cs, &
sn);
}
}
if (*n1 == 2) {
/* Standardize new 2-by-2 block T22 */
j3 = *j1 + *n2;
j4 = j3 + 1;
slanv2_(&t_ref(j3, j3), &t_ref(j3, j4), &t_ref(j4, j3), &t_ref(j4,
j4), &wr1, &wi1, &wr2, &wi2, &cs, &sn);
if (j3 + 2 <= *n) {
i__1 = *n - j3 - 1;
srot_(&i__1, &t_ref(j3, j3 + 2), ldt, &t_ref(j4, j3 + 2), ldt,
&cs, &sn);
}
i__1 = j3 - 1;
srot_(&i__1, &t_ref(1, j3), &c__1, &t_ref(1, j4), &c__1, &cs, &sn)
;
if (*wantq) {
srot_(n, &q_ref(1, j3), &c__1, &q_ref(1, j4), &c__1, &cs, &sn)
;
}
}
}
return 0;
/* Exit with INFO = 1 if swap was rejected. */
L50:
*info = 1;
return 0;
/* End of SLAEXC */
} /* slaexc_ */
/* Subroutine */ int ssbtrd_(char *vect, char *uplo, integer *n, integer *kd,
real *ab, integer *ldab, real *d__, real *e, real *q, integer *ldq,
real *work, integer *info)
{
/* System generated locals */
integer ab_dim1, ab_offset, q_dim1, q_offset, i__1, i__2, i__3, i__4,
i__5;
/* Local variables */
integer i__, j, k, l, i2, j1, j2, nq, nr, kd1, ibl, iqb, kdn, jin, nrt,
kdm1, inca, jend, lend, jinc, incx, last;
real temp;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
integer j1end, j1inc, iqend;
extern logical lsame_(char *, char *);
logical initq, wantq, upper;
extern /* Subroutine */ int slar2v_(integer *, real *, real *, real *,
integer *, real *, real *, integer *);
integer iqaend;
extern /* Subroutine */ int xerbla_(char *, integer *), slaset_(
char *, integer *, integer *, real *, real *, real *, integer *), slartg_(real *, real *, real *, real *, real *), slargv_(
integer *, real *, integer *, real *, integer *, real *, integer *
), slartv_(integer *, real *, integer *, real *, integer *, real *
, real *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SSBTRD reduces a real symmetric band matrix A to symmetric */
/* tridiagonal form T by an orthogonal similarity transformation: */
/* Q**T * A * Q = T. */
/* Arguments */
/* ========= */
/* VECT (input) CHARACTER*1 */
/* = 'N': do not form Q; */
/* = 'V': form Q; */
/* = 'U': update a matrix X, by forming X*Q. */
/* UPLO (input) CHARACTER*1 */
/* = 'U': Upper triangle of A is stored; */
/* = 'L': Lower triangle of A is stored. */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* KD (input) INTEGER */
/* The number of superdiagonals of the matrix A if UPLO = 'U', */
/* or the number of subdiagonals if UPLO = 'L'. KD >= 0. */
/* AB (input/output) REAL array, dimension (LDAB,N) */
/* On entry, the upper or lower triangle of the symmetric band */
/* matrix A, stored in the first KD+1 rows of the array. The */
/* j-th column of A is stored in the j-th column of the array AB */
/* as follows: */
/* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). */
/* On exit, the diagonal elements of AB are overwritten by the */
/* diagonal elements of the tridiagonal matrix T; if KD > 0, the */
/* elements on the first superdiagonal (if UPLO = 'U') or the */
/* first subdiagonal (if UPLO = 'L') are overwritten by the */
/* off-diagonal elements of T; the rest of AB is overwritten by */
/* values generated during the reduction. */
/* LDAB (input) INTEGER */
/* The leading dimension of the array AB. LDAB >= KD+1. */
/* D (output) REAL array, dimension (N) */
/* The diagonal elements of the tridiagonal matrix T. */
/* E (output) REAL array, dimension (N-1) */
/* The off-diagonal elements of the tridiagonal matrix T: */
/* E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. */
/* Q (input/output) REAL array, dimension (LDQ,N) */
/* On entry, if VECT = 'U', then Q must contain an N-by-N */
/* matrix X; if VECT = 'N' or 'V', then Q need not be set. */
/* On exit: */
/* if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; */
/* if VECT = 'U', Q contains the product X*Q; */
/* if VECT = 'N', the array Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. */
/* LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. */
/* WORK (workspace) REAL array, dimension (N) */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* Modified by Linda Kaufman, Bell Labs. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters */
/* Parameter adjustments */
ab_dim1 = *ldab;
ab_offset = 1 + ab_dim1;
ab -= ab_offset;
--d__;
--e;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--work;
/* Function Body */
initq = lsame_(vect, "V");
wantq = initq || lsame_(vect, "U");
upper = lsame_(uplo, "U");
kd1 = *kd + 1;
kdm1 = *kd - 1;
incx = *ldab - 1;
iqend = 1;
*info = 0;
if (! wantq && ! lsame_(vect, "N")) {
*info = -1;
} else if (! upper && ! lsame_(uplo, "L")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*kd < 0) {
*info = -4;
} else if (*ldab < kd1) {
*info = -6;
} else if (*ldq < max(1,*n) && wantq) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SSBTRD", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Initialize Q to the unit matrix, if needed */
if (initq) {
slaset_("Full", n, n, &c_b9, &c_b10, &q[q_offset], ldq);
}
/* Wherever possible, plane rotations are generated and applied in */
/* vector operations of length NR over the index set J1:J2:KD1. */
/* The cosines and sines of the plane rotations are stored in the */
/* arrays D and WORK. */
inca = kd1 * *ldab;
/* Computing MIN */
i__1 = *n - 1;
kdn = min(i__1,*kd);
if (upper) {
if (*kd > 1) {
/* Reduce to tridiagonal form, working with upper triangle */
nr = 0;
j1 = kdn + 2;
j2 = 1;
i__1 = *n - 2;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th row of matrix to tridiagonal form */
for (k = kdn + 1; k >= 2; --k) {
j1 += kdn;
j2 += kdn;
if (nr > 0) {
/* generate plane rotations to annihilate nonzero */
/* elements which have been created outside the band */
slargv_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &inca, &
work[j1], &kd1, &d__[j1], &kd1);
/* apply rotations from the right */
/* Dependent on the the number of diagonals either */
/* SLARTV or SROT is used */
if (nr >= (*kd << 1) - 1) {
i__2 = *kd - 1;
for (l = 1; l <= i__2; ++l) {
slartv_(&nr, &ab[l + 1 + (j1 - 1) * ab_dim1],
&inca, &ab[l + j1 * ab_dim1], &inca, &
d__[j1], &work[j1], &kd1);
/* L10: */
}
} else {
jend = j1 + (nr - 1) * kd1;
i__2 = jend;
i__3 = kd1;
for (jinc = j1; i__3 < 0 ? jinc >= i__2 : jinc <=
i__2; jinc += i__3) {
srot_(&kdm1, &ab[(jinc - 1) * ab_dim1 + 2], &
c__1, &ab[jinc * ab_dim1 + 1], &c__1,
&d__[jinc], &work[jinc]);
/* L20: */
}
}
}
if (k > 2) {
if (k <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i,i+k-1) */
/* within the band */
slartg_(&ab[*kd - k + 3 + (i__ + k - 2) * ab_dim1]
, &ab[*kd - k + 2 + (i__ + k - 1) *
ab_dim1], &d__[i__ + k - 1], &work[i__ +
k - 1], &temp);
ab[*kd - k + 3 + (i__ + k - 2) * ab_dim1] = temp;
/* apply rotation from the right */
i__3 = k - 3;
srot_(&i__3, &ab[*kd - k + 4 + (i__ + k - 2) *
ab_dim1], &c__1, &ab[*kd - k + 3 + (i__ +
k - 1) * ab_dim1], &c__1, &d__[i__ + k -
1], &work[i__ + k - 1]);
}
++nr;
j1 = j1 - kdn - 1;
}
/* apply plane rotations from both sides to diagonal */
/* blocks */
if (nr > 0) {
slar2v_(&nr, &ab[kd1 + (j1 - 1) * ab_dim1], &ab[kd1 +
j1 * ab_dim1], &ab[*kd + j1 * ab_dim1], &inca,
&d__[j1], &work[j1], &kd1);
}
/* apply plane rotations from the left */
if (nr > 0) {
if ((*kd << 1) - 1 < nr) {
/* Dependent on the the number of diagonals either */
/* SLARTV or SROT is used */
i__3 = *kd - 1;
for (l = 1; l <= i__3; ++l) {
if (j2 + l > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab[*kd - l + (j1 + l) *
ab_dim1], &inca, &ab[*kd - l + 1
+ (j1 + l) * ab_dim1], &inca, &
d__[j1], &work[j1], &kd1);
}
/* L30: */
}
} else {
j1end = j1 + kd1 * (nr - 2);
if (j1end >= j1) {
i__3 = j1end;
i__2 = kd1;
for (jin = j1; i__2 < 0 ? jin >= i__3 : jin <=
i__3; jin += i__2) {
i__4 = *kd - 1;
srot_(&i__4, &ab[*kd - 1 + (jin + 1) *
ab_dim1], &incx, &ab[*kd + (jin +
1) * ab_dim1], &incx, &d__[jin], &
work[jin]);
/* L40: */
}
}
/* Computing MIN */
i__2 = kdm1, i__3 = *n - j2;
lend = min(i__2,i__3);
last = j1end + kd1;
if (lend > 0) {
srot_(&lend, &ab[*kd - 1 + (last + 1) *
ab_dim1], &incx, &ab[*kd + (last + 1)
* ab_dim1], &incx, &d__[last], &work[
last]);
}
}
}
if (wantq) {
/* accumulate product of plane rotations in Q */
if (initq) {
/* take advantage of the fact that Q was */
/* initially the Identity matrix */
iqend = max(iqend,j2);
/* Computing MAX */
i__2 = 0, i__3 = k - 3;
i2 = max(i__2,i__3);
iqaend = i__ * *kd + 1;
if (k == 2) {
iqaend += *kd;
}
iqaend = min(iqaend,iqend);
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j
+= i__3) {
ibl = i__ - i2 / kdm1;
++i2;
/* Computing MAX */
i__4 = 1, i__5 = j - ibl;
iqb = max(i__4,i__5);
nq = iqaend + 1 - iqb;
/* Computing MIN */
i__4 = iqaend + *kd;
iqaend = min(i__4,iqend);
srot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1,
&q[iqb + j * q_dim1], &c__1, &d__[j],
&work[j]);
/* L50: */
}
} else {
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j
+= i__2) {
srot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[
j * q_dim1 + 1], &c__1, &d__[j], &
work[j]);
/* L60: */
}
}
}
if (j2 + kdn > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 = j2 - kdn - 1;
}
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j += i__3)
{
/* create nonzero element a(j-1,j+kd) outside the band */
/* and store it in WORK */
work[j + *kd] = work[j] * ab[(j + *kd) * ab_dim1 + 1];
ab[(j + *kd) * ab_dim1 + 1] = d__[j] * ab[(j + *kd) *
ab_dim1 + 1];
/* L70: */
}
/* L80: */
}
/* L90: */
}
}
if (*kd > 0) {
/* copy off-diagonal elements to E */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab[*kd + (i__ + 1) * ab_dim1];
/* L100: */
}
} else {
/* set E to zero if original matrix was diagonal */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.f;
/* L110: */
}
}
/* copy diagonal elements to D */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab[kd1 + i__ * ab_dim1];
/* L120: */
}
} else {
if (*kd > 1) {
/* Reduce to tridiagonal form, working with lower triangle */
nr = 0;
j1 = kdn + 2;
j2 = 1;
i__1 = *n - 2;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Reduce i-th column of matrix to tridiagonal form */
for (k = kdn + 1; k >= 2; --k) {
j1 += kdn;
j2 += kdn;
if (nr > 0) {
/* generate plane rotations to annihilate nonzero */
/* elements which have been created outside the band */
slargv_(&nr, &ab[kd1 + (j1 - kd1) * ab_dim1], &inca, &
work[j1], &kd1, &d__[j1], &kd1);
/* apply plane rotations from one side */
/* Dependent on the the number of diagonals either */
/* SLARTV or SROT is used */
if (nr > (*kd << 1) - 1) {
i__3 = *kd - 1;
for (l = 1; l <= i__3; ++l) {
slartv_(&nr, &ab[kd1 - l + (j1 - kd1 + l) *
ab_dim1], &inca, &ab[kd1 - l + 1 + (
j1 - kd1 + l) * ab_dim1], &inca, &d__[
j1], &work[j1], &kd1);
/* L130: */
}
} else {
jend = j1 + kd1 * (nr - 1);
i__3 = jend;
i__2 = kd1;
for (jinc = j1; i__2 < 0 ? jinc >= i__3 : jinc <=
i__3; jinc += i__2) {
srot_(&kdm1, &ab[*kd + (jinc - *kd) * ab_dim1]
, &incx, &ab[kd1 + (jinc - *kd) *
ab_dim1], &incx, &d__[jinc], &work[
jinc]);
/* L140: */
}
}
}
if (k > 2) {
if (k <= *n - i__ + 1) {
/* generate plane rotation to annihilate a(i+k-1,i) */
/* within the band */
slartg_(&ab[k - 1 + i__ * ab_dim1], &ab[k + i__ *
ab_dim1], &d__[i__ + k - 1], &work[i__ +
k - 1], &temp);
ab[k - 1 + i__ * ab_dim1] = temp;
/* apply rotation from the left */
i__2 = k - 3;
i__3 = *ldab - 1;
i__4 = *ldab - 1;
srot_(&i__2, &ab[k - 2 + (i__ + 1) * ab_dim1], &
i__3, &ab[k - 1 + (i__ + 1) * ab_dim1], &
i__4, &d__[i__ + k - 1], &work[i__ + k -
1]);
}
++nr;
j1 = j1 - kdn - 1;
}
/* apply plane rotations from both sides to diagonal */
/* blocks */
if (nr > 0) {
slar2v_(&nr, &ab[(j1 - 1) * ab_dim1 + 1], &ab[j1 *
ab_dim1 + 1], &ab[(j1 - 1) * ab_dim1 + 2], &
inca, &d__[j1], &work[j1], &kd1);
}
/* apply plane rotations from the right */
/* Dependent on the the number of diagonals either */
/* SLARTV or SROT is used */
if (nr > 0) {
if (nr > (*kd << 1) - 1) {
i__2 = *kd - 1;
for (l = 1; l <= i__2; ++l) {
if (j2 + l > *n) {
nrt = nr - 1;
} else {
nrt = nr;
}
if (nrt > 0) {
slartv_(&nrt, &ab[l + 2 + (j1 - 1) *
ab_dim1], &inca, &ab[l + 1 + j1 *
ab_dim1], &inca, &d__[j1], &work[
j1], &kd1);
}
/* L150: */
}
} else {
j1end = j1 + kd1 * (nr - 2);
if (j1end >= j1) {
i__2 = j1end;
i__3 = kd1;
for (j1inc = j1; i__3 < 0 ? j1inc >= i__2 :
j1inc <= i__2; j1inc += i__3) {
srot_(&kdm1, &ab[(j1inc - 1) * ab_dim1 +
3], &c__1, &ab[j1inc * ab_dim1 +
2], &c__1, &d__[j1inc], &work[
j1inc]);
/* L160: */
}
}
/* Computing MIN */
i__3 = kdm1, i__2 = *n - j2;
lend = min(i__3,i__2);
last = j1end + kd1;
if (lend > 0) {
srot_(&lend, &ab[(last - 1) * ab_dim1 + 3], &
c__1, &ab[last * ab_dim1 + 2], &c__1,
&d__[last], &work[last]);
}
}
}
if (wantq) {
/* accumulate product of plane rotations in Q */
if (initq) {
/* take advantage of the fact that Q was */
/* initially the Identity matrix */
iqend = max(iqend,j2);
/* Computing MAX */
i__3 = 0, i__2 = k - 3;
i2 = max(i__3,i__2);
iqaend = i__ * *kd + 1;
if (k == 2) {
iqaend += *kd;
}
iqaend = min(iqaend,iqend);
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j
+= i__2) {
ibl = i__ - i2 / kdm1;
++i2;
/* Computing MAX */
i__4 = 1, i__5 = j - ibl;
iqb = max(i__4,i__5);
nq = iqaend + 1 - iqb;
/* Computing MIN */
i__4 = iqaend + *kd;
iqaend = min(i__4,iqend);
srot_(&nq, &q[iqb + (j - 1) * q_dim1], &c__1,
&q[iqb + j * q_dim1], &c__1, &d__[j],
&work[j]);
/* L170: */
}
} else {
i__2 = j2;
i__3 = kd1;
for (j = j1; i__3 < 0 ? j >= i__2 : j <= i__2; j
+= i__3) {
srot_(n, &q[(j - 1) * q_dim1 + 1], &c__1, &q[
j * q_dim1 + 1], &c__1, &d__[j], &
work[j]);
/* L180: */
}
}
}
if (j2 + kdn > *n) {
/* adjust J2 to keep within the bounds of the matrix */
--nr;
j2 = j2 - kdn - 1;
}
i__3 = j2;
i__2 = kd1;
for (j = j1; i__2 < 0 ? j >= i__3 : j <= i__3; j += i__2)
{
/* create nonzero element a(j+kd,j-1) outside the */
/* band and store it in WORK */
work[j + *kd] = work[j] * ab[kd1 + j * ab_dim1];
ab[kd1 + j * ab_dim1] = d__[j] * ab[kd1 + j * ab_dim1]
;
/* L190: */
}
/* L200: */
}
/* L210: */
}
}
if (*kd > 0) {
/* copy off-diagonal elements to E */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = ab[i__ * ab_dim1 + 2];
/* L220: */
}
} else {
/* set E to zero if original matrix was diagonal */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
e[i__] = 0.f;
/* L230: */
}
}
/* copy diagonal elements to D */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
d__[i__] = ab[i__ * ab_dim1 + 1];
/* L240: */
}
}
return 0;
/* End of SSBTRD */
} /* ssbtrd_ */
/* Subroutine */ int sbdsqr_(char *uplo, integer *n, integer *ncvt, integer *
nru, integer *ncc, real *d__, real *e, real *vt, integer *ldvt, real *
u, integer *ldu, real *c__, integer *ldc, real *work, integer *info)
{
/* System generated locals */
integer c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1,
i__2;
real r__1, r__2, r__3, r__4;
doublereal d__1;
/* Builtin functions */
double pow_dd(doublereal *, doublereal *), sqrt(doublereal), r_sign(real *
, real *);
/* Local variables */
static real abse;
static integer idir;
static real abss;
static integer oldm;
static real cosl;
static integer isub, iter;
static real unfl, sinl, cosr, smin, smax, sinr;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *), slas2_(real *, real *, real *, real *,
real *);
static real f, g, h__;
static integer i__, j, m;
static real r__;
extern logical lsame_(char *, char *);
static real oldcs;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
static integer oldll;
static real shift, sigmn, oldsn;
static integer maxit;
static real sminl;
extern /* Subroutine */ int slasr_(char *, char *, char *, integer *,
integer *, real *, real *, real *, integer *);
static real sigmx;
static logical lower;
extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *,
integer *), slasq1_(integer *, real *, real *, real *, integer *),
slasv2_(real *, real *, real *, real *, real *, real *, real *,
real *, real *);
static real cs;
static integer ll;
static real sn, mu;
extern doublereal slamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
static real sminoa;
extern /* Subroutine */ int slartg_(real *, real *, real *, real *, real *
);
static real thresh;
static logical rotate;
static real sminlo;
static integer nm1;
static real tolmul;
static integer nm12, nm13, lll;
static real eps, sll, tol;
#define c___ref(a_1,a_2) c__[(a_2)*c_dim1 + a_1]
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
SBDSQR computes the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B: B = Q * S * P' (P'
denotes the transpose of P), where S is a diagonal matrix with
non-negative diagonal elements (the singular values of B), and Q
and P are orthogonal matrices.
The routine computes S, and optionally computes U * Q, P' * VT,
or Q' * C, for given real input matrices U, VT, and C.
See "Computing Small Singular Values of Bidiagonal Matrices With
Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan,
LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
no. 5, pp. 873-912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by
B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
Department, University of California at Berkeley, July 1992
for a detailed description of the algorithm.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
N (input) INTEGER
The order of the matrix B. N >= 0.
NCVT (input) INTEGER
The number of columns of the matrix VT. NCVT >= 0.
NRU (input) INTEGER
The number of rows of the matrix U. NRU >= 0.
NCC (input) INTEGER
The number of columns of the matrix C. NCC >= 0.
D (input/output) REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B in decreasing
order.
E (input/output) REAL array, dimension (N)
On entry, the elements of E contain the
offdiagonal elements of the bidiagonal matrix whose SVD
is desired. On normal exit (INFO = 0), E is destroyed.
If the algorithm does not converge (INFO > 0), D and E
will contain the diagonal and superdiagonal elements of a
bidiagonal matrix orthogonally equivalent to the one given
as input. E(N) is used for workspace.
VT (input/output) REAL array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT.
On exit, VT is overwritten by P' * VT.
VT is not referenced if NCVT = 0.
LDVT (input) INTEGER
The leading dimension of the array VT.
LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.
U (input/output) REAL array, dimension (LDU, N)
On entry, an NRU-by-N matrix U.
On exit, U is overwritten by U * Q.
U is not referenced if NRU = 0.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,NRU).
C (input/output) REAL array, dimension (LDC, NCC)
On entry, an N-by-NCC matrix C.
On exit, C is overwritten by Q' * C.
C is not referenced if NCC = 0.
LDC (input) INTEGER
The leading dimension of the array C.
LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.
WORK (workspace) REAL array, dimension (4*N)
INFO (output) INTEGER
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0: the algorithm did not converge; D and E contain the
elements of a bidiagonal matrix which is orthogonally
similar to the input matrix B; if INFO = i, i
elements of E have not converged to zero.
Internal Parameters
===================
TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop.
If it is positive, TOLMUL*EPS is the desired relative
precision in the computed singular values.
If it is negative, abs(TOLMUL*EPS*sigma_max) is the
desired absolute accuracy in the computed singular
values (corresponds to relative accuracy
abs(TOLMUL*EPS) in the largest singular value.
abs(TOLMUL) should be between 1 and 1/EPS, and preferably
between 10 (for fast convergence) and .1/EPS
(for there to be some accuracy in the results).
Default is to lose at either one eighth or 2 of the
available decimal digits in each computed singular value
(whichever is smaller).
MAXITR INTEGER, default = 6
MAXITR controls the maximum number of passes of the
algorithm through its inner loop. The algorithms stops
(and so fails to converge) if the number of passes
through the inner loop exceeds MAXITR*N**2.
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1 * 1;
c__ -= c_offset;
--work;
/* Function Body */
*info = 0;
lower = lsame_(uplo, "L");
if (! lsame_(uplo, "U") && ! lower) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*ncvt < 0) {
*info = -3;
} else if (*nru < 0) {
*info = -4;
} else if (*ncc < 0) {
*info = -5;
} else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < max(1,*n)) {
*info = -9;
} else if (*ldu < max(1,*nru)) {
*info = -11;
} else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < max(1,*n)) {
*info = -13;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSQR", &i__1);
return 0;
}
if (*n == 0) {
return 0;
}
if (*n == 1) {
goto L160;
}
/* ROTATE is true if any singular vectors desired, false otherwise */
rotate = *ncvt > 0 || *nru > 0 || *ncc > 0;
/* If no singular vectors desired, use qd algorithm */
if (! rotate) {
slasq1_(n, &d__[1], &e[1], &work[1], info);
return 0;
}
nm1 = *n - 1;
nm12 = nm1 + nm1;
nm13 = nm12 + nm1;
idir = 0;
/* Get machine constants */
eps = slamch_("Epsilon");
unfl = slamch_("Safe minimum");
/* If matrix lower bidiagonal, rotate to be upper bidiagonal
by applying Givens rotations on the left */
if (lower) {
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
work[i__] = cs;
work[nm1 + i__] = sn;
/* L10: */
}
/* Update singular vectors if desired */
if (*nru > 0) {
slasr_("R", "V", "F", nru, n, &work[1], &work[*n], &u[u_offset],
ldu);
}
if (*ncc > 0) {
slasr_("L", "V", "F", n, ncc, &work[1], &work[*n], &c__[c_offset],
ldc);
}
}
/* Compute singular values to relative accuracy TOL
(By setting TOL to be negative, algorithm will compute
singular values to absolute accuracy ABS(TOL)*norm(input matrix))
Computing MAX
Computing MIN */
d__1 = (doublereal) eps;
r__3 = 100.f, r__4 = pow_dd(&d__1, &c_b15);
r__1 = 10.f, r__2 = dmin(r__3,r__4);
tolmul = dmax(r__1,r__2);
tol = tolmul * eps;
/* Compute approximate maximum, minimum singular values */
smax = 0.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__2 = smax, r__3 = (r__1 = d__[i__], dabs(r__1));
smax = dmax(r__2,r__3);
/* L20: */
}
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__2 = smax, r__3 = (r__1 = e[i__], dabs(r__1));
smax = dmax(r__2,r__3);
/* L30: */
}
sminl = 0.f;
if (tol >= 0.f) {
/* Relative accuracy desired */
sminoa = dabs(d__[1]);
if (sminoa == 0.f) {
goto L50;
}
mu = sminoa;
i__1 = *n;
for (i__ = 2; i__ <= i__1; ++i__) {
mu = (r__2 = d__[i__], dabs(r__2)) * (mu / (mu + (r__1 = e[i__ -
1], dabs(r__1))));
sminoa = dmin(sminoa,mu);
if (sminoa == 0.f) {
goto L50;
}
/* L40: */
}
L50:
sminoa /= sqrt((real) (*n));
/* Computing MAX */
r__1 = tol * sminoa, r__2 = *n * 6 * *n * unfl;
thresh = dmax(r__1,r__2);
} else {
/* Absolute accuracy desired
Computing MAX */
r__1 = dabs(tol) * smax, r__2 = *n * 6 * *n * unfl;
thresh = dmax(r__1,r__2);
}
/* Prepare for main iteration loop for the singular values
(MAXIT is the maximum number of passes through the inner
loop permitted before nonconvergence signalled.) */
maxit = *n * 6 * *n;
iter = 0;
oldll = -1;
oldm = -1;
/* M points to last element of unconverged part of matrix */
m = *n;
/* Begin main iteration loop */
L60:
/* Check for convergence or exceeding iteration count */
if (m <= 1) {
goto L160;
}
if (iter > maxit) {
goto L200;
}
/* Find diagonal block of matrix to work on */
if (tol < 0.f && (r__1 = d__[m], dabs(r__1)) <= thresh) {
d__[m] = 0.f;
}
smax = (r__1 = d__[m], dabs(r__1));
smin = smax;
i__1 = m - 1;
for (lll = 1; lll <= i__1; ++lll) {
ll = m - lll;
abss = (r__1 = d__[ll], dabs(r__1));
abse = (r__1 = e[ll], dabs(r__1));
if (tol < 0.f && abss <= thresh) {
d__[ll] = 0.f;
}
if (abse <= thresh) {
goto L80;
}
smin = dmin(smin,abss);
/* Computing MAX */
r__1 = max(smax,abss);
smax = dmax(r__1,abse);
/* L70: */
}
ll = 0;
goto L90;
L80:
e[ll] = 0.f;
/* Matrix splits since E(LL) = 0 */
if (ll == m - 1) {
/* Convergence of bottom singular value, return to top of loop */
--m;
goto L60;
}
L90:
++ll;
/* E(LL) through E(M-1) are nonzero, E(LL-1) is zero */
if (ll == m - 1) {
/* 2 by 2 block, handle separately */
slasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr,
&sinl, &cosl);
d__[m - 1] = sigmx;
e[m - 1] = 0.f;
d__[m] = sigmn;
/* Compute singular vectors, if desired */
if (*ncvt > 0) {
srot_(ncvt, &vt_ref(m - 1, 1), ldvt, &vt_ref(m, 1), ldvt, &cosr, &
sinr);
}
if (*nru > 0) {
srot_(nru, &u_ref(1, m - 1), &c__1, &u_ref(1, m), &c__1, &cosl, &
sinl);
}
if (*ncc > 0) {
srot_(ncc, &c___ref(m - 1, 1), ldc, &c___ref(m, 1), ldc, &cosl, &
sinl);
}
m += -2;
goto L60;
}
/* If working on new submatrix, choose shift direction
(from larger end diagonal element towards smaller) */
if (ll > oldm || m < oldll) {
if ((r__1 = d__[ll], dabs(r__1)) >= (r__2 = d__[m], dabs(r__2))) {
/* Chase bulge from top (big end) to bottom (small end) */
idir = 1;
} else {
/* Chase bulge from bottom (big end) to top (small end) */
idir = 2;
}
}
/* Apply convergence tests */
if (idir == 1) {
/* Run convergence test in forward direction
First apply standard test to bottom of matrix */
if ((r__2 = e[m - 1], dabs(r__2)) <= dabs(tol) * (r__1 = d__[m], dabs(
r__1)) || tol < 0.f && (r__3 = e[m - 1], dabs(r__3)) <=
thresh) {
e[m - 1] = 0.f;
goto L60;
}
if (tol >= 0.f) {
/* If relative accuracy desired,
apply convergence criterion forward */
mu = (r__1 = d__[ll], dabs(r__1));
sminl = mu;
i__1 = m - 1;
for (lll = ll; lll <= i__1; ++lll) {
if ((r__1 = e[lll], dabs(r__1)) <= tol * mu) {
e[lll] = 0.f;
goto L60;
}
sminlo = sminl;
mu = (r__2 = d__[lll + 1], dabs(r__2)) * (mu / (mu + (r__1 =
e[lll], dabs(r__1))));
sminl = dmin(sminl,mu);
/* L100: */
}
}
} else {
/* Run convergence test in backward direction
First apply standard test to top of matrix */
if ((r__2 = e[ll], dabs(r__2)) <= dabs(tol) * (r__1 = d__[ll], dabs(
r__1)) || tol < 0.f && (r__3 = e[ll], dabs(r__3)) <= thresh) {
e[ll] = 0.f;
goto L60;
}
if (tol >= 0.f) {
/* If relative accuracy desired,
apply convergence criterion backward */
mu = (r__1 = d__[m], dabs(r__1));
sminl = mu;
i__1 = ll;
for (lll = m - 1; lll >= i__1; --lll) {
if ((r__1 = e[lll], dabs(r__1)) <= tol * mu) {
e[lll] = 0.f;
goto L60;
}
sminlo = sminl;
mu = (r__2 = d__[lll], dabs(r__2)) * (mu / (mu + (r__1 = e[
lll], dabs(r__1))));
sminl = dmin(sminl,mu);
/* L110: */
}
}
}
oldll = ll;
oldm = m;
/* Compute shift. First, test if shifting would ruin relative
accuracy, and if so set the shift to zero.
Computing MAX */
r__1 = eps, r__2 = tol * .01f;
if (tol >= 0.f && *n * tol * (sminl / smax) <= dmax(r__1,r__2)) {
/* Use a zero shift to avoid loss of relative accuracy */
shift = 0.f;
} else {
/* Compute the shift from 2-by-2 block at end of matrix */
if (idir == 1) {
sll = (r__1 = d__[ll], dabs(r__1));
slas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__);
} else {
sll = (r__1 = d__[m], dabs(r__1));
slas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__);
}
/* Test if shift negligible, and if so set to zero */
if (sll > 0.f) {
/* Computing 2nd power */
r__1 = shift / sll;
if (r__1 * r__1 < eps) {
shift = 0.f;
}
}
}
/* Increment iteration count */
iter = iter + m - ll;
/* If SHIFT = 0, do simplified QR iteration */
if (shift == 0.f) {
if (idir == 1) {
/* Chase bulge from top to bottom
Save cosines and sines for later singular vector updates */
cs = 1.f;
oldcs = 1.f;
i__1 = m - 1;
for (i__ = ll; i__ <= i__1; ++i__) {
r__1 = d__[i__] * cs;
slartg_(&r__1, &e[i__], &cs, &sn, &r__);
if (i__ > ll) {
e[i__ - 1] = oldsn * r__;
}
r__1 = oldcs * r__;
r__2 = d__[i__ + 1] * sn;
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
work[i__ - ll + 1] = cs;
work[i__ - ll + 1 + nm1] = sn;
work[i__ - ll + 1 + nm12] = oldcs;
work[i__ - ll + 1 + nm13] = oldsn;
/* L120: */
}
h__ = d__[m] * cs;
d__[m] = h__ * oldcs;
e[m - 1] = h__ * oldsn;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &
vt_ref(ll, 1), ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
+ 1], &u_ref(1, ll), ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
+ 1], &c___ref(ll, 1), ldc);
}
/* Test convergence */
if ((r__1 = e[m - 1], dabs(r__1)) <= thresh) {
e[m - 1] = 0.f;
}
} else {
/* Chase bulge from bottom to top
Save cosines and sines for later singular vector updates */
cs = 1.f;
oldcs = 1.f;
i__1 = ll + 1;
for (i__ = m; i__ >= i__1; --i__) {
r__1 = d__[i__] * cs;
slartg_(&r__1, &e[i__ - 1], &cs, &sn, &r__);
if (i__ < m) {
e[i__] = oldsn * r__;
}
r__1 = oldcs * r__;
r__2 = d__[i__ - 1] * sn;
slartg_(&r__1, &r__2, &oldcs, &oldsn, &d__[i__]);
work[i__ - ll] = cs;
work[i__ - ll + nm1] = -sn;
work[i__ - ll + nm12] = oldcs;
work[i__ - ll + nm13] = -oldsn;
/* L130: */
}
h__ = d__[ll] * cs;
d__[ll] = h__ * oldcs;
e[ll] = h__ * oldsn;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
nm13 + 1], &vt_ref(ll, 1), ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u_ref(
1, ll), ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &
c___ref(ll, 1), ldc);
}
/* Test convergence */
if ((r__1 = e[ll], dabs(r__1)) <= thresh) {
e[ll] = 0.f;
}
}
} else {
/* Use nonzero shift */
if (idir == 1) {
/* Chase bulge from top to bottom
Save cosines and sines for later singular vector updates */
f = ((r__1 = d__[ll], dabs(r__1)) - shift) * (r_sign(&c_b49, &d__[
ll]) + shift / d__[ll]);
g = e[ll];
i__1 = m - 1;
for (i__ = ll; i__ <= i__1; ++i__) {
slartg_(&f, &g, &cosr, &sinr, &r__);
if (i__ > ll) {
e[i__ - 1] = r__;
}
f = cosr * d__[i__] + sinr * e[i__];
e[i__] = cosr * e[i__] - sinr * d__[i__];
g = sinr * d__[i__ + 1];
d__[i__ + 1] = cosr * d__[i__ + 1];
slartg_(&f, &g, &cosl, &sinl, &r__);
d__[i__] = r__;
f = cosl * e[i__] + sinl * d__[i__ + 1];
d__[i__ + 1] = cosl * d__[i__ + 1] - sinl * e[i__];
if (i__ < m - 1) {
g = sinl * e[i__ + 1];
e[i__ + 1] = cosl * e[i__ + 1];
}
work[i__ - ll + 1] = cosr;
work[i__ - ll + 1 + nm1] = sinr;
work[i__ - ll + 1 + nm12] = cosl;
work[i__ - ll + 1 + nm13] = sinl;
/* L140: */
}
e[m - 1] = f;
/* Update singular vectors */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncvt, &work[1], &work[*n], &
vt_ref(ll, 1), ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "F", nru, &i__1, &work[nm12 + 1], &work[nm13
+ 1], &u_ref(1, ll), ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "F", &i__1, ncc, &work[nm12 + 1], &work[nm13
+ 1], &c___ref(ll, 1), ldc);
}
/* Test convergence */
if ((r__1 = e[m - 1], dabs(r__1)) <= thresh) {
e[m - 1] = 0.f;
}
} else {
/* Chase bulge from bottom to top
Save cosines and sines for later singular vector updates */
f = ((r__1 = d__[m], dabs(r__1)) - shift) * (r_sign(&c_b49, &d__[
m]) + shift / d__[m]);
g = e[m - 1];
i__1 = ll + 1;
for (i__ = m; i__ >= i__1; --i__) {
slartg_(&f, &g, &cosr, &sinr, &r__);
if (i__ < m) {
e[i__] = r__;
}
f = cosr * d__[i__] + sinr * e[i__ - 1];
e[i__ - 1] = cosr * e[i__ - 1] - sinr * d__[i__];
g = sinr * d__[i__ - 1];
d__[i__ - 1] = cosr * d__[i__ - 1];
slartg_(&f, &g, &cosl, &sinl, &r__);
d__[i__] = r__;
f = cosl * e[i__ - 1] + sinl * d__[i__ - 1];
d__[i__ - 1] = cosl * d__[i__ - 1] - sinl * e[i__ - 1];
if (i__ > ll + 1) {
g = sinl * e[i__ - 2];
e[i__ - 2] = cosl * e[i__ - 2];
}
work[i__ - ll] = cosr;
work[i__ - ll + nm1] = -sinr;
work[i__ - ll + nm12] = cosl;
work[i__ - ll + nm13] = -sinl;
/* L150: */
}
e[ll] = f;
/* Test convergence */
if ((r__1 = e[ll], dabs(r__1)) <= thresh) {
e[ll] = 0.f;
}
/* Update singular vectors if desired */
if (*ncvt > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncvt, &work[nm12 + 1], &work[
nm13 + 1], &vt_ref(ll, 1), ldvt);
}
if (*nru > 0) {
i__1 = m - ll + 1;
slasr_("R", "V", "B", nru, &i__1, &work[1], &work[*n], &u_ref(
1, ll), ldu);
}
if (*ncc > 0) {
i__1 = m - ll + 1;
slasr_("L", "V", "B", &i__1, ncc, &work[1], &work[*n], &
c___ref(ll, 1), ldc);
}
}
}
/* QR iteration finished, go back and check convergence */
goto L60;
/* All singular values converged, so make them positive */
L160:
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (d__[i__] < 0.f) {
d__[i__] = -d__[i__];
/* Change sign of singular vectors, if desired */
if (*ncvt > 0) {
sscal_(ncvt, &c_b72, &vt_ref(i__, 1), ldvt);
}
}
/* L170: */
}
/* Sort the singular values into decreasing order (insertion sort on
singular values, but only one transposition per singular vector) */
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Scan for smallest D(I) */
isub = 1;
smin = d__[1];
i__2 = *n + 1 - i__;
for (j = 2; j <= i__2; ++j) {
if (d__[j] <= smin) {
isub = j;
smin = d__[j];
}
/* L180: */
}
if (isub != *n + 1 - i__) {
/* Swap singular values and vectors */
d__[isub] = d__[*n + 1 - i__];
d__[*n + 1 - i__] = smin;
if (*ncvt > 0) {
sswap_(ncvt, &vt_ref(isub, 1), ldvt, &vt_ref(*n + 1 - i__, 1),
ldvt);
}
if (*nru > 0) {
sswap_(nru, &u_ref(1, isub), &c__1, &u_ref(1, *n + 1 - i__), &
c__1);
}
if (*ncc > 0) {
sswap_(ncc, &c___ref(isub, 1), ldc, &c___ref(*n + 1 - i__, 1),
ldc);
}
}
/* L190: */
}
goto L220;
/* Maximum number of iterations exceeded, failure to converge */
L200:
*info = 0;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
if (e[i__] != 0.f) {
++(*info);
}
/* L210: */
}
L220:
return 0;
/* End of SBDSQR */
} /* sbdsqr_ */
int sbdsdc_(char *uplo, char *compq, int *n, float *d__,
float *e, float *u, int *ldu, float *vt, int *ldvt, float *q,
int *iq, float *work, int *iwork, int *info)
{
/* System generated locals */
int u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
float r__1;
/* Builtin functions */
double r_sign(float *, float *), log(double);
/* Local variables */
int i__, j, k;
float p, r__;
int z__, ic, ii, kk;
float cs;
int is, iu;
float sn;
int nm1;
float eps;
int ivt, difl, difr, ierr, perm, mlvl, sqre;
extern int lsame_(char *, char *);
int poles;
extern int slasr_(char *, char *, char *, int *,
int *, float *, float *, float *, int *);
int iuplo, nsize, start;
extern int scopy_(int *, float *, int *, float *,
int *), sswap_(int *, float *, int *, float *, int *
), slasd0_(int *, int *, float *, float *, float *, int *
, float *, int *, int *, int *, float *, int *);
extern double slamch_(char *);
extern int slasda_(int *, int *, int *,
int *, float *, float *, float *, int *, float *, int *,
float *, float *, float *, float *, int *, int *, int *,
int *, float *, float *, float *, float *, int *, int *),
xerbla_(char *, int *);
extern int ilaenv_(int *, char *, char *, int *, int *,
int *, int *);
extern int slascl_(char *, int *, int *, float *,
float *, int *, int *, float *, int *, int *);
int givcol;
extern int slasdq_(char *, int *, int *, int
*, int *, int *, float *, float *, float *, int *, float *
, int *, float *, int *, float *, int *);
int icompq;
extern int slaset_(char *, int *, int *, float *,
float *, float *, int *), slartg_(float *, float *, float *
, float *, float *);
float orgnrm;
int givnum;
extern double slanst_(char *, int *, float *, float *);
int givptr, qstart, smlsiz, wstart, smlszp;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SBDSDC computes the singular value decomposition (SVD) of a float */
/* N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT, */
/* using a divide and conquer method, where S is a diagonal matrix */
/* with non-negative diagonal elements (the singular values of B), and */
/* U and VT are orthogonal matrices of left and right singular vectors, */
/* respectively. SBDSDC can be used to compute all singular values, */
/* and optionally, singular vectors or singular vectors in compact form. */
/* This code makes very mild assumptions about floating point */
/* arithmetic. It will work on machines with a guard digit in */
/* add/subtract, or on those binary machines without guard digits */
/* which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. */
/* It could conceivably fail on hexadecimal or decimal machines */
/* without guard digits, but we know of none. See SLASD3 for details. */
/* The code currently calls SLASDQ if singular values only are desired. */
/* However, it can be slightly modified to compute singular values */
/* using the divide and conquer method. */
/* Arguments */
/* ========= */
/* UPLO (input) CHARACTER*1 */
/* = 'U': B is upper bidiagonal. */
/* = 'L': B is lower bidiagonal. */
/* COMPQ (input) CHARACTER*1 */
/* Specifies whether singular vectors are to be computed */
/* as follows: */
/* = 'N': Compute singular values only; */
/* = 'P': Compute singular values and compute singular */
/* vectors in compact form; */
/* = 'I': Compute singular values and singular vectors. */
/* N (input) INTEGER */
/* The order of the matrix B. N >= 0. */
/* D (input/output) REAL array, dimension (N) */
/* On entry, the n diagonal elements of the bidiagonal matrix B. */
/* On exit, if INFO=0, the singular values of B. */
/* E (input/output) REAL array, dimension (N-1) */
/* On entry, the elements of E contain the offdiagonal */
/* elements of the bidiagonal matrix whose SVD is desired. */
/* On exit, E has been destroyed. */
/* U (output) REAL array, dimension (LDU,N) */
/* If COMPQ = 'I', then: */
/* On exit, if INFO = 0, U contains the left singular vectors */
/* of the bidiagonal matrix. */
/* For other values of COMPQ, U is not referenced. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= 1. */
/* If singular vectors are desired, then LDU >= MAX( 1, N ). */
/* VT (output) REAL array, dimension (LDVT,N) */
/* If COMPQ = 'I', then: */
/* On exit, if INFO = 0, VT' contains the right singular */
/* vectors of the bidiagonal matrix. */
/* For other values of COMPQ, VT is not referenced. */
/* LDVT (input) INTEGER */
/* The leading dimension of the array VT. LDVT >= 1. */
/* If singular vectors are desired, then LDVT >= MAX( 1, N ). */
/* Q (output) REAL array, dimension (LDQ) */
/* If COMPQ = 'P', then: */
/* On exit, if INFO = 0, Q and IQ contain the left */
/* and right singular vectors in a compact form, */
/* requiring O(N log N) space instead of 2*N**2. */
/* In particular, Q contains all the REAL data in */
/* LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) */
/* words of memory, where SMLSIZ is returned by ILAENV and */
/* is equal to the maximum size of the subproblems at the */
/* bottom of the computation tree (usually about 25). */
/* For other values of COMPQ, Q is not referenced. */
/* IQ (output) INTEGER array, dimension (LDIQ) */
/* If COMPQ = 'P', then: */
/* On exit, if INFO = 0, Q and IQ contain the left */
/* and right singular vectors in a compact form, */
/* requiring O(N log N) space instead of 2*N**2. */
/* In particular, IQ contains all INTEGER data in */
/* LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) */
/* words of memory, where SMLSIZ is returned by ILAENV and */
/* is equal to the maximum size of the subproblems at the */
/* bottom of the computation tree (usually about 25). */
/* For other values of COMPQ, IQ is not referenced. */
/* WORK (workspace) REAL array, dimension (MAX(1,LWORK)) */
/* If COMPQ = 'N' then LWORK >= (4 * N). */
/* If COMPQ = 'P' then LWORK >= (6 * N). */
/* If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). */
/* IWORK (workspace) INTEGER array, dimension (8*N) */
/* INFO (output) INTEGER */
/* = 0: successful exit. */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: The algorithm failed to compute an singular value. */
/* The update process of divide and conquer failed. */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* Changed dimension statement in comment describing E from (N) to */
/* (N-1). Sven, 17 Feb 05. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters. */
/* Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1;
vt -= vt_offset;
--q;
--iq;
--work;
--iwork;
/* Function Body */
*info = 0;
iuplo = 0;
if (lsame_(uplo, "U")) {
iuplo = 1;
}
if (lsame_(uplo, "L")) {
iuplo = 2;
}
if (lsame_(compq, "N")) {
icompq = 0;
} else if (lsame_(compq, "P")) {
icompq = 1;
} else if (lsame_(compq, "I")) {
icompq = 2;
} else {
icompq = -1;
}
if (iuplo == 0) {
*info = -1;
} else if (icompq < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ldu < 1 || icompq == 2 && *ldu < *n) {
*info = -7;
} else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSDC", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0);
if (*n == 1) {
if (icompq == 1) {
q[1] = r_sign(&c_b15, &d__[1]);
q[smlsiz * *n + 1] = 1.f;
} else if (icompq == 2) {
u[u_dim1 + 1] = r_sign(&c_b15, &d__[1]);
vt[vt_dim1 + 1] = 1.f;
}
d__[1] = ABS(d__[1]);
return 0;
}
nm1 = *n - 1;
/* If matrix lower bidiagonal, rotate to be upper bidiagonal */
/* by applying Givens rotations on the left */
wstart = 1;
qstart = 3;
if (icompq == 1) {
scopy_(n, &d__[1], &c__1, &q[1], &c__1);
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1);
}
if (iuplo == 2) {
qstart = 5;
wstart = (*n << 1) - 1;
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (icompq == 1) {
q[i__ + (*n << 1)] = cs;
q[i__ + *n * 3] = sn;
} else if (icompq == 2) {
work[i__] = cs;
work[nm1 + i__] = -sn;
}
/* L10: */
}
}
/* If ICOMPQ = 0, use SLASDQ to compute the singular values. */
if (icompq == 0) {
slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
goto L40;
}
/* If N is smaller than the minimum divide size SMLSIZ, then solve */
/* the problem with another solver. */
if (*n <= smlsiz) {
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
} else if (icompq == 1) {
iu = 1;
ivt = iu + *n;
slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n);
slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + (
qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[
iu + (qstart - 1) * *n], n, &work[wstart], info);
}
goto L40;
}
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
}
/* Scale. */
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
return 0;
}
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, &
ierr);
eps = slamch_("Epsilon");
mlvl = (int) (log((float) (*n) / (float) (smlsiz + 1)) / log(2.f)) + 1;
smlszp = smlsiz + 1;
if (icompq == 1) {
iu = 1;
ivt = smlsiz + 1;
difl = ivt + smlszp;
difr = difl + mlvl;
z__ = difr + (mlvl << 1);
ic = z__ + mlvl;
is = ic + 1;
poles = is + 1;
givnum = poles + (mlvl << 1);
k = 1;
givptr = 2;
perm = 3;
givcol = perm + mlvl;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], ABS(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L20: */
}
start = 1;
sqre = 0;
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], ABS(r__1)) < eps || i__ == nm1) {
/* Subproblem found. First determine its size and then */
/* apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - start + 1;
} else if ((r__1 = e[i__], ABS(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - start + 1;
} else {
/* A subproblem with E(NM1) small. This implies an */
/* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem */
/* first. */
nsize = i__ - start + 1;
if (icompq == 2) {
u[*n + *n * u_dim1] = r_sign(&c_b15, &d__[*n]);
vt[*n + *n * vt_dim1] = 1.f;
} else if (icompq == 1) {
q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]);
q[*n + (smlsiz + qstart - 1) * *n] = 1.f;
}
d__[*n] = (r__1 = d__[*n], ABS(r__1));
}
if (icompq == 2) {
slasd0_(&nsize, &sqre, &d__[start], &e[start], &u[start +
start * u_dim1], ldu, &vt[start + start * vt_dim1],
ldvt, &smlsiz, &iwork[1], &work[wstart], info);
} else {
slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[
start], &q[start + (iu + qstart - 2) * *n], n, &q[
start + (ivt + qstart - 2) * *n], &iq[start + k * *n],
&q[start + (difl + qstart - 2) * *n], &q[start + (
difr + qstart - 2) * *n], &q[start + (z__ + qstart -
2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[
start + givptr * *n], &iq[start + givcol * *n], n, &
iq[start + perm * *n], &q[start + (givnum + qstart -
2) * *n], &q[start + (ic + qstart - 2) * *n], &q[
start + (is + qstart - 2) * *n], &work[wstart], &
iwork[1], info);
if (*info != 0) {
return 0;
}
}
start = i__ + 1;
}
/* L30: */
}
/* Unscale */
slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr);
L40:
/* Use Selection Sort to minimize swaps of singular vectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
kk = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] > p) {
kk = j;
p = d__[j];
}
/* L50: */
}
if (kk != i__) {
d__[kk] = d__[i__];
d__[i__] = p;
if (icompq == 1) {
iq[i__] = kk;
} else if (icompq == 2) {
sswap_(n, &u[i__ * u_dim1 + 1], &c__1, &u[kk * u_dim1 + 1], &
c__1);
sswap_(n, &vt[i__ + vt_dim1], ldvt, &vt[kk + vt_dim1], ldvt);
}
} else if (icompq == 1) {
iq[i__] = i__;
}
/* L60: */
}
/* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */
if (icompq == 1) {
if (iuplo == 1) {
iq[*n] = 1;
} else {
iq[*n] = 0;
}
}
/* If B is lower bidiagonal, update U by those Givens rotations */
/* which rotated B to be upper bidiagonal */
if (iuplo == 2 && icompq == 2) {
slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu);
}
return 0;
/* End of SBDSDC */
} /* sbdsdc_ */
/* Subroutine */ int sgeevx_(char *balanc, char *jobvl, char *jobvr, char *
sense, integer *n, real *a, integer *lda, real *wr, real *wi, real *
vl, integer *ldvl, real *vr, integer *ldvr, integer *ilo, integer *
ihi, real *scale, real *abnrm, real *rconde, real *rcondv, real *work,
integer *lwork, integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1,
i__2, i__3;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, k;
real r__, cs, sn;
char job[1];
real scl, dum[1], eps;
char side[1];
real anrm;
integer ierr, itau, iwrk, nout;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
extern doublereal snrm2_(integer *, real *, integer *);
integer icond;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
extern doublereal slapy2_(real *, real *);
extern /* Subroutine */ int slabad_(real *, real *);
logical scalea;
real cscale;
extern /* Subroutine */ int sgebak_(char *, char *, integer *, integer *,
integer *, real *, integer *, real *, integer *, integer *), sgebal_(char *, integer *, real *, integer *,
integer *, integer *, real *, integer *);
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
extern /* Subroutine */ int sgehrd_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *, integer *), xerbla_(char
*, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
logical select[1];
real bignum;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
extern integer isamax_(integer *, real *, integer *);
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), slartg_(real *, real *,
real *, real *, real *), sorghr_(integer *, integer *, integer *,
real *, integer *, real *, real *, integer *, integer *), shseqr_(
char *, char *, integer *, integer *, integer *, real *, integer *
, real *, real *, real *, integer *, real *, integer *, integer *), strevc_(char *, char *, logical *, integer *,
real *, integer *, real *, integer *, real *, integer *, integer *
, integer *, real *, integer *);
integer minwrk, maxwrk;
extern /* Subroutine */ int strsna_(char *, char *, logical *, integer *,
real *, integer *, real *, integer *, real *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *,
integer *);
logical wantvl, wntsnb;
integer hswork;
logical wntsne;
real smlnum;
logical lquery, wantvr, wntsnn, wntsnv;
/* -- LAPACK driver routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGEEVX computes for an N-by-N real nonsymmetric matrix A, the */
/* eigenvalues and, optionally, the left and/or right eigenvectors. */
/* Optionally also, it computes a balancing transformation to improve */
/* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, */
/* SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues */
/* (RCONDE), and reciprocal condition numbers for the right */
/* eigenvectors (RCONDV). */
/* The right eigenvector v(j) of A satisfies */
/* A * v(j) = lambda(j) * v(j) */
/* where lambda(j) is its eigenvalue. */
/* The left eigenvector u(j) of A satisfies */
/* u(j)**H * A = lambda(j) * u(j)**H */
/* where u(j)**H denotes the conjugate transpose of u(j). */
/* The computed eigenvectors are normalized to have Euclidean norm */
/* equal to 1 and largest component real. */
/* Balancing a matrix means permuting the rows and columns to make it */
/* more nearly upper triangular, and applying a diagonal similarity */
/* transformation D * A * D**(-1), where D is a diagonal matrix, to */
/* make its rows and columns closer in norm and the condition numbers */
/* of its eigenvalues and eigenvectors smaller. The computed */
/* reciprocal condition numbers correspond to the balanced matrix. */
/* Permuting rows and columns will not change the condition numbers */
/* (in exact arithmetic) but diagonal scaling will. For further */
/* explanation of balancing, see section 4.10.2 of the LAPACK */
/* Users' Guide. */
/* Arguments */
/* ========= */
/* BALANC (input) CHARACTER*1 */
/* Indicates how the input matrix should be diagonally scaled */
/* and/or permuted to improve the conditioning of its */
/* eigenvalues. */
/* = 'N': Do not diagonally scale or permute; */
/* = 'P': Perform permutations to make the matrix more nearly */
/* upper triangular. Do not diagonally scale; */
/* = 'S': Diagonally scale the matrix, i.e. replace A by */
/* D*A*D**(-1), where D is a diagonal matrix chosen */
/* to make the rows and columns of A more equal in */
/* norm. Do not permute; */
/* = 'B': Both diagonally scale and permute A. */
/* Computed reciprocal condition numbers will be for the matrix */
/* after balancing and/or permuting. Permuting does not change */
/* condition numbers (in exact arithmetic), but balancing does. */
/* JOBVL (input) CHARACTER*1 */
/* = 'N': left eigenvectors of A are not computed; */
/* = 'V': left eigenvectors of A are computed. */
/* If SENSE = 'E' or 'B', JOBVL must = 'V'. */
/* JOBVR (input) CHARACTER*1 */
/* = 'N': right eigenvectors of A are not computed; */
/* = 'V': right eigenvectors of A are computed. */
/* If SENSE = 'E' or 'B', JOBVR must = 'V'. */
/* SENSE (input) CHARACTER*1 */
/* Determines which reciprocal condition numbers are computed. */
/* = 'N': None are computed; */
/* = 'E': Computed for eigenvalues only; */
/* = 'V': Computed for right eigenvectors only; */
/* = 'B': Computed for eigenvalues and right eigenvectors. */
/* If SENSE = 'E' or 'B', both left and right eigenvectors */
/* must also be computed (JOBVL = 'V' and JOBVR = 'V'). */
/* N (input) INTEGER */
/* The order of the matrix A. N >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the N-by-N matrix A. */
/* On exit, A has been overwritten. If JOBVL = 'V' or */
/* JOBVR = 'V', A contains the real Schur form of the balanced */
/* version of the input matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* WR (output) REAL array, dimension (N) */
/* WI (output) REAL array, dimension (N) */
/* WR and WI contain the real and imaginary parts, */
/* respectively, of the computed eigenvalues. Complex */
/* conjugate pairs of eigenvalues will appear consecutively */
/* with the eigenvalue having the positive imaginary part */
/* first. */
/* VL (output) REAL array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left eigenvectors u(j) are stored one */
/* after another in the columns of VL, in the same order */
/* as their eigenvalues. */
/* If JOBVL = 'N', VL is not referenced. */
/* If the j-th eigenvalue is real, then u(j) = VL(:,j), */
/* the j-th column of VL. */
/* If the j-th and (j+1)-st eigenvalues form a complex */
/* conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and */
/* u(j+1) = VL(:,j) - i*VL(:,j+1). */
/* LDVL (input) INTEGER */
/* The leading dimension of the array VL. LDVL >= 1; if */
/* JOBVL = 'V', LDVL >= N. */
/* VR (output) REAL array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right eigenvectors v(j) are stored one */
/* after another in the columns of VR, in the same order */
/* as their eigenvalues. */
/* If JOBVR = 'N', VR is not referenced. */
/* If the j-th eigenvalue is real, then v(j) = VR(:,j), */
/* the j-th column of VR. */
/* If the j-th and (j+1)-st eigenvalues form a complex */
/* conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and */
/* v(j+1) = VR(:,j) - i*VR(:,j+1). */
/* LDVR (input) INTEGER */
/* The leading dimension of the array VR. LDVR >= 1, and if */
/* JOBVR = 'V', LDVR >= N. */
/* ILO (output) INTEGER */
/* IHI (output) INTEGER */
/* ILO and IHI are integer values determined when A was */
/* balanced. The balanced A(i,j) = 0 if I > J and */
/* J = 1,...,ILO-1 or I = IHI+1,...,N. */
/* SCALE (output) REAL array, dimension (N) */
/* Details of the permutations and scaling factors applied */
/* when balancing A. If P(j) is the index of the row and column */
/* interchanged with row and column j, and D(j) is the scaling */
/* factor applied to row and column j, then */
/* SCALE(J) = P(J), for J = 1,...,ILO-1 */
/* = D(J), for J = ILO,...,IHI */
/* = P(J) for J = IHI+1,...,N. */
/* The order in which the interchanges are made is N to IHI+1, */
/* then 1 to ILO-1. */
/* ABNRM (output) REAL */
/* The one-norm of the balanced matrix (the maximum */
/* of the sum of absolute values of elements of any column). */
/* RCONDE (output) REAL array, dimension (N) */
/* RCONDE(j) is the reciprocal condition number of the j-th */
/* eigenvalue. */
/* RCONDV (output) REAL array, dimension (N) */
/* RCONDV(j) is the reciprocal condition number of the j-th */
/* right eigenvector. */
/* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. If SENSE = 'N' or 'E', */
/* LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', */
/* LWORK >= 3*N. If SENSE = 'V' or 'B', LWORK >= N*(N+6). */
/* For good performance, LWORK must generally be larger. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace) INTEGER array, dimension (2*N-2) */
/* If SENSE = 'N' or 'E', not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = i, the QR algorithm failed to compute all the */
/* eigenvalues, and no eigenvectors or condition numbers */
/* have been computed; elements 1:ILO-1 and i+1:N of WR */
/* and WI contain eigenvalues which have converged. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--wr;
--wi;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--scale;
--rconde;
--rcondv;
--work;
--iwork;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
wantvl = lsame_(jobvl, "V");
wantvr = lsame_(jobvr, "V");
wntsnn = lsame_(sense, "N");
wntsne = lsame_(sense, "E");
wntsnv = lsame_(sense, "V");
wntsnb = lsame_(sense, "B");
if (! (lsame_(balanc, "N") || lsame_(balanc, "S") || lsame_(balanc, "P")
|| lsame_(balanc, "B"))) {
*info = -1;
} else if (! wantvl && ! lsame_(jobvl, "N")) {
*info = -2;
} else if (! wantvr && ! lsame_(jobvr, "N")) {
*info = -3;
} else if (! (wntsnn || wntsne || wntsnb || wntsnv) || (wntsne || wntsnb)
&& ! (wantvl && wantvr)) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldvl < 1 || wantvl && *ldvl < *n) {
*info = -11;
} else if (*ldvr < 1 || wantvr && *ldvr < *n) {
*info = -13;
}
/* Compute workspace */
/* (Note: Comments in the code beginning "Workspace:" describe the */
/* minimal amount of workspace needed at that point in the code, */
/* as well as the preferred amount for good performance. */
/* NB refers to the optimal block size for the immediately */
/* following subroutine, as returned by ILAENV. */
/* HSWORK refers to the workspace preferred by SHSEQR, as */
/* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, */
/* the worst case.) */
if (*info == 0) {
if (*n == 0) {
minwrk = 1;
maxwrk = 1;
} else {
maxwrk = *n + *n * ilaenv_(&c__1, "SGEHRD", " ", n, &c__1, n, &
c__0);
if (wantvl) {
shseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[
1], &vl[vl_offset], ldvl, &work[1], &c_n1, info);
} else if (wantvr) {
shseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[
1], &vr[vr_offset], ldvr, &work[1], &c_n1, info);
} else {
if (wntsnn) {
shseqr_("E", "N", n, &c__1, n, &a[a_offset], lda, &wr[1],
&wi[1], &vr[vr_offset], ldvr, &work[1], &c_n1,
info);
} else {
shseqr_("S", "N", n, &c__1, n, &a[a_offset], lda, &wr[1],
&wi[1], &vr[vr_offset], ldvr, &work[1], &c_n1,
info);
}
}
hswork = work[1];
if (! wantvl && ! wantvr) {
minwrk = *n << 1;
if (! wntsnn) {
/* Computing MAX */
i__1 = minwrk, i__2 = *n * *n + *n * 6;
minwrk = max(i__1,i__2);
}
maxwrk = max(maxwrk,hswork);
if (! wntsnn) {
/* Computing MAX */
i__1 = maxwrk, i__2 = *n * *n + *n * 6;
maxwrk = max(i__1,i__2);
}
} else {
minwrk = *n * 3;
if (! wntsnn && ! wntsne) {
/* Computing MAX */
i__1 = minwrk, i__2 = *n * *n + *n * 6;
minwrk = max(i__1,i__2);
}
maxwrk = max(maxwrk,hswork);
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + (*n - 1) * ilaenv_(&c__1, "SORGHR",
" ", n, &c__1, n, &c_n1);
maxwrk = max(i__1,i__2);
if (! wntsnn && ! wntsne) {
/* Computing MAX */
i__1 = maxwrk, i__2 = *n * *n + *n * 6;
maxwrk = max(i__1,i__2);
}
/* Computing MAX */
i__1 = maxwrk, i__2 = *n * 3;
maxwrk = max(i__1,i__2);
}
maxwrk = max(maxwrk,minwrk);
}
work[1] = (real) maxwrk;
if (*lwork < minwrk && ! lquery) {
*info = -21;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGEEVX", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1.f / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
icond = 0;
anrm = slange_("M", n, n, &a[a_offset], lda, dum);
scalea = FALSE_;
if (anrm > 0.f && anrm < smlnum) {
scalea = TRUE_;
cscale = smlnum;
} else if (anrm > bignum) {
scalea = TRUE_;
cscale = bignum;
}
if (scalea) {
slascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, &
ierr);
}
/* Balance the matrix and compute ABNRM */
sgebal_(balanc, n, &a[a_offset], lda, ilo, ihi, &scale[1], &ierr);
*abnrm = slange_("1", n, n, &a[a_offset], lda, dum);
if (scalea) {
dum[0] = *abnrm;
slascl_("G", &c__0, &c__0, &cscale, &anrm, &c__1, &c__1, dum, &c__1, &
ierr);
*abnrm = dum[0];
}
/* Reduce to upper Hessenberg form */
/* (Workspace: need 2*N, prefer N+N*NB) */
itau = 1;
iwrk = itau + *n;
i__1 = *lwork - iwrk + 1;
sgehrd_(n, ilo, ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1, &
ierr);
if (wantvl) {
/* Want left eigenvectors */
/* Copy Householder vectors to VL */
*(unsigned char *)side = 'L';
slacpy_("L", n, n, &a[a_offset], lda, &vl[vl_offset], ldvl)
;
/* Generate orthogonal matrix in VL */
/* (Workspace: need 2*N-1, prefer N+(N-1)*NB) */
i__1 = *lwork - iwrk + 1;
sorghr_(n, ilo, ihi, &vl[vl_offset], ldvl, &work[itau], &work[iwrk], &
i__1, &ierr);
/* Perform QR iteration, accumulating Schur vectors in VL */
/* (Workspace: need 1, prefer HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_("S", "V", n, ilo, ihi, &a[a_offset], lda, &wr[1], &wi[1], &vl[
vl_offset], ldvl, &work[iwrk], &i__1, info);
if (wantvr) {
/* Want left and right eigenvectors */
/* Copy Schur vectors to VR */
*(unsigned char *)side = 'B';
slacpy_("F", n, n, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr);
}
} else if (wantvr) {
/* Want right eigenvectors */
/* Copy Householder vectors to VR */
*(unsigned char *)side = 'R';
slacpy_("L", n, n, &a[a_offset], lda, &vr[vr_offset], ldvr)
;
/* Generate orthogonal matrix in VR */
/* (Workspace: need 2*N-1, prefer N+(N-1)*NB) */
i__1 = *lwork - iwrk + 1;
sorghr_(n, ilo, ihi, &vr[vr_offset], ldvr, &work[itau], &work[iwrk], &
i__1, &ierr);
/* Perform QR iteration, accumulating Schur vectors in VR */
/* (Workspace: need 1, prefer HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_("S", "V", n, ilo, ihi, &a[a_offset], lda, &wr[1], &wi[1], &vr[
vr_offset], ldvr, &work[iwrk], &i__1, info);
} else {
/* Compute eigenvalues only */
/* If condition numbers desired, compute Schur form */
if (wntsnn) {
*(unsigned char *)job = 'E';
} else {
*(unsigned char *)job = 'S';
}
/* (Workspace: need 1, prefer HSWORK (see comments) ) */
iwrk = itau;
i__1 = *lwork - iwrk + 1;
shseqr_(job, "N", n, ilo, ihi, &a[a_offset], lda, &wr[1], &wi[1], &vr[
vr_offset], ldvr, &work[iwrk], &i__1, info);
}
/* If INFO > 0 from SHSEQR, then quit */
if (*info > 0) {
goto L50;
}
if (wantvl || wantvr) {
/* Compute left and/or right eigenvectors */
/* (Workspace: need 3*N) */
strevc_(side, "B", select, n, &a[a_offset], lda, &vl[vl_offset], ldvl,
&vr[vr_offset], ldvr, n, &nout, &work[iwrk], &ierr);
}
/* Compute condition numbers if desired */
/* (Workspace: need N*N+6*N unless SENSE = 'E') */
if (! wntsnn) {
strsna_(sense, "A", select, n, &a[a_offset], lda, &vl[vl_offset],
ldvl, &vr[vr_offset], ldvr, &rconde[1], &rcondv[1], n, &nout,
&work[iwrk], n, &iwork[1], &icond);
}
if (wantvl) {
/* Undo balancing of left eigenvectors */
sgebak_(balanc, "L", n, ilo, ihi, &scale[1], n, &vl[vl_offset], ldvl,
&ierr);
/* Normalize left eigenvectors and make largest component real */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (wi[i__] == 0.f) {
scl = 1.f / snrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1);
sscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1);
} else if (wi[i__] > 0.f) {
r__1 = snrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1);
r__2 = snrm2_(n, &vl[(i__ + 1) * vl_dim1 + 1], &c__1);
scl = 1.f / slapy2_(&r__1, &r__2);
sscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1);
sscal_(n, &scl, &vl[(i__ + 1) * vl_dim1 + 1], &c__1);
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
r__1 = vl[k + i__ * vl_dim1];
/* Computing 2nd power */
r__2 = vl[k + (i__ + 1) * vl_dim1];
work[k] = r__1 * r__1 + r__2 * r__2;
/* L10: */
}
k = isamax_(n, &work[1], &c__1);
slartg_(&vl[k + i__ * vl_dim1], &vl[k + (i__ + 1) * vl_dim1],
&cs, &sn, &r__);
srot_(n, &vl[i__ * vl_dim1 + 1], &c__1, &vl[(i__ + 1) *
vl_dim1 + 1], &c__1, &cs, &sn);
vl[k + (i__ + 1) * vl_dim1] = 0.f;
}
/* L20: */
}
}
if (wantvr) {
/* Undo balancing of right eigenvectors */
sgebak_(balanc, "R", n, ilo, ihi, &scale[1], n, &vr[vr_offset], ldvr,
&ierr);
/* Normalize right eigenvectors and make largest component real */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (wi[i__] == 0.f) {
scl = 1.f / snrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1);
sscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1);
} else if (wi[i__] > 0.f) {
r__1 = snrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1);
r__2 = snrm2_(n, &vr[(i__ + 1) * vr_dim1 + 1], &c__1);
scl = 1.f / slapy2_(&r__1, &r__2);
sscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1);
sscal_(n, &scl, &vr[(i__ + 1) * vr_dim1 + 1], &c__1);
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
/* Computing 2nd power */
r__1 = vr[k + i__ * vr_dim1];
/* Computing 2nd power */
r__2 = vr[k + (i__ + 1) * vr_dim1];
work[k] = r__1 * r__1 + r__2 * r__2;
/* L30: */
}
k = isamax_(n, &work[1], &c__1);
slartg_(&vr[k + i__ * vr_dim1], &vr[k + (i__ + 1) * vr_dim1],
&cs, &sn, &r__);
srot_(n, &vr[i__ * vr_dim1 + 1], &c__1, &vr[(i__ + 1) *
vr_dim1 + 1], &c__1, &cs, &sn);
vr[k + (i__ + 1) * vr_dim1] = 0.f;
}
/* L40: */
}
}
/* Undo scaling if necessary */
L50:
if (scalea) {
i__1 = *n - *info;
/* Computing MAX */
i__3 = *n - *info;
i__2 = max(i__3,1);
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[*info +
1], &i__2, &ierr);
i__1 = *n - *info;
/* Computing MAX */
i__3 = *n - *info;
i__2 = max(i__3,1);
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[*info +
1], &i__2, &ierr);
if (*info == 0) {
if ((wntsnv || wntsnb) && icond == 0) {
slascl_("G", &c__0, &c__0, &cscale, &anrm, n, &c__1, &rcondv[
1], n, &ierr);
}
} else {
i__1 = *ilo - 1;
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[1],
n, &ierr);
i__1 = *ilo - 1;
slascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[1],
n, &ierr);
}
}
work[1] = (real) maxwrk;
return 0;
/* End of SGEEVX */
} /* sgeevx_ */
/* Subroutine */ int ssapps_(integer *n, integer *kev, integer *np, real *
shift, real *v, integer *ldv, real *h__, integer *ldh, real *resid,
real *q, integer *ldq, real *workd)
{
/* Initialized data */
static logical first = TRUE_;
/* System generated locals */
integer h_dim1, h_offset, q_dim1, q_offset, v_dim1, v_offset, i__1, i__2,
i__3, i__4;
real r__1, r__2;
/* Local variables */
static real c__, f, g;
static integer i__, j;
static real r__, s, a1, a2, a3, a4, t0, t1;
static integer jj;
static real big;
static integer iend, itop;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *),
sgemv_(char *, integer *, integer *, real *, real *, integer *,
real *, integer *, real *, real *, integer *, ftnlen), scopy_(
integer *, real *, integer *, real *, integer *), saxpy_(integer *
, real *, real *, integer *, real *, integer *), ivout_(integer *,
integer *, integer *, integer *, char *, ftnlen), svout_(integer
*, integer *, real *, integer *, char *, ftnlen);
extern doublereal slamch_(char *, ftnlen);
extern /* Subroutine */ int arscnd_(real *);
static real epsmch;
static integer istart, kplusp, msglvl;
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *, ftnlen), slartg_(real *, real *,
real *, real *, real *), slaset_(char *, integer *, integer *,
real *, real *, real *, integer *, ftnlen);
/* %----------------------------------------------------% */
/* | Include files for debugging and timing information | */
/* %----------------------------------------------------% */
/* \SCCS Information: @(#) */
/* FILE: debug.h SID: 2.3 DATE OF SID: 11/16/95 RELEASE: 2 */
/* %---------------------------------% */
/* | See debug.doc for documentation | */
/* %---------------------------------% */
/* %------------------% */
/* | Scalar Arguments | */
/* %------------------% */
/* %--------------------------------% */
/* | See stat.doc for documentation | */
/* %--------------------------------% */
/* \SCCS Information: @(#) */
/* FILE: stat.h SID: 2.2 DATE OF SID: 11/16/95 RELEASE: 2 */
/* %-----------------% */
/* | Array Arguments | */
/* %-----------------% */
/* %------------% */
/* | Parameters | */
/* %------------% */
/* %---------------% */
/* | Local Scalars | */
/* %---------------% */
/* %----------------------% */
/* | External Subroutines | */
/* %----------------------% */
/* %--------------------% */
/* | External Functions | */
/* %--------------------% */
/* %----------------------% */
/* | Intrinsics Functions | */
/* %----------------------% */
/* %----------------% */
/* | Data statments | */
/* %----------------% */
/* Parameter adjustments */
--workd;
--resid;
--shift;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
h_dim1 = *ldh;
h_offset = 1 + h_dim1;
h__ -= h_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
/* Function Body */
/* %-----------------------% */
/* | Executable Statements | */
/* %-----------------------% */
if (first) {
epsmch = slamch_("Epsilon-Machine", (ftnlen)15);
first = FALSE_;
}
itop = 1;
/* %-------------------------------% */
/* | Initialize timing statistics | */
/* | & message level for debugging | */
/* %-------------------------------% */
arscnd_(&t0);
msglvl = debug_1.msapps;
kplusp = *kev + *np;
/* %----------------------------------------------% */
/* | Initialize Q to the identity matrix of order | */
/* | kplusp used to accumulate the rotations. | */
/* %----------------------------------------------% */
slaset_("All", &kplusp, &kplusp, &c_b4, &c_b5, &q[q_offset], ldq, (ftnlen)
3);
/* %----------------------------------------------% */
/* | Quick return if there are no shifts to apply | */
/* %----------------------------------------------% */
if (*np == 0) {
goto L9000;
}
/* %----------------------------------------------------------% */
/* | Apply the np shifts implicitly. Apply each shift to the | */
/* | whole matrix and not just to the submatrix from which it | */
/* | comes. | */
/* %----------------------------------------------------------% */
i__1 = *np;
for (jj = 1; jj <= i__1; ++jj) {
istart = itop;
/* %----------------------------------------------------------% */
/* | Check for splitting and deflation. Currently we consider | */
/* | an off-diagonal element h(i+1,1) negligible if | */
/* | h(i+1,1) .le. epsmch*( |h(i,2)| + |h(i+1,2)| ) | */
/* | for i=1:KEV+NP-1. | */
/* | If above condition tests true then we set h(i+1,1) = 0. | */
/* | Note that h(1:KEV+NP,1) are assumed to be non negative. | */
/* %----------------------------------------------------------% */
L20:
/* %------------------------------------------------% */
/* | The following loop exits early if we encounter | */
/* | a negligible off diagonal element. | */
/* %------------------------------------------------% */
i__2 = kplusp - 1;
for (i__ = istart; i__ <= i__2; ++i__) {
big = (r__1 = h__[i__ + (h_dim1 << 1)], dabs(r__1)) + (r__2 = h__[
i__ + 1 + (h_dim1 << 1)], dabs(r__2));
if (h__[i__ + 1 + h_dim1] <= epsmch * big) {
if (msglvl > 0) {
ivout_(&debug_1.logfil, &c__1, &i__, &debug_1.ndigit,
"_sapps: deflation at row/column no.", (ftnlen)35)
;
ivout_(&debug_1.logfil, &c__1, &jj, &debug_1.ndigit,
"_sapps: occured before shift number.", (ftnlen)
36);
svout_(&debug_1.logfil, &c__1, &h__[i__ + 1 + h_dim1], &
debug_1.ndigit, "_sapps: the corresponding off d"
"iagonal element", (ftnlen)46);
}
h__[i__ + 1 + h_dim1] = 0.f;
iend = i__;
goto L40;
}
/* L30: */
}
iend = kplusp;
L40:
if (istart < iend) {
/* %--------------------------------------------------------% */
/* | Construct the plane rotation G'(istart,istart+1,theta) | */
/* | that attempts to drive h(istart+1,1) to zero. | */
/* %--------------------------------------------------------% */
f = h__[istart + (h_dim1 << 1)] - shift[jj];
g = h__[istart + 1 + h_dim1];
slartg_(&f, &g, &c__, &s, &r__);
/* %-------------------------------------------------------% */
/* | Apply rotation to the left and right of H; | */
/* | H <- G' * H * G, where G = G(istart,istart+1,theta). | */
/* | This will create a "bulge". | */
/* %-------------------------------------------------------% */
a1 = c__ * h__[istart + (h_dim1 << 1)] + s * h__[istart + 1 +
h_dim1];
a2 = c__ * h__[istart + 1 + h_dim1] + s * h__[istart + 1 + (
h_dim1 << 1)];
a4 = c__ * h__[istart + 1 + (h_dim1 << 1)] - s * h__[istart + 1 +
h_dim1];
a3 = c__ * h__[istart + 1 + h_dim1] - s * h__[istart + (h_dim1 <<
1)];
h__[istart + (h_dim1 << 1)] = c__ * a1 + s * a2;
h__[istart + 1 + (h_dim1 << 1)] = c__ * a4 - s * a3;
h__[istart + 1 + h_dim1] = c__ * a3 + s * a4;
/* %----------------------------------------------------% */
/* | Accumulate the rotation in the matrix Q; Q <- Q*G | */
/* %----------------------------------------------------% */
/* Computing MIN */
i__3 = istart + jj;
i__2 = min(i__3,kplusp);
for (j = 1; j <= i__2; ++j) {
a1 = c__ * q[j + istart * q_dim1] + s * q[j + (istart + 1) *
q_dim1];
q[j + (istart + 1) * q_dim1] = -s * q[j + istart * q_dim1] +
c__ * q[j + (istart + 1) * q_dim1];
q[j + istart * q_dim1] = a1;
/* L60: */
}
/* %----------------------------------------------% */
/* | The following loop chases the bulge created. | */
/* | Note that the previous rotation may also be | */
/* | done within the following loop. But it is | */
/* | kept separate to make the distinction among | */
/* | the bulge chasing sweeps and the first plane | */
/* | rotation designed to drive h(istart+1,1) to | */
/* | zero. | */
/* %----------------------------------------------% */
i__2 = iend - 1;
for (i__ = istart + 1; i__ <= i__2; ++i__) {
/* %----------------------------------------------% */
/* | Construct the plane rotation G'(i,i+1,theta) | */
/* | that zeros the i-th bulge that was created | */
/* | by G(i-1,i,theta). g represents the bulge. | */
/* %----------------------------------------------% */
f = h__[i__ + h_dim1];
g = s * h__[i__ + 1 + h_dim1];
/* %----------------------------------% */
/* | Final update with G(i-1,i,theta) | */
/* %----------------------------------% */
h__[i__ + 1 + h_dim1] = c__ * h__[i__ + 1 + h_dim1];
slartg_(&f, &g, &c__, &s, &r__);
/* %-------------------------------------------% */
/* | The following ensures that h(1:iend-1,1), | */
/* | the first iend-2 off diagonal of elements | */
/* | H, remain non negative. | */
/* %-------------------------------------------% */
if (r__ < 0.f) {
r__ = -r__;
c__ = -c__;
s = -s;
}
/* %--------------------------------------------% */
/* | Apply rotation to the left and right of H; | */
/* | H <- G * H * G', where G = G(i,i+1,theta) | */
/* %--------------------------------------------% */
h__[i__ + h_dim1] = r__;
a1 = c__ * h__[i__ + (h_dim1 << 1)] + s * h__[i__ + 1 +
h_dim1];
a2 = c__ * h__[i__ + 1 + h_dim1] + s * h__[i__ + 1 + (h_dim1
<< 1)];
a3 = c__ * h__[i__ + 1 + h_dim1] - s * h__[i__ + (h_dim1 << 1)
];
a4 = c__ * h__[i__ + 1 + (h_dim1 << 1)] - s * h__[i__ + 1 +
h_dim1];
h__[i__ + (h_dim1 << 1)] = c__ * a1 + s * a2;
h__[i__ + 1 + (h_dim1 << 1)] = c__ * a4 - s * a3;
h__[i__ + 1 + h_dim1] = c__ * a3 + s * a4;
/* %----------------------------------------------------% */
/* | Accumulate the rotation in the matrix Q; Q <- Q*G | */
/* %----------------------------------------------------% */
/* Computing MIN */
i__4 = i__ + jj;
i__3 = min(i__4,kplusp);
for (j = 1; j <= i__3; ++j) {
a1 = c__ * q[j + i__ * q_dim1] + s * q[j + (i__ + 1) *
q_dim1];
q[j + (i__ + 1) * q_dim1] = -s * q[j + i__ * q_dim1] +
c__ * q[j + (i__ + 1) * q_dim1];
q[j + i__ * q_dim1] = a1;
/* L50: */
}
/* L70: */
}
}
/* %--------------------------% */
/* | Update the block pointer | */
/* %--------------------------% */
istart = iend + 1;
/* %------------------------------------------% */
/* | Make sure that h(iend,1) is non-negative | */
/* | If not then set h(iend,1) <-- -h(iend,1) | */
/* | and negate the last column of Q. | */
/* | We have effectively carried out a | */
/* | similarity on transformation H | */
/* %------------------------------------------% */
if (h__[iend + h_dim1] < 0.f) {
h__[iend + h_dim1] = -h__[iend + h_dim1];
sscal_(&kplusp, &c_b20, &q[iend * q_dim1 + 1], &c__1);
}
/* %--------------------------------------------------------% */
/* | Apply the same shift to the next block if there is any | */
/* %--------------------------------------------------------% */
if (iend < kplusp) {
goto L20;
}
/* %-----------------------------------------------------% */
/* | Check if we can increase the the start of the block | */
/* %-----------------------------------------------------% */
i__2 = kplusp - 1;
for (i__ = itop; i__ <= i__2; ++i__) {
if (h__[i__ + 1 + h_dim1] > 0.f) {
goto L90;
}
++itop;
/* L80: */
}
/* %-----------------------------------% */
/* | Finished applying the jj-th shift | */
/* %-----------------------------------% */
L90:
;
}
/* %------------------------------------------% */
/* | All shifts have been applied. Check for | */
/* | more possible deflation that might occur | */
/* | after the last shift is applied. | */
/* %------------------------------------------% */
i__1 = kplusp - 1;
for (i__ = itop; i__ <= i__1; ++i__) {
big = (r__1 = h__[i__ + (h_dim1 << 1)], dabs(r__1)) + (r__2 = h__[i__
+ 1 + (h_dim1 << 1)], dabs(r__2));
if (h__[i__ + 1 + h_dim1] <= epsmch * big) {
if (msglvl > 0) {
ivout_(&debug_1.logfil, &c__1, &i__, &debug_1.ndigit, "_sapp"
"s: deflation at row/column no.", (ftnlen)35);
svout_(&debug_1.logfil, &c__1, &h__[i__ + 1 + h_dim1], &
debug_1.ndigit, "_sapps: the corresponding off diago"
"nal element", (ftnlen)46);
}
h__[i__ + 1 + h_dim1] = 0.f;
}
/* L100: */
}
/* %-------------------------------------------------% */
/* | Compute the (kev+1)-st column of (V*Q) and | */
/* | temporarily store the result in WORKD(N+1:2*N). | */
/* | This is not necessary if h(kev+1,1) = 0. | */
/* %-------------------------------------------------% */
if (h__[*kev + 1 + h_dim1] > 0.f) {
sgemv_("N", n, &kplusp, &c_b5, &v[v_offset], ldv, &q[(*kev + 1) *
q_dim1 + 1], &c__1, &c_b4, &workd[*n + 1], &c__1, (ftnlen)1);
}
/* %-------------------------------------------------------% */
/* | Compute column 1 to kev of (V*Q) in backward order | */
/* | taking advantage that Q is an upper triangular matrix | */
/* | with lower bandwidth np. | */
/* | Place results in v(:,kplusp-kev:kplusp) temporarily. | */
/* %-------------------------------------------------------% */
i__1 = *kev;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = kplusp - i__ + 1;
sgemv_("N", n, &i__2, &c_b5, &v[v_offset], ldv, &q[(*kev - i__ + 1) *
q_dim1 + 1], &c__1, &c_b4, &workd[1], &c__1, (ftnlen)1);
scopy_(n, &workd[1], &c__1, &v[(kplusp - i__ + 1) * v_dim1 + 1], &
c__1);
/* L130: */
}
/* %-------------------------------------------------% */
/* | Move v(:,kplusp-kev+1:kplusp) into v(:,1:kev). | */
/* %-------------------------------------------------% */
slacpy_("All", n, kev, &v[(*np + 1) * v_dim1 + 1], ldv, &v[v_offset], ldv,
(ftnlen)3);
/* %--------------------------------------------% */
/* | Copy the (kev+1)-st column of (V*Q) in the | */
/* | appropriate place if h(kev+1,1) .ne. zero. | */
/* %--------------------------------------------% */
if (h__[*kev + 1 + h_dim1] > 0.f) {
scopy_(n, &workd[*n + 1], &c__1, &v[(*kev + 1) * v_dim1 + 1], &c__1);
}
/* %-------------------------------------% */
/* | Update the residual vector: | */
/* | r <- sigmak*r + betak*v(:,kev+1) | */
/* | where | */
/* | sigmak = (e_{kev+p}'*Q)*e_{kev} | */
/* | betak = e_{kev+1}'*H*e_{kev} | */
/* %-------------------------------------% */
sscal_(n, &q[kplusp + *kev * q_dim1], &resid[1], &c__1);
if (h__[*kev + 1 + h_dim1] > 0.f) {
saxpy_(n, &h__[*kev + 1 + h_dim1], &v[(*kev + 1) * v_dim1 + 1], &c__1,
&resid[1], &c__1);
}
if (msglvl > 1) {
svout_(&debug_1.logfil, &c__1, &q[kplusp + *kev * q_dim1], &
debug_1.ndigit, "_sapps: sigmak of the updated residual vect"
"or", (ftnlen)45);
svout_(&debug_1.logfil, &c__1, &h__[*kev + 1 + h_dim1], &
debug_1.ndigit, "_sapps: betak of the updated residual vector"
, (ftnlen)44);
svout_(&debug_1.logfil, kev, &h__[(h_dim1 << 1) + 1], &debug_1.ndigit,
"_sapps: updated main diagonal of H for next iteration", (
ftnlen)53);
if (*kev > 1) {
i__1 = *kev - 1;
svout_(&debug_1.logfil, &i__1, &h__[h_dim1 + 2], &debug_1.ndigit,
"_sapps: updated sub diagonal of H for next iteration", (
ftnlen)52);
}
}
arscnd_(&t1);
timing_1.tsapps += t1 - t0;
L9000:
return 0;
/* %---------------% */
/* | End of ssapps | */
/* %---------------% */
} /* ssapps_ */
/* Subroutine */ int stgsja_(char *jobu, char *jobv, char *jobq, integer *m,
integer *p, integer *n, integer *k, integer *l, real *a, integer *lda,
real *b, integer *ldb, real *tola, real *tolb, real *alpha, real *
beta, real *u, integer *ldu, real *v, integer *ldv, real *q, integer *
ldq, real *work, integer *ncycle, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1,
u_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4;
real r__1;
/* Local variables */
integer i__, j;
real a1, a2, a3, b1, b2, b3, csq, csu, csv, snq, rwk, snu, snv;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
real gamma;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
logical initq, initu, initv, wantq, upper;
real error, ssmin;
logical wantu, wantv;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slags2_(logical *, real *, real *, real *, real *,
real *, real *, real *, real *, real *, real *, real *, real *);
integer kcycle;
extern /* Subroutine */ int xerbla_(char *, integer *), slapll_(
integer *, real *, integer *, real *, integer *, real *), slartg_(
real *, real *, real *, real *, real *), slaset_(char *, integer *
, integer *, real *, real *, real *, integer *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* STGSJA computes the generalized singular value decomposition (GSVD) */
/* of two real upper triangular (or trapezoidal) matrices A and B. */
/* On entry, it is assumed that matrices A and B have the following */
/* forms, which may be obtained by the preprocessing subroutine SGGSVP */
/* from a general M-by-N matrix A and P-by-N matrix B: */
/* N-K-L K L */
/* A = K ( 0 A12 A13 ) if M-K-L >= 0; */
/* L ( 0 0 A23 ) */
/* M-K-L ( 0 0 0 ) */
/* N-K-L K L */
/* A = K ( 0 A12 A13 ) if M-K-L < 0; */
/* M-K ( 0 0 A23 ) */
/* N-K-L K L */
/* B = L ( 0 0 B13 ) */
/* P-L ( 0 0 0 ) */
/* where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular */
/* upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, */
/* otherwise A23 is (M-K)-by-L upper trapezoidal. */
/* On exit, */
/* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ), */
/* where U, V and Q are orthogonal matrices, Z' denotes the transpose */
/* of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are */
/* ``diagonal'' matrices, which are of the following structures: */
/* If M-K-L >= 0, */
/* K L */
/* D1 = K ( I 0 ) */
/* L ( 0 C ) */
/* M-K-L ( 0 0 ) */
/* K L */
/* D2 = L ( 0 S ) */
/* P-L ( 0 0 ) */
/* N-K-L K L */
/* ( 0 R ) = K ( 0 R11 R12 ) K */
/* L ( 0 0 R22 ) L */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */
/* S = diag( BETA(K+1), ... , BETA(K+L) ), */
/* C**2 + S**2 = I. */
/* R is stored in A(1:K+L,N-K-L+1:N) on exit. */
/* If M-K-L < 0, */
/* K M-K K+L-M */
/* D1 = K ( I 0 0 ) */
/* M-K ( 0 C 0 ) */
/* K M-K K+L-M */
/* D2 = M-K ( 0 S 0 ) */
/* K+L-M ( 0 0 I ) */
/* P-L ( 0 0 0 ) */
/* N-K-L K M-K K+L-M */
/* ( 0 R ) = K ( 0 R11 R12 R13 ) */
/* M-K ( 0 0 R22 R23 ) */
/* K+L-M ( 0 0 0 R33 ) */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */
/* S = diag( BETA(K+1), ... , BETA(M) ), */
/* C**2 + S**2 = I. */
/* R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored */
/* ( 0 R22 R23 ) */
/* in B(M-K+1:L,N+M-K-L+1:N) on exit. */
/* The computation of the orthogonal transformation matrices U, V or Q */
/* is optional. These matrices may either be formed explicitly, or they */
/* may be postmultiplied into input matrices U1, V1, or Q1. */
/* Arguments */
/* ========= */
/* JOBU (input) CHARACTER*1 */
/* = 'U': U must contain an orthogonal matrix U1 on entry, and */
/* the product U1*U is returned; */
/* = 'I': U is initialized to the unit matrix, and the */
/* orthogonal matrix U is returned; */
/* = 'N': U is not computed. */
/* JOBV (input) CHARACTER*1 */
/* = 'V': V must contain an orthogonal matrix V1 on entry, and */
/* the product V1*V is returned; */
/* = 'I': V is initialized to the unit matrix, and the */
/* orthogonal matrix V is returned; */
/* = 'N': V is not computed. */
/* JOBQ (input) CHARACTER*1 */
/* = 'Q': Q must contain an orthogonal matrix Q1 on entry, and */
/* the product Q1*Q is returned; */
/* = 'I': Q is initialized to the unit matrix, and the */
/* orthogonal matrix Q is returned; */
/* = 'N': Q is not computed. */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* P (input) INTEGER */
/* The number of rows of the matrix B. P >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrices A and B. N >= 0. */
/* K (input) INTEGER */
/* L (input) INTEGER */
/* K and L specify the subblocks in the input matrices A and B: */
/* A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) */
/* of A and B, whose GSVD is going to be computed by STGSJA. */
/* See Further details. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular */
/* matrix R or part of R. See Purpose for details. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* B (input/output) REAL array, dimension (LDB,N) */
/* On entry, the P-by-N matrix B. */
/* On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains */
/* a part of R. See Purpose for details. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,P). */
/* TOLA (input) REAL */
/* TOLB (input) REAL */
/* TOLA and TOLB are the convergence criteria for the Jacobi- */
/* Kogbetliantz iteration procedure. Generally, they are the */
/* same as used in the preprocessing step, say */
/* TOLA = max(M,N)*norm(A)*MACHEPS, */
/* TOLB = max(P,N)*norm(B)*MACHEPS. */
/* ALPHA (output) REAL array, dimension (N) */
/* BETA (output) REAL array, dimension (N) */
/* On exit, ALPHA and BETA contain the generalized singular */
/* value pairs of A and B; */
/* ALPHA(1:K) = 1, */
/* BETA(1:K) = 0, */
/* and if M-K-L >= 0, */
/* ALPHA(K+1:K+L) = diag(C), */
/* BETA(K+1:K+L) = diag(S), */
/* or if M-K-L < 0, */
/* ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0 */
/* BETA(K+1:M) = S, BETA(M+1:K+L) = 1. */
/* Furthermore, if K+L < N, */
/* ALPHA(K+L+1:N) = 0 and */
/* BETA(K+L+1:N) = 0. */
/* U (input/output) REAL array, dimension (LDU,M) */
/* On entry, if JOBU = 'U', U must contain a matrix U1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBU = 'I', U contains the orthogonal matrix U; */
/* if JOBU = 'U', U contains the product U1*U. */
/* If JOBU = 'N', U is not referenced. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= max(1,M) if */
/* JOBU = 'U'; LDU >= 1 otherwise. */
/* V (input/output) REAL array, dimension (LDV,P) */
/* On entry, if JOBV = 'V', V must contain a matrix V1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBV = 'I', V contains the orthogonal matrix V; */
/* if JOBV = 'V', V contains the product V1*V. */
/* If JOBV = 'N', V is not referenced. */
/* LDV (input) INTEGER */
/* The leading dimension of the array V. LDV >= max(1,P) if */
/* JOBV = 'V'; LDV >= 1 otherwise. */
/* Q (input/output) REAL array, dimension (LDQ,N) */
/* On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually */
/* the orthogonal matrix returned by SGGSVP). */
/* On exit, */
/* if JOBQ = 'I', Q contains the orthogonal matrix Q; */
/* if JOBQ = 'Q', Q contains the product Q1*Q. */
/* If JOBQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= max(1,N) if */
/* JOBQ = 'Q'; LDQ >= 1 otherwise. */
/* WORK (workspace) REAL array, dimension (2*N) */
/* NCYCLE (output) INTEGER */
/* The number of cycles required for convergence. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* = 1: the procedure does not converge after MAXIT cycles. */
/* Internal Parameters */
/* =================== */
/* MAXIT INTEGER */
/* MAXIT specifies the total loops that the iterative procedure */
/* may take. If after MAXIT cycles, the routine fails to */
/* converge, we return INFO = 1. */
/* Further Details */
/* =============== */
/* STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce */
/* min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L */
/* matrix B13 to the form: */
/* U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1, */
/* where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose */
/* of Z. C1 and S1 are diagonal matrices satisfying */
/* C1**2 + S1**2 = I, */
/* and R1 is an L-by-L nonsingular upper triangular matrix. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--work;
/* Function Body */
initu = lsame_(jobu, "I");
wantu = initu || lsame_(jobu, "U");
initv = lsame_(jobv, "I");
wantv = initv || lsame_(jobv, "V");
initq = lsame_(jobq, "I");
wantq = initq || lsame_(jobq, "Q");
*info = 0;
if (! (initu || wantu || lsame_(jobu, "N"))) {
*info = -1;
} else if (! (initv || wantv || lsame_(jobv, "N")))
{
*info = -2;
} else if (! (initq || wantq || lsame_(jobq, "N")))
{
*info = -3;
} else if (*m < 0) {
*info = -4;
} else if (*p < 0) {
*info = -5;
} else if (*n < 0) {
*info = -6;
} else if (*lda < max(1,*m)) {
*info = -10;
} else if (*ldb < max(1,*p)) {
*info = -12;
} else if (*ldu < 1 || wantu && *ldu < *m) {
*info = -18;
} else if (*ldv < 1 || wantv && *ldv < *p) {
*info = -20;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -22;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("STGSJA", &i__1);
return 0;
}
/* Initialize U, V and Q, if necessary */
if (initu) {
slaset_("Full", m, m, &c_b13, &c_b14, &u[u_offset], ldu);
}
if (initv) {
slaset_("Full", p, p, &c_b13, &c_b14, &v[v_offset], ldv);
}
if (initq) {
slaset_("Full", n, n, &c_b13, &c_b14, &q[q_offset], ldq);
}
/* Loop until convergence */
upper = FALSE_;
for (kcycle = 1; kcycle <= 40; ++kcycle) {
upper = ! upper;
i__1 = *l - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *l;
for (j = i__ + 1; j <= i__2; ++j) {
a1 = 0.f;
a2 = 0.f;
a3 = 0.f;
if (*k + i__ <= *m) {
a1 = a[*k + i__ + (*n - *l + i__) * a_dim1];
}
if (*k + j <= *m) {
a3 = a[*k + j + (*n - *l + j) * a_dim1];
}
b1 = b[i__ + (*n - *l + i__) * b_dim1];
b3 = b[j + (*n - *l + j) * b_dim1];
if (upper) {
if (*k + i__ <= *m) {
a2 = a[*k + i__ + (*n - *l + j) * a_dim1];
}
b2 = b[i__ + (*n - *l + j) * b_dim1];
} else {
if (*k + j <= *m) {
a2 = a[*k + j + (*n - *l + i__) * a_dim1];
}
b2 = b[j + (*n - *l + i__) * b_dim1];
}
slags2_(&upper, &a1, &a2, &a3, &b1, &b2, &b3, &csu, &snu, &
csv, &snv, &csq, &snq);
/* Update (K+I)-th and (K+J)-th rows of matrix A: U'*A */
if (*k + j <= *m) {
srot_(l, &a[*k + j + (*n - *l + 1) * a_dim1], lda, &a[*k
+ i__ + (*n - *l + 1) * a_dim1], lda, &csu, &snu);
}
/* Update I-th and J-th rows of matrix B: V'*B */
srot_(l, &b[j + (*n - *l + 1) * b_dim1], ldb, &b[i__ + (*n - *
l + 1) * b_dim1], ldb, &csv, &snv);
/* Update (N-L+I)-th and (N-L+J)-th columns of matrices */
/* A and B: A*Q and B*Q */
/* Computing MIN */
i__4 = *k + *l;
i__3 = min(i__4,*m);
srot_(&i__3, &a[(*n - *l + j) * a_dim1 + 1], &c__1, &a[(*n - *
l + i__) * a_dim1 + 1], &c__1, &csq, &snq);
srot_(l, &b[(*n - *l + j) * b_dim1 + 1], &c__1, &b[(*n - *l +
i__) * b_dim1 + 1], &c__1, &csq, &snq);
if (upper) {
if (*k + i__ <= *m) {
a[*k + i__ + (*n - *l + j) * a_dim1] = 0.f;
}
b[i__ + (*n - *l + j) * b_dim1] = 0.f;
} else {
if (*k + j <= *m) {
a[*k + j + (*n - *l + i__) * a_dim1] = 0.f;
}
b[j + (*n - *l + i__) * b_dim1] = 0.f;
}
/* Update orthogonal matrices U, V, Q, if desired. */
if (wantu && *k + j <= *m) {
srot_(m, &u[(*k + j) * u_dim1 + 1], &c__1, &u[(*k + i__) *
u_dim1 + 1], &c__1, &csu, &snu);
}
if (wantv) {
srot_(p, &v[j * v_dim1 + 1], &c__1, &v[i__ * v_dim1 + 1],
&c__1, &csv, &snv);
}
if (wantq) {
srot_(n, &q[(*n - *l + j) * q_dim1 + 1], &c__1, &q[(*n - *
l + i__) * q_dim1 + 1], &c__1, &csq, &snq);
}
/* L10: */
}
/* L20: */
}
if (! upper) {
/* The matrices A13 and B13 were lower triangular at the start */
/* of the cycle, and are now upper triangular. */
/* Convergence test: test the parallelism of the corresponding */
/* rows of A and B. */
error = 0.f;
/* Computing MIN */
i__2 = *l, i__3 = *m - *k;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *l - i__ + 1;
scopy_(&i__2, &a[*k + i__ + (*n - *l + i__) * a_dim1], lda, &
work[1], &c__1);
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &work[*
l + 1], &c__1);
i__2 = *l - i__ + 1;
slapll_(&i__2, &work[1], &c__1, &work[*l + 1], &c__1, &ssmin);
error = dmax(error,ssmin);
/* L30: */
}
if (dabs(error) <= dmin(*tola,*tolb)) {
goto L50;
}
}
/* End of cycle loop */
/* L40: */
}
/* The algorithm has not converged after MAXIT cycles. */
*info = 1;
goto L100;
L50:
/* If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged. */
/* Compute the generalized singular value pairs (ALPHA, BETA), and */
/* set the triangular matrix R to array A. */
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
alpha[i__] = 1.f;
beta[i__] = 0.f;
/* L60: */
}
/* Computing MIN */
i__2 = *l, i__3 = *m - *k;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
a1 = a[*k + i__ + (*n - *l + i__) * a_dim1];
b1 = b[i__ + (*n - *l + i__) * b_dim1];
if (a1 != 0.f) {
gamma = b1 / a1;
/* change sign if necessary */
if (gamma < 0.f) {
i__2 = *l - i__ + 1;
sscal_(&i__2, &c_b43, &b[i__ + (*n - *l + i__) * b_dim1], ldb)
;
if (wantv) {
sscal_(p, &c_b43, &v[i__ * v_dim1 + 1], &c__1);
}
}
r__1 = dabs(gamma);
slartg_(&r__1, &c_b14, &beta[*k + i__], &alpha[*k + i__], &rwk);
if (alpha[*k + i__] >= beta[*k + i__]) {
i__2 = *l - i__ + 1;
r__1 = 1.f / alpha[*k + i__];
sscal_(&i__2, &r__1, &a[*k + i__ + (*n - *l + i__) * a_dim1],
lda);
} else {
i__2 = *l - i__ + 1;
r__1 = 1.f / beta[*k + i__];
sscal_(&i__2, &r__1, &b[i__ + (*n - *l + i__) * b_dim1], ldb);
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k
+ i__ + (*n - *l + i__) * a_dim1], lda);
}
} else {
alpha[*k + i__] = 0.f;
beta[*k + i__] = 1.f;
i__2 = *l - i__ + 1;
scopy_(&i__2, &b[i__ + (*n - *l + i__) * b_dim1], ldb, &a[*k +
i__ + (*n - *l + i__) * a_dim1], lda);
}
/* L70: */
}
/* Post-assignment */
i__1 = *k + *l;
for (i__ = *m + 1; i__ <= i__1; ++i__) {
alpha[i__] = 0.f;
beta[i__] = 1.f;
/* L80: */
}
if (*k + *l < *n) {
i__1 = *n;
for (i__ = *k + *l + 1; i__ <= i__1; ++i__) {
alpha[i__] = 0.f;
beta[i__] = 0.f;
/* L90: */
}
}
L100:
*ncycle = kcycle;
return 0;
/* End of STGSJA */
} /* stgsja_ */
/* Subroutine */ int stgex2_(logical *wantq, logical *wantz, integer *n, real
*a, integer *lda, real *b, integer *ldb, real *q, integer *ldq, real *
z__, integer *ldz, integer *j1, integer *n1, integer *n2, real *work,
integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
real f, g;
integer i__, m;
real s[16] /* was [4][4] */, t[16] /* was [4][4] */, be[2], ai[2], ar[2],
sa, sb, li[16] /* was [4][4] */, ir[16] /* was [4][4]
*/, ss, ws, eps;
logical weak;
real ddum;
integer idum;
real taul[4], dsum, taur[4], scpy[16] /* was [4][4] */, tcpy[16]
/* was [4][4] */;
extern /* Subroutine */ int srot_(integer *, real *, integer *, real *,
integer *, real *, real *);
real scale, bqra21, brqa21;
extern /* Subroutine */ int sscal_(integer *, real *, real *, integer *);
real licop[16] /* was [4][4] */;
integer linfo;
extern /* Subroutine */ int sgemm_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, real *, integer *, real *,
real *, integer *);
real ircop[16] /* was [4][4] */, dnorm;
integer iwork[4];
extern /* Subroutine */ int slagv2_(real *, integer *, real *, integer *,
real *, real *, real *, real *, real *, real *, real *), sgeqr2_(
integer *, integer *, real *, integer *, real *, real *, integer *
), sgerq2_(integer *, integer *, real *, integer *, real *, real *
, integer *), sorg2r_(integer *, integer *, integer *, real *,
integer *, real *, real *, integer *), sorgr2_(integer *, integer
*, integer *, real *, integer *, real *, real *, integer *),
sorm2r_(char *, char *, integer *, integer *, integer *, real *,
integer *, real *, real *, integer *, real *, integer *), sormr2_(char *, char *, integer *, integer *, integer *,
real *, integer *, real *, real *, integer *, real *, integer *);
real dscale;
extern /* Subroutine */ int stgsy2_(char *, integer *, integer *, integer
*, real *, integer *, real *, integer *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *, real *, real *,
real *, integer *, integer *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *,
integer *, real *, integer *), slartg_(real *, real *,
real *, real *, real *);
real thresh;
extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
real *, real *, integer *), slassq_(integer *, real *,
integer *, real *, real *);
real smlnum;
logical strong;
/* -- LAPACK auxiliary routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* STGEX2 swaps adjacent diagonal blocks (A11, B11) and (A22, B22) */
/* of size 1-by-1 or 2-by-2 in an upper (quasi) triangular matrix pair */
/* (A, B) by an orthogonal equivalence transformation. */
/* (A, B) must be in generalized real Schur canonical form (as returned */
/* by SGGES), i.e. A is block upper triangular with 1-by-1 and 2-by-2 */
/* diagonal blocks. B is upper triangular. */
/* Optionally, the matrices Q and Z of generalized Schur vectors are */
/* updated. */
/* Q(in) * A(in) * Z(in)' = Q(out) * A(out) * Z(out)' */
/* Q(in) * B(in) * Z(in)' = Q(out) * B(out) * Z(out)' */
/* Arguments */
/* ========= */
/* WANTQ (input) LOGICAL */
/* .TRUE. : update the left transformation matrix Q; */
/* .FALSE.: do not update Q. */
/* WANTZ (input) LOGICAL */
/* .TRUE. : update the right transformation matrix Z; */
/* .FALSE.: do not update Z. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* A (input/output) REAL arrays, dimensions (LDA,N) */
/* On entry, the matrix A in the pair (A, B). */
/* On exit, the updated matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,N). */
/* B (input/output) REAL arrays, dimensions (LDB,N) */
/* On entry, the matrix B in the pair (A, B). */
/* On exit, the updated matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,N). */
/* Q (input/output) REAL array, dimension (LDZ,N) */
/* On entry, if WANTQ = .TRUE., the orthogonal matrix Q. */
/* On exit, the updated matrix Q. */
/* Not referenced if WANTQ = .FALSE.. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= 1. */
/* If WANTQ = .TRUE., LDQ >= N. */
/* Z (input/output) REAL array, dimension (LDZ,N) */
/* On entry, if WANTZ =.TRUE., the orthogonal matrix Z. */
/* On exit, the updated matrix Z. */
/* Not referenced if WANTZ = .FALSE.. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1. */
/* If WANTZ = .TRUE., LDZ >= N. */
/* J1 (input) INTEGER */
/* The index to the first block (A11, B11). 1 <= J1 <= N. */
/* N1 (input) INTEGER */
/* The order of the first block (A11, B11). N1 = 0, 1 or 2. */
/* N2 (input) INTEGER */
/* The order of the second block (A22, B22). N2 = 0, 1 or 2. */
/* WORK (workspace) REAL array, dimension (MAX(1,LWORK)). */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. */
/* LWORK >= MAX( N*(N2+N1), (N2+N1)*(N2+N1)*2 ) */
/* INFO (output) INTEGER */
/* =0: Successful exit */
/* >0: If INFO = 1, the transformed matrix (A, B) would be */
/* too far from generalized Schur form; the blocks are */
/* not swapped and (A, B) and (Q, Z) are unchanged. */
/* The problem of swapping is too ill-conditioned. */
/* <0: If INFO = -16: LWORK is too small. Appropriate value */
/* for LWORK is returned in WORK(1). */
/* Further Details */
/* =============== */
/* Based on contributions by */
/* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
/* Umea University, S-901 87 Umea, Sweden. */
/* In the current code both weak and strong stability tests are */
/* performed. The user can omit the strong stability test by changing */
/* the internal logical parameter WANDS to .FALSE.. See ref. [2] for */
/* details. */
/* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
/* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
/* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
/* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
/* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
/* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
/* Estimation: Theory, Algorithms and Software, */
/* Report UMINF - 94.04, Department of Computing Science, Umea */
/* University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working */
/* Note 87. To appear in Numerical Algorithms, 1996. */
/* ===================================================================== */
/* Replaced various illegal calls to SCOPY by calls to SLASET, or by DO */
/* loops. Sven Hammarling, 1/5/02. */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--work;
/* Function Body */
*info = 0;
/* Quick return if possible */
if (*n <= 1 || *n1 <= 0 || *n2 <= 0) {
return 0;
}
if (*n1 > *n || *j1 + *n1 > *n) {
return 0;
}
m = *n1 + *n2;
/* Computing MAX */
i__1 = *n * m, i__2 = m * m << 1;
if (*lwork < max(i__1,i__2)) {
*info = -16;
/* Computing MAX */
i__1 = *n * m, i__2 = m * m << 1;
work[1] = (real) max(i__1,i__2);
return 0;
}
weak = FALSE_;
strong = FALSE_;
/* Make a local copy of selected block */
slaset_("Full", &c__4, &c__4, &c_b5, &c_b5, li, &c__4);
slaset_("Full", &c__4, &c__4, &c_b5, &c_b5, ir, &c__4);
slacpy_("Full", &m, &m, &a[*j1 + *j1 * a_dim1], lda, s, &c__4);
slacpy_("Full", &m, &m, &b[*j1 + *j1 * b_dim1], ldb, t, &c__4);
/* Compute threshold for testing acceptance of swapping. */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
dscale = 0.f;
dsum = 1.f;
slacpy_("Full", &m, &m, s, &c__4, &work[1], &m);
i__1 = m * m;
slassq_(&i__1, &work[1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, t, &c__4, &work[1], &m);
i__1 = m * m;
slassq_(&i__1, &work[1], &c__1, &dscale, &dsum);
dnorm = dscale * sqrt(dsum);
/* Computing MAX */
r__1 = eps * 10.f * dnorm;
thresh = dmax(r__1,smlnum);
if (m == 2) {
/* CASE 1: Swap 1-by-1 and 1-by-1 blocks. */
/* Compute orthogonal QL and RQ that swap 1-by-1 and 1-by-1 blocks */
/* using Givens rotations and perform the swap tentatively. */
f = s[5] * t[0] - t[5] * s[0];
g = s[5] * t[4] - t[5] * s[4];
sb = dabs(t[5]);
sa = dabs(s[5]);
slartg_(&f, &g, &ir[4], ir, &ddum);
ir[1] = -ir[4];
ir[5] = ir[0];
srot_(&c__2, s, &c__1, &s[4], &c__1, ir, &ir[1]);
srot_(&c__2, t, &c__1, &t[4], &c__1, ir, &ir[1]);
if (sa >= sb) {
slartg_(s, &s[1], li, &li[1], &ddum);
} else {
slartg_(t, &t[1], li, &li[1], &ddum);
}
srot_(&c__2, s, &c__4, &s[1], &c__4, li, &li[1]);
srot_(&c__2, t, &c__4, &t[1], &c__4, li, &li[1]);
li[5] = li[0];
li[4] = -li[1];
/* Weak stability test: */
/* |S21| + |T21| <= O(EPS * F-norm((S, T))) */
ws = dabs(s[1]) + dabs(t[1]);
weak = ws <= thresh;
if (! weak) {
goto L70;
}
if (TRUE_) {
/* Strong stability test: */
/* F-norm((A-QL'*S*QR, B-QL'*T*QR)) <= O(EPS*F-norm((A,B))) */
slacpy_("Full", &m, &m, &a[*j1 + *j1 * a_dim1], lda, &work[m * m
+ 1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b42, li, &c__4, s, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b48, &work[1], &m, ir, &c__4, &
c_b42, &work[m * m + 1], &m);
dscale = 0.f;
dsum = 1.f;
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, &b[*j1 + *j1 * b_dim1], ldb, &work[m * m
+ 1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b42, li, &c__4, t, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b48, &work[1], &m, ir, &c__4, &
c_b42, &work[m * m + 1], &m);
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
ss = dscale * sqrt(dsum);
strong = ss <= thresh;
if (! strong) {
goto L70;
}
}
/* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and */
/* (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). */
i__1 = *j1 + 1;
srot_(&i__1, &a[*j1 * a_dim1 + 1], &c__1, &a[(*j1 + 1) * a_dim1 + 1],
&c__1, ir, &ir[1]);
i__1 = *j1 + 1;
srot_(&i__1, &b[*j1 * b_dim1 + 1], &c__1, &b[(*j1 + 1) * b_dim1 + 1],
&c__1, ir, &ir[1]);
i__1 = *n - *j1 + 1;
srot_(&i__1, &a[*j1 + *j1 * a_dim1], lda, &a[*j1 + 1 + *j1 * a_dim1],
lda, li, &li[1]);
i__1 = *n - *j1 + 1;
srot_(&i__1, &b[*j1 + *j1 * b_dim1], ldb, &b[*j1 + 1 + *j1 * b_dim1],
ldb, li, &li[1]);
/* Set N1-by-N2 (2,1) - blocks to ZERO. */
a[*j1 + 1 + *j1 * a_dim1] = 0.f;
b[*j1 + 1 + *j1 * b_dim1] = 0.f;
/* Accumulate transformations into Q and Z if requested. */
if (*wantz) {
srot_(n, &z__[*j1 * z_dim1 + 1], &c__1, &z__[(*j1 + 1) * z_dim1 +
1], &c__1, ir, &ir[1]);
}
if (*wantq) {
srot_(n, &q[*j1 * q_dim1 + 1], &c__1, &q[(*j1 + 1) * q_dim1 + 1],
&c__1, li, &li[1]);
}
/* Exit with INFO = 0 if swap was successfully performed. */
return 0;
} else {
/* CASE 2: Swap 1-by-1 and 2-by-2 blocks, or 2-by-2 */
/* and 2-by-2 blocks. */
/* Solve the generalized Sylvester equation */
/* S11 * R - L * S22 = SCALE * S12 */
/* T11 * R - L * T22 = SCALE * T12 */
/* for R and L. Solutions in LI and IR. */
slacpy_("Full", n1, n2, &t[(*n1 + 1 << 2) - 4], &c__4, li, &c__4);
slacpy_("Full", n1, n2, &s[(*n1 + 1 << 2) - 4], &c__4, &ir[*n2 + 1 + (
*n1 + 1 << 2) - 5], &c__4);
stgsy2_("N", &c__0, n1, n2, s, &c__4, &s[*n1 + 1 + (*n1 + 1 << 2) - 5]
, &c__4, &ir[*n2 + 1 + (*n1 + 1 << 2) - 5], &c__4, t, &c__4, &
t[*n1 + 1 + (*n1 + 1 << 2) - 5], &c__4, li, &c__4, &scale, &
dsum, &dscale, iwork, &idum, &linfo);
/* Compute orthogonal matrix QL: */
/* QL' * LI = [ TL ] */
/* [ 0 ] */
/* where */
/* LI = [ -L ] */
/* [ SCALE * identity(N2) ] */
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
sscal_(n1, &c_b48, &li[(i__ << 2) - 4], &c__1);
li[*n1 + i__ + (i__ << 2) - 5] = scale;
/* L10: */
}
sgeqr2_(&m, n2, li, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorg2r_(&m, &m, n2, li, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
/* Compute orthogonal matrix RQ: */
/* IR * RQ' = [ 0 TR], */
/* where IR = [ SCALE * identity(N1), R ] */
i__1 = *n1;
for (i__ = 1; i__ <= i__1; ++i__) {
ir[*n2 + i__ + (i__ << 2) - 5] = scale;
/* L20: */
}
sgerq2_(n1, &m, &ir[*n2], &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorgr2_(&m, &m, n1, ir, &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
/* Perform the swapping tentatively: */
sgemm_("T", "N", &m, &m, &m, &c_b42, li, &c__4, s, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b42, &work[1], &m, ir, &c__4, &c_b5,
s, &c__4);
sgemm_("T", "N", &m, &m, &m, &c_b42, li, &c__4, t, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "T", &m, &m, &m, &c_b42, &work[1], &m, ir, &c__4, &c_b5,
t, &c__4);
slacpy_("F", &m, &m, s, &c__4, scpy, &c__4);
slacpy_("F", &m, &m, t, &c__4, tcpy, &c__4);
slacpy_("F", &m, &m, ir, &c__4, ircop, &c__4);
slacpy_("F", &m, &m, li, &c__4, licop, &c__4);
/* Triangularize the B-part by an RQ factorization. */
/* Apply transformation (from left) to A-part, giving S. */
sgerq2_(&m, &m, t, &c__4, taur, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sormr2_("R", "T", &m, &m, &m, t, &c__4, taur, s, &c__4, &work[1], &
linfo);
if (linfo != 0) {
goto L70;
}
sormr2_("L", "N", &m, &m, &m, t, &c__4, taur, ir, &c__4, &work[1], &
linfo);
if (linfo != 0) {
goto L70;
}
/* Compute F-norm(S21) in BRQA21. (T21 is 0.) */
dscale = 0.f;
dsum = 1.f;
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
slassq_(n1, &s[*n2 + 1 + (i__ << 2) - 5], &c__1, &dscale, &dsum);
/* L30: */
}
brqa21 = dscale * sqrt(dsum);
/* Triangularize the B-part by a QR factorization. */
/* Apply transformation (from right) to A-part, giving S. */
sgeqr2_(&m, &m, tcpy, &c__4, taul, &work[1], &linfo);
if (linfo != 0) {
goto L70;
}
sorm2r_("L", "T", &m, &m, &m, tcpy, &c__4, taul, scpy, &c__4, &work[1]
, info);
sorm2r_("R", "N", &m, &m, &m, tcpy, &c__4, taul, licop, &c__4, &work[
1], info);
if (linfo != 0) {
goto L70;
}
/* Compute F-norm(S21) in BQRA21. (T21 is 0.) */
dscale = 0.f;
dsum = 1.f;
i__1 = *n2;
for (i__ = 1; i__ <= i__1; ++i__) {
slassq_(n1, &scpy[*n2 + 1 + (i__ << 2) - 5], &c__1, &dscale, &
dsum);
/* L40: */
}
bqra21 = dscale * sqrt(dsum);
/* Decide which method to use. */
/* Weak stability test: */
/* F-norm(S21) <= O(EPS * F-norm((S, T))) */
if (bqra21 <= brqa21 && bqra21 <= thresh) {
slacpy_("F", &m, &m, scpy, &c__4, s, &c__4);
slacpy_("F", &m, &m, tcpy, &c__4, t, &c__4);
slacpy_("F", &m, &m, ircop, &c__4, ir, &c__4);
slacpy_("F", &m, &m, licop, &c__4, li, &c__4);
} else if (brqa21 >= thresh) {
goto L70;
}
/* Set lower triangle of B-part to zero */
i__1 = m - 1;
i__2 = m - 1;
slaset_("Lower", &i__1, &i__2, &c_b5, &c_b5, &t[1], &c__4);
if (TRUE_) {
/* Strong stability test: */
/* F-norm((A-QL*S*QR', B-QL*T*QR')) <= O(EPS*F-norm((A,B))) */
slacpy_("Full", &m, &m, &a[*j1 + *j1 * a_dim1], lda, &work[m * m
+ 1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b42, li, &c__4, s, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b48, &work[1], &m, ir, &c__4, &
c_b42, &work[m * m + 1], &m);
dscale = 0.f;
dsum = 1.f;
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
slacpy_("Full", &m, &m, &b[*j1 + *j1 * b_dim1], ldb, &work[m * m
+ 1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b42, li, &c__4, t, &c__4, &c_b5, &
work[1], &m);
sgemm_("N", "N", &m, &m, &m, &c_b48, &work[1], &m, ir, &c__4, &
c_b42, &work[m * m + 1], &m);
i__1 = m * m;
slassq_(&i__1, &work[m * m + 1], &c__1, &dscale, &dsum);
ss = dscale * sqrt(dsum);
strong = ss <= thresh;
if (! strong) {
goto L70;
}
}
/* If the swap is accepted ("weakly" and "strongly"), apply the */
/* transformations and set N1-by-N2 (2,1)-block to zero. */
slaset_("Full", n1, n2, &c_b5, &c_b5, &s[*n2], &c__4);
/* copy back M-by-M diagonal block starting at index J1 of (A, B) */
slacpy_("F", &m, &m, s, &c__4, &a[*j1 + *j1 * a_dim1], lda)
;
slacpy_("F", &m, &m, t, &c__4, &b[*j1 + *j1 * b_dim1], ldb)
;
slaset_("Full", &c__4, &c__4, &c_b5, &c_b5, t, &c__4);
/* Standardize existing 2-by-2 blocks. */
i__1 = m * m;
for (i__ = 1; i__ <= i__1; ++i__) {
work[i__] = 0.f;
/* L50: */
}
work[1] = 1.f;
t[0] = 1.f;
idum = *lwork - m * m - 2;
if (*n2 > 1) {
slagv2_(&a[*j1 + *j1 * a_dim1], lda, &b[*j1 + *j1 * b_dim1], ldb,
ar, ai, be, &work[1], &work[2], t, &t[1]);
work[m + 1] = -work[2];
work[m + 2] = work[1];
t[*n2 + (*n2 << 2) - 5] = t[0];
t[4] = -t[1];
}
work[m * m] = 1.f;
t[m + (m << 2) - 5] = 1.f;
if (*n1 > 1) {
slagv2_(&a[*j1 + *n2 + (*j1 + *n2) * a_dim1], lda, &b[*j1 + *n2 +
(*j1 + *n2) * b_dim1], ldb, taur, taul, &work[m * m + 1],
&work[*n2 * m + *n2 + 1], &work[*n2 * m + *n2 + 2], &t[*
n2 + 1 + (*n2 + 1 << 2) - 5], &t[m + (m - 1 << 2) - 5]);
work[m * m] = work[*n2 * m + *n2 + 1];
work[m * m - 1] = -work[*n2 * m + *n2 + 2];
t[m + (m << 2) - 5] = t[*n2 + 1 + (*n2 + 1 << 2) - 5];
t[m - 1 + (m << 2) - 5] = -t[m + (m - 1 << 2) - 5];
}
sgemm_("T", "N", n2, n1, n2, &c_b42, &work[1], &m, &a[*j1 + (*j1 + *
n2) * a_dim1], lda, &c_b5, &work[m * m + 1], n2);
slacpy_("Full", n2, n1, &work[m * m + 1], n2, &a[*j1 + (*j1 + *n2) *
a_dim1], lda);
sgemm_("T", "N", n2, n1, n2, &c_b42, &work[1], &m, &b[*j1 + (*j1 + *
n2) * b_dim1], ldb, &c_b5, &work[m * m + 1], n2);
slacpy_("Full", n2, n1, &work[m * m + 1], n2, &b[*j1 + (*j1 + *n2) *
b_dim1], ldb);
sgemm_("N", "N", &m, &m, &m, &c_b42, li, &c__4, &work[1], &m, &c_b5, &
work[m * m + 1], &m);
slacpy_("Full", &m, &m, &work[m * m + 1], &m, li, &c__4);
sgemm_("N", "N", n2, n1, n1, &c_b42, &a[*j1 + (*j1 + *n2) * a_dim1],
lda, &t[*n2 + 1 + (*n2 + 1 << 2) - 5], &c__4, &c_b5, &work[1],
n2);
slacpy_("Full", n2, n1, &work[1], n2, &a[*j1 + (*j1 + *n2) * a_dim1],
lda);
sgemm_("N", "N", n2, n1, n1, &c_b42, &b[*j1 + (*j1 + *n2) * b_dim1],
ldb, &t[*n2 + 1 + (*n2 + 1 << 2) - 5], &c__4, &c_b5, &work[1],
n2);
slacpy_("Full", n2, n1, &work[1], n2, &b[*j1 + (*j1 + *n2) * b_dim1],
ldb);
sgemm_("T", "N", &m, &m, &m, &c_b42, ir, &c__4, t, &c__4, &c_b5, &
work[1], &m);
slacpy_("Full", &m, &m, &work[1], &m, ir, &c__4);
/* Accumulate transformations into Q and Z if requested. */
if (*wantq) {
sgemm_("N", "N", n, &m, &m, &c_b42, &q[*j1 * q_dim1 + 1], ldq, li,
&c__4, &c_b5, &work[1], n);
slacpy_("Full", n, &m, &work[1], n, &q[*j1 * q_dim1 + 1], ldq);
}
if (*wantz) {
sgemm_("N", "N", n, &m, &m, &c_b42, &z__[*j1 * z_dim1 + 1], ldz,
ir, &c__4, &c_b5, &work[1], n);
slacpy_("Full", n, &m, &work[1], n, &z__[*j1 * z_dim1 + 1], ldz);
}
/* Update (A(J1:J1+M-1, M+J1:N), B(J1:J1+M-1, M+J1:N)) and */
/* (A(1:J1-1, J1:J1+M), B(1:J1-1, J1:J1+M)). */
i__ = *j1 + m;
if (i__ <= *n) {
i__1 = *n - i__ + 1;
sgemm_("T", "N", &m, &i__1, &m, &c_b42, li, &c__4, &a[*j1 + i__ *
a_dim1], lda, &c_b5, &work[1], &m);
i__1 = *n - i__ + 1;
slacpy_("Full", &m, &i__1, &work[1], &m, &a[*j1 + i__ * a_dim1],
lda);
i__1 = *n - i__ + 1;
sgemm_("T", "N", &m, &i__1, &m, &c_b42, li, &c__4, &b[*j1 + i__ *
b_dim1], ldb, &c_b5, &work[1], &m);
i__1 = *n - i__ + 1;
slacpy_("Full", &m, &i__1, &work[1], &m, &b[*j1 + i__ * b_dim1],
ldb);
}
i__ = *j1 - 1;
if (i__ > 0) {
sgemm_("N", "N", &i__, &m, &m, &c_b42, &a[*j1 * a_dim1 + 1], lda,
ir, &c__4, &c_b5, &work[1], &i__);
slacpy_("Full", &i__, &m, &work[1], &i__, &a[*j1 * a_dim1 + 1],
lda);
sgemm_("N", "N", &i__, &m, &m, &c_b42, &b[*j1 * b_dim1 + 1], ldb,
ir, &c__4, &c_b5, &work[1], &i__);
slacpy_("Full", &i__, &m, &work[1], &i__, &b[*j1 * b_dim1 + 1],
ldb);
}
/* Exit with INFO = 0 if swap was successfully performed. */
return 0;
}
/* Exit with INFO = 1 if swap was rejected. */
L70:
*info = 1;
return 0;
/* End of STGEX2 */
} /* stgex2_ */
/* Subroutine */ int sbdsdc_(char *uplo, char *compq, integer *n, real *d__,
real *e, real *u, integer *ldu, real *vt, integer *ldvt, real *q,
integer *iq, real *work, integer *iwork, integer *info)
{
/* System generated locals */
integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
real r__1;
/* Builtin functions */
double r_sign(real *, real *), log(doublereal);
/* Local variables */
static integer difl, difr, ierr, perm, mlvl, sqre, i__, j, k;
static real p, r__;
static integer z__;
extern logical lsame_(char *, char *);
static integer poles;
extern /* Subroutine */ int slasr_(char *, char *, char *, integer *,
integer *, real *, real *, real *, integer *);
static integer iuplo, nsize, start;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), sswap_(integer *, real *, integer *, real *, integer *
), slasd0_(integer *, integer *, real *, real *, real *, integer *
, real *, integer *, integer *, integer *, real *, integer *);
static integer ic, ii, kk;
static real cs;
static integer is, iu;
static real sn;
extern doublereal slamch_(char *);
extern /* Subroutine */ int slasda_(integer *, integer *, integer *,
integer *, real *, real *, real *, integer *, real *, integer *,
real *, real *, real *, real *, integer *, integer *, integer *,
integer *, real *, real *, real *, real *, integer *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *);
static integer givcol;
extern /* Subroutine */ int slasdq_(char *, integer *, integer *, integer
*, integer *, integer *, real *, real *, real *, integer *, real *
, integer *, real *, integer *, real *, integer *);
static integer icompq;
extern /* Subroutine */ int slaset_(char *, integer *, integer *, real *,
real *, real *, integer *), slartg_(real *, real *, real *
, real *, real *);
static real orgnrm;
static integer givnum;
extern doublereal slanst_(char *, integer *, real *, real *);
static integer givptr, nm1, qstart, smlsiz, wstart, smlszp;
static real eps;
static integer ivt;
#define u_ref(a_1,a_2) u[(a_2)*u_dim1 + a_1]
#define vt_ref(a_1,a_2) vt[(a_2)*vt_dim1 + a_1]
/* -- LAPACK routine (instrumented to count ops, version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1999
Purpose
=======
SBDSDC computes the singular value decomposition (SVD) of a real
N-by-N (upper or lower) bidiagonal matrix B: B = U * S * VT,
using a divide and conquer method, where S is a diagonal matrix
with non-negative diagonal elements (the singular values of B), and
U and VT are orthogonal matrices of left and right singular vectors,
respectively. SBDSDC can be used to compute all singular values,
and optionally, singular vectors or singular vectors in compact form.
This code makes very mild assumptions about floating point
arithmetic. It will work on machines with a guard digit in
add/subtract, or on those binary machines without guard digits
which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
It could conceivably fail on hexadecimal or decimal machines
without guard digits, but we know of none. See SLASD3 for details.
The code currently call SLASDQ if singular values only are desired.
However, it can be slightly modified to compute singular values
using the divide and conquer method.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal.
= 'L': B is lower bidiagonal.
COMPQ (input) CHARACTER*1
Specifies whether singular vectors are to be computed
as follows:
= 'N': Compute singular values only;
= 'P': Compute singular values and compute singular
vectors in compact form;
= 'I': Compute singular values and singular vectors.
N (input) INTEGER
The order of the matrix B. N >= 0.
D (input/output) REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B.
On exit, if INFO=0, the singular values of B.
E (input/output) REAL array, dimension (N)
On entry, the elements of E contain the offdiagonal
elements of the bidiagonal matrix whose SVD is desired.
On exit, E has been destroyed.
U (output) REAL array, dimension (LDU,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, U contains the left singular vectors
of the bidiagonal matrix.
For other values of COMPQ, U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1.
If singular vectors are desired, then LDU >= max( 1, N ).
VT (output) REAL array, dimension (LDVT,N)
If COMPQ = 'I', then:
On exit, if INFO = 0, VT' contains the right singular
vectors of the bidiagonal matrix.
For other values of COMPQ, VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1.
If singular vectors are desired, then LDVT >= max( 1, N ).
Q (output) REAL array, dimension (LDQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, Q contains all the REAL data in
LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, Q is not referenced.
IQ (output) INTEGER array, dimension (LDIQ)
If COMPQ = 'P', then:
On exit, if INFO = 0, Q and IQ contain the left
and right singular vectors in a compact form,
requiring O(N log N) space instead of 2*N**2.
In particular, IQ contains all INTEGER data in
LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
words of memory, where SMLSIZ is returned by ILAENV and
is equal to the maximum size of the subproblems at the
bottom of the computation tree (usually about 25).
For other values of COMPQ, IQ is not referenced.
WORK (workspace) REAL array, dimension (LWORK)
If COMPQ = 'N' then LWORK >= (4 * N).
If COMPQ = 'P' then LWORK >= (6 * N).
If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
IWORK (workspace) INTEGER array, dimension (7*N)
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: The algorithm failed to compute an singular value.
The update process of divide and conquer failed.
Further Details
===============
Based on contributions by
Ming Gu and Huan Ren, Computer Science Division, University of
California at Berkeley, USA
=====================================================================
Test the input parameters.
Parameter adjustments */
--d__;
--e;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
vt_dim1 = *ldvt;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
--q;
--iq;
--work;
--iwork;
/* Function Body */
*info = 0;
iuplo = 0;
if (lsame_(uplo, "U")) {
iuplo = 1;
}
if (lsame_(uplo, "L")) {
iuplo = 2;
}
if (lsame_(compq, "N")) {
icompq = 0;
} else if (lsame_(compq, "P")) {
icompq = 1;
} else if (lsame_(compq, "I")) {
icompq = 2;
} else {
icompq = -1;
}
if (iuplo == 0) {
*info = -1;
} else if (icompq < 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*ldu < 1 || icompq == 2 && *ldu < *n) {
*info = -7;
} else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SBDSDC", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
smlsiz = ilaenv_(&c__9, "SBDSDC", " ", &c__0, &c__0, &c__0, &c__0, (
ftnlen)6, (ftnlen)1);
if (*n == 1) {
if (icompq == 1) {
q[1] = r_sign(&c_b15, &d__[1]);
q[smlsiz * *n + 1] = 1.f;
} else if (icompq == 2) {
u_ref(1, 1) = r_sign(&c_b15, &d__[1]);
vt_ref(1, 1) = 1.f;
}
d__[1] = dabs(d__[1]);
return 0;
}
nm1 = *n - 1;
/* If matrix lower bidiagonal, rotate to be upper bidiagonal
by applying Givens rotations on the left */
wstart = 1;
qstart = 3;
if (icompq == 1) {
scopy_(n, &d__[1], &c__1, &q[1], &c__1);
i__1 = *n - 1;
scopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1);
}
if (iuplo == 2) {
qstart = 5;
wstart = (*n << 1) - 1;
latime_1.ops += (real) (*n - 1 << 3);
i__1 = *n - 1;
for (i__ = 1; i__ <= i__1; ++i__) {
slartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
d__[i__] = r__;
e[i__] = sn * d__[i__ + 1];
d__[i__ + 1] = cs * d__[i__ + 1];
if (icompq == 1) {
q[i__ + (*n << 1)] = cs;
q[i__ + *n * 3] = sn;
} else if (icompq == 2) {
work[i__] = cs;
work[nm1 + i__] = -sn;
}
/* L10: */
}
}
/* If ICOMPQ = 0, use SLASDQ to compute the singular values. */
if (icompq == 0) {
slasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
goto L40;
}
/* If N is smaller than the minimum divide size SMLSIZ, then solve
the problem with another solver. */
if (*n <= smlsiz) {
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
wstart], info);
} else if (icompq == 1) {
iu = 1;
ivt = iu + *n;
slaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n);
slaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n);
slasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + (
qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[
iu + (qstart - 1) * *n], n, &work[wstart], info);
}
goto L40;
}
if (icompq == 2) {
slaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
slaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
}
/* Scale. */
orgnrm = slanst_("M", n, &d__[1], &e[1]);
if (orgnrm == 0.f) {
return 0;
}
latime_1.ops += (real) (*n + nm1);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr);
slascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, &
ierr);
eps = slamch_("Epsilon");
mlvl = (integer) (log((real) (*n) / (real) (smlsiz + 1)) / log(2.f)) + 1;
smlszp = smlsiz + 1;
if (icompq == 1) {
iu = 1;
ivt = smlsiz + 1;
difl = ivt + smlszp;
difr = difl + mlvl;
z__ = difr + (mlvl << 1);
ic = z__ + mlvl;
is = ic + 1;
poles = is + 1;
givnum = poles + (mlvl << 1);
k = 1;
givptr = 2;
perm = 3;
givcol = perm + mlvl;
}
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = d__[i__], dabs(r__1)) < eps) {
d__[i__] = r_sign(&eps, &d__[i__]);
}
/* L20: */
}
start = 1;
sqre = 0;
i__1 = nm1;
for (i__ = 1; i__ <= i__1; ++i__) {
if ((r__1 = e[i__], dabs(r__1)) < eps || i__ == nm1) {
/* Subproblem found. First determine its size and then
apply divide and conquer on it. */
if (i__ < nm1) {
/* A subproblem with E(I) small for I < NM1. */
nsize = i__ - start + 1;
} else if ((r__1 = e[i__], dabs(r__1)) >= eps) {
/* A subproblem with E(NM1) not too small but I = NM1. */
nsize = *n - start + 1;
} else {
/* A subproblem with E(NM1) small. This implies an
1-by-1 subproblem at D(N). Solve this 1-by-1 problem
first. */
nsize = i__ - start + 1;
if (icompq == 2) {
u_ref(*n, *n) = r_sign(&c_b15, &d__[*n]);
vt_ref(*n, *n) = 1.f;
} else if (icompq == 1) {
q[*n + (qstart - 1) * *n] = r_sign(&c_b15, &d__[*n]);
q[*n + (smlsiz + qstart - 1) * *n] = 1.f;
}
d__[*n] = (r__1 = d__[*n], dabs(r__1));
}
if (icompq == 2) {
slasd0_(&nsize, &sqre, &d__[start], &e[start], &u_ref(start,
start), ldu, &vt_ref(start, start), ldvt, &smlsiz, &
iwork[1], &work[wstart], info);
} else {
slasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[
start], &q[start + (iu + qstart - 2) * *n], n, &q[
start + (ivt + qstart - 2) * *n], &iq[start + k * *n],
&q[start + (difl + qstart - 2) * *n], &q[start + (
difr + qstart - 2) * *n], &q[start + (z__ + qstart -
2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[
start + givptr * *n], &iq[start + givcol * *n], n, &
iq[start + perm * *n], &q[start + (givnum + qstart -
2) * *n], &q[start + (ic + qstart - 2) * *n], &q[
start + (is + qstart - 2) * *n], &work[wstart], &
iwork[1], info);
if (*info != 0) {
return 0;
}
}
start = i__ + 1;
}
/* L30: */
}
/* Unscale */
latime_1.ops += (real) (*n);
slascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr);
L40:
/* Use Selection Sort to minimize swaps of singular vectors */
i__1 = *n;
for (ii = 2; ii <= i__1; ++ii) {
i__ = ii - 1;
kk = i__;
p = d__[i__];
i__2 = *n;
for (j = ii; j <= i__2; ++j) {
if (d__[j] > p) {
kk = j;
p = d__[j];
}
/* L50: */
}
if (kk != i__) {
d__[kk] = d__[i__];
d__[i__] = p;
if (icompq == 1) {
iq[i__] = kk;
} else if (icompq == 2) {
sswap_(n, &u_ref(1, i__), &c__1, &u_ref(1, kk), &c__1);
sswap_(n, &vt_ref(i__, 1), ldvt, &vt_ref(kk, 1), ldvt);
}
} else if (icompq == 1) {
iq[i__] = i__;
}
/* L60: */
}
/* If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */
if (icompq == 1) {
if (iuplo == 1) {
iq[*n] = 1;
} else {
iq[*n] = 0;
}
}
/* If B is lower bidiagonal, update U by those Givens rotations
which rotated B to be upper bidiagonal */
if (iuplo == 2 && icompq == 2) {
latime_1.ops += (real) ((*n - 1) * 6 * *n);
slasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu);
}
return 0;
/* End of SBDSDC */
} /* sbdsdc_ */
