/* 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 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 sorgrq_(integer *m, integer *n, integer *k, real *a,
integer *lda, real *tau, real *work, integer *lwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
integer i__, j, l, ib, nb, ii, kk, nx, iws, nbmin, iinfo;
extern /* Subroutine */ int sorgr2_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *), slarfb_(char *, char *,
char *, char *, integer *, integer *, integer *, real *, integer *
, real *, integer *, real *, integer *, real *, integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
extern /* Subroutine */ int slarft_(char *, char *, integer *, integer *,
real *, integer *, real *, real *, integer *);
integer ldwork, lwkopt;
logical lquery;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SORGRQ generates an M-by-N real matrix Q with orthonormal rows, */
/* which is defined as the last M rows of a product of K elementary */
/* reflectors of order N */
/* Q = H(1) H(2) . . . H(k) */
/* as returned by SGERQF. */
/* Arguments */
/* ========= */
/* M (input) INTEGER */
/* The number of rows of the matrix Q. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrix Q. N >= M. */
/* K (input) INTEGER */
/* The number of elementary reflectors whose product defines the */
/* matrix Q. M >= K >= 0. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the (m-k+i)-th row must contain the vector which */
/* defines the elementary reflector H(i), for i = 1,2,...,k, as */
/* returned by SGERQF in the last k rows of its array argument */
/* A. */
/* On exit, the M-by-N matrix Q. */
/* LDA (input) INTEGER */
/* The first dimension of the array A. LDA >= max(1,M). */
/* TAU (input) REAL array, dimension (K) */
/* TAU(i) must contain the scalar factor of the elementary */
/* reflector H(i), as returned by SGERQF. */
/* 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,M). */
/* For optimum performance LWORK >= M*NB, where NB is the */
/* optimal blocksize. */
/* 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 has an illegal value */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < *m) {
*info = -2;
} else if (*k < 0 || *k > *m) {
*info = -3;
} else if (*lda < max(1,*m)) {
*info = -5;
}
if (*info == 0) {
if (*m <= 0) {
lwkopt = 1;
} else {
nb = ilaenv_(&c__1, "SORGRQ", " ", m, n, k, &c_n1);
lwkopt = *m * nb;
}
work[1] = (real) lwkopt;
if (*lwork < max(1,*m) && ! lquery) {
*info = -8;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SORGRQ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*m <= 0) {
return 0;
}
nbmin = 2;
nx = 0;
iws = *m;
if (nb > 1 && nb < *k) {
/* Determine when to cross over from blocked to unblocked code. */
/* Computing MAX */
i__1 = 0, i__2 = ilaenv_(&c__3, "SORGRQ", " ", m, n, k, &c_n1);
nx = max(i__1,i__2);
if (nx < *k) {
/* Determine if workspace is large enough for blocked code. */
ldwork = *m;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: reduce NB and */
/* determine the minimum value of NB. */
nb = *lwork / ldwork;
/* Computing MAX */
i__1 = 2, i__2 = ilaenv_(&c__2, "SORGRQ", " ", m, n, k, &c_n1);
nbmin = max(i__1,i__2);
}
}
}
if (nb >= nbmin && nb < *k && nx < *k) {
/* Use blocked code after the first block. */
/* The last kk rows are handled by the block method. */
/* Computing MIN */
i__1 = *k, i__2 = (*k - nx + nb - 1) / nb * nb;
kk = min(i__1,i__2);
/* Set A(1:m-kk,n-kk+1:n) to zero. */
i__1 = *n;
for (j = *n - kk + 1; j <= i__1; ++j) {
i__2 = *m - kk;
for (i__ = 1; i__ <= i__2; ++i__) {
a[i__ + j * a_dim1] = 0.f;
/* L10: */
}
/* L20: */
}
} else {
kk = 0;
}
/* Use unblocked code for the first or only block. */
i__1 = *m - kk;
i__2 = *n - kk;
i__3 = *k - kk;
sorgr2_(&i__1, &i__2, &i__3, &a[a_offset], lda, &tau[1], &work[1], &iinfo)
;
if (kk > 0) {
/* Use blocked code */
i__1 = *k;
i__2 = nb;
for (i__ = *k - kk + 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ +=
i__2) {
/* Computing MIN */
i__3 = nb, i__4 = *k - i__ + 1;
ib = min(i__3,i__4);
ii = *m - *k + i__;
if (ii > 1) {
/* Form the triangular factor of the block reflector */
/* H = H(i+ib-1) . . . H(i+1) H(i) */
i__3 = *n - *k + i__ + ib - 1;
slarft_("Backward", "Rowwise", &i__3, &ib, &a[ii + a_dim1],
lda, &tau[i__], &work[1], &ldwork);
/* Apply H' to A(1:m-k+i-1,1:n-k+i+ib-1) from the right */
i__3 = ii - 1;
i__4 = *n - *k + i__ + ib - 1;
slarfb_("Right", "Transpose", "Backward", "Rowwise", &i__3, &
i__4, &ib, &a[ii + a_dim1], lda, &work[1], &ldwork, &
a[a_offset], lda, &work[ib + 1], &ldwork);
}
/* Apply H' to columns 1:n-k+i+ib-1 of current block */
i__3 = *n - *k + i__ + ib - 1;
sorgr2_(&ib, &i__3, &ib, &a[ii + a_dim1], lda, &tau[i__], &work[1]
, &iinfo);
/* Set columns n-k+i+ib:n of current block to zero */
i__3 = *n;
for (l = *n - *k + i__ + ib; l <= i__3; ++l) {
i__4 = ii + ib - 1;
for (j = ii; j <= i__4; ++j) {
a[j + l * a_dim1] = 0.f;
/* L30: */
}
/* L40: */
}
/* L50: */
}
}
work[1] = (real) iws;
return 0;
/* End of SORGRQ */
} /* sorgrq_ */
/* Subroutine */ int sorgrq_(integer *m, integer *n, integer *k, real *a,
integer *lda, real *tau, real *work, integer *lwork, integer *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
=======
SORGRQ generates an M-by-N real matrix Q with orthonormal rows,
which is defined as the last M rows of a product of K elementary
reflectors of order N
Q = H(1) H(2) . . . H(k)
as returned by SGERQF.
Arguments
=========
M (input) INTEGER
The number of rows of the matrix Q. M >= 0.
N (input) INTEGER
The number of columns of the matrix Q. N >= M.
K (input) INTEGER
The number of elementary reflectors whose product defines the
matrix Q. M >= K >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the (m-k+i)-th row must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by SGERQF in the last k rows of its array argument
A.
On exit, the M-by-N matrix Q.
LDA (input) INTEGER
The first dimension of the array A. LDA >= max(1,M).
TAU (input) REAL array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by SGERQF.
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,M).
For optimum performance LWORK >= M*NB, where NB is the
optimal blocksize.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
=====================================================================
Test the input arguments
Parameter adjustments
Function Body */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static integer c__3 = 3;
static integer c__2 = 2;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
static integer i, j, l, nbmin, iinfo, ib;
extern /* Subroutine */ int sorgr2_(integer *, integer *, integer *, real
*, integer *, real *, real *, integer *);
static integer nb, ii, kk, nx;
extern /* Subroutine */ int slarfb_(char *, char *, char *, char *,
integer *, integer *, integer *, real *, integer *, real *,
integer *, real *, integer *, real *, integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int slarft_(char *, char *, integer *, integer *,
real *, integer *, real *, real *, integer *);
static integer ldwork, iws;
#define TAU(I) tau[(I)-1]
#define WORK(I) work[(I)-1]
#define A(I,J) a[(I)-1 + ((J)-1)* ( *lda)]
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < *m) {
*info = -2;
} else if (*k < 0 || *k > *m) {
*info = -3;
} else if (*lda < max(1,*m)) {
*info = -5;
} else if (*lwork < max(1,*m)) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SORGRQ", &i__1);
return 0;
}
/* Quick return if possible */
if (*m <= 0) {
WORK(1) = 1.f;
return 0;
}
/* Determine the block size. */
nb = ilaenv_(&c__1, "SORGRQ", " ", m, n, k, &c_n1, 6L, 1L);
nbmin = 2;
nx = 0;
iws = *m;
if (nb > 1 && nb < *k) {
/* Determine when to cross over from blocked to unblocked code.
Computing MAX */
i__1 = 0, i__2 = ilaenv_(&c__3, "SORGRQ", " ", m, n, k, &c_n1, 6L, 1L)
;
nx = max(i__1,i__2);
if (nx < *k) {
/* Determine if workspace is large enough for blocked co
de. */
ldwork = *m;
iws = ldwork * nb;
if (*lwork < iws) {
/* Not enough workspace to use optimal NB: reduc
e NB and
determine the minimum value of NB. */
nb = *lwork / ldwork;
/* Computing MAX */
i__1 = 2, i__2 = ilaenv_(&c__2, "SORGRQ", " ", m, n, k, &c_n1,
6L, 1L);
nbmin = max(i__1,i__2);
}
}
}
if (nb >= nbmin && nb < *k && nx < *k) {
/* Use blocked code after the first block.
The last kk rows are handled by the block method.
Computing MIN */
i__1 = *k, i__2 = (*k - nx + nb - 1) / nb * nb;
kk = min(i__1,i__2);
/* Set A(1:m-kk,n-kk+1:n) to zero. */
i__1 = *n;
for (j = *n - kk + 1; j <= *n; ++j) {
i__2 = *m - kk;
for (i = 1; i <= *m-kk; ++i) {
A(i,j) = 0.f;
/* L10: */
}
/* L20: */
}
} else {
kk = 0;
}
/* Use unblocked code for the first or only block. */
i__1 = *m - kk;
i__2 = *n - kk;
i__3 = *k - kk;
sorgr2_(&i__1, &i__2, &i__3, &A(1,1), lda, &TAU(1), &WORK(1), &iinfo)
;
if (kk > 0) {
/* Use blocked code */
i__1 = *k;
i__2 = nb;
for (i = *k - kk + 1; nb < 0 ? i >= *k : i <= *k; i += nb) {
/* Computing MIN */
i__3 = nb, i__4 = *k - i + 1;
ib = min(i__3,i__4);
ii = *m - *k + i;
if (ii > 1) {
/* Form the triangular factor of the block reflec
tor
H = H(i+ib-1) . . . H(i+1) H(i) */
i__3 = *n - *k + i + ib - 1;
slarft_("Backward", "Rowwise", &i__3, &ib, &A(ii,1),
lda, &TAU(i), &WORK(1), &ldwork);
/* Apply H' to A(1:m-k+i-1,1:n-k+i+ib-1) from the
right */
i__3 = ii - 1;
i__4 = *n - *k + i + ib - 1;
slarfb_("Right", "Transpose", "Backward", "Rowwise", &i__3, &
i__4, &ib, &A(ii,1), lda, &WORK(1), &ldwork, &
A(1,1), lda, &WORK(ib + 1), &ldwork);
}
/* Apply H' to columns 1:n-k+i+ib-1 of current block */
i__3 = *n - *k + i + ib - 1;
sorgr2_(&ib, &i__3, &ib, &A(ii,1), lda, &TAU(i), &WORK(1),
&iinfo);
/* Set columns n-k+i+ib:n of current block to zero */
i__3 = *n;
for (l = *n - *k + i + ib; l <= *n; ++l) {
i__4 = ii + ib - 1;
for (j = ii; j <= ii+ib-1; ++j) {
A(j,l) = 0.f;
/* L30: */
}
/* L40: */
}
/* L50: */
}
}
WORK(1) = (real) iws;
return 0;
/* End of SORGRQ */
} /* sorgrq_ */
