/* Subroutine */ int csytri_(char *uplo, integer *n, complex *a, integer *lda,
integer *ipiv, complex *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
=======
CSYTRI computes the inverse of a complex symmetric indefinite matrix
A using the factorization A = U*D*U**T or A = L*D*L**T computed by
CSYTRF.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**T;
= 'L': Lower triangular, form is A = L*D*L**T.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by CSYTRF.
On exit, if INFO = 0, the (symmetric) inverse of the original
matrix. If UPLO = 'U', the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if UPLO = 'L' the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF.
WORK (workspace) COMPLEX array, dimension (2*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {1.f,0.f};
static complex c_b2 = {0.f,0.f};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
complex q__1, q__2, q__3;
/* Builtin functions */
void c_div(complex *, complex *, complex *);
/* Local variables */
static complex temp, akkp1, d__;
static integer k;
static complex t;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *);
extern /* Complex */ VOID cdotu_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern /* Subroutine */ int cswap_(integer *, complex *, integer *,
complex *, integer *);
static integer kstep;
static logical upper;
extern /* Subroutine */ int csymv_(char *, integer *, complex *, complex *
, integer *, complex *, integer *, complex *, complex *, integer *
);
static complex ak;
static integer kp;
extern /* Subroutine */ int xerbla_(char *, integer *);
static complex akp1;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipiv;
--work;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CSYTRI", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Check that the diagonal matrix D is nonsingular. */
if (upper) {
/* Upper triangular storage: examine D from bottom to top */
for (*info = *n; *info >= 1; --(*info)) {
i__1 = a_subscr(*info, *info);
if (ipiv[*info] > 0 && (a[i__1].r == 0.f && a[i__1].i == 0.f)) {
return 0;
}
/* L10: */
}
} else {
/* Lower triangular storage: examine D from top to bottom. */
i__1 = *n;
for (*info = 1; *info <= i__1; ++(*info)) {
i__2 = a_subscr(*info, *info);
if (ipiv[*info] > 0 && (a[i__2].r == 0.f && a[i__2].i == 0.f)) {
return 0;
}
/* L20: */
}
}
*info = 0;
if (upper) {
/* Compute inv(A) from the factorization A = U*D*U'.
K is the main loop index, increasing from 1 to N in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = 1;
L30:
/* If K > N, exit from loop. */
if (k > *n) {
goto L40;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Invert the diagonal block. */
i__1 = a_subscr(k, k);
c_div(&q__1, &c_b1, &a_ref(k, k));
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
/* Compute column K of the inverse. */
if (k > 1) {
i__1 = k - 1;
ccopy_(&i__1, &a_ref(1, k), &c__1, &work[1], &c__1);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a[a_offset], lda, &work[1], &c__1,
&c_b2, &a_ref(1, k), &c__1);
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
i__3 = k - 1;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(1, k), &c__1);
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
}
kstep = 1;
} else {
/* 2 x 2 diagonal block
Invert the diagonal block. */
i__1 = a_subscr(k, k + 1);
t.r = a[i__1].r, t.i = a[i__1].i;
c_div(&q__1, &a_ref(k, k), &t);
ak.r = q__1.r, ak.i = q__1.i;
c_div(&q__1, &a_ref(k + 1, k + 1), &t);
akp1.r = q__1.r, akp1.i = q__1.i;
c_div(&q__1, &a_ref(k, k + 1), &t);
akkp1.r = q__1.r, akkp1.i = q__1.i;
q__3.r = ak.r * akp1.r - ak.i * akp1.i, q__3.i = ak.r * akp1.i +
ak.i * akp1.r;
q__2.r = q__3.r - 1.f, q__2.i = q__3.i + 0.f;
q__1.r = t.r * q__2.r - t.i * q__2.i, q__1.i = t.r * q__2.i + t.i
* q__2.r;
d__.r = q__1.r, d__.i = q__1.i;
i__1 = a_subscr(k, k);
c_div(&q__1, &akp1, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k + 1, k + 1);
c_div(&q__1, &ak, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k, k + 1);
q__2.r = -akkp1.r, q__2.i = -akkp1.i;
c_div(&q__1, &q__2, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
/* Compute columns K and K+1 of the inverse. */
if (k > 1) {
i__1 = k - 1;
ccopy_(&i__1, &a_ref(1, k), &c__1, &work[1], &c__1);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a[a_offset], lda, &work[1], &c__1,
&c_b2, &a_ref(1, k), &c__1);
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
i__3 = k - 1;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(1, k), &c__1);
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k, k + 1);
i__2 = a_subscr(k, k + 1);
i__3 = k - 1;
cdotu_(&q__2, &i__3, &a_ref(1, k), &c__1, &a_ref(1, k + 1), &
c__1);
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = k - 1;
ccopy_(&i__1, &a_ref(1, k + 1), &c__1, &work[1], &c__1);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a[a_offset], lda, &work[1], &c__1,
&c_b2, &a_ref(1, k + 1), &c__1);
i__1 = a_subscr(k + 1, k + 1);
i__2 = a_subscr(k + 1, k + 1);
i__3 = k - 1;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(1, k + 1), &c__1)
;
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
}
kstep = 2;
}
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k) {
/* Interchange rows and columns K and KP in the leading
submatrix A(1:k+1,1:k+1) */
i__1 = kp - 1;
cswap_(&i__1, &a_ref(1, k), &c__1, &a_ref(1, kp), &c__1);
i__1 = k - kp - 1;
cswap_(&i__1, &a_ref(kp + 1, k), &c__1, &a_ref(kp, kp + 1), lda);
i__1 = a_subscr(k, k);
temp.r = a[i__1].r, temp.i = a[i__1].i;
i__1 = a_subscr(k, k);
i__2 = a_subscr(kp, kp);
a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i;
i__1 = a_subscr(kp, kp);
a[i__1].r = temp.r, a[i__1].i = temp.i;
if (kstep == 2) {
i__1 = a_subscr(k, k + 1);
temp.r = a[i__1].r, temp.i = a[i__1].i;
i__1 = a_subscr(k, k + 1);
i__2 = a_subscr(kp, k + 1);
a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i;
i__1 = a_subscr(kp, k + 1);
a[i__1].r = temp.r, a[i__1].i = temp.i;
}
}
k += kstep;
goto L30;
L40:
;
} else {
/* Compute inv(A) from the factorization A = L*D*L'.
K is the main loop index, increasing from 1 to N in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = *n;
L50:
/* If K < 1, exit from loop. */
if (k < 1) {
goto L60;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Invert the diagonal block. */
i__1 = a_subscr(k, k);
c_div(&q__1, &c_b1, &a_ref(k, k));
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
/* Compute column K of the inverse. */
if (k < *n) {
i__1 = *n - k;
ccopy_(&i__1, &a_ref(k + 1, k), &c__1, &work[1], &c__1);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a_ref(k + 1, k + 1), lda, &work[1]
, &c__1, &c_b2, &a_ref(k + 1, k), &c__1);
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
i__3 = *n - k;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(k + 1, k), &c__1)
;
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
}
kstep = 1;
} else {
/* 2 x 2 diagonal block
Invert the diagonal block. */
i__1 = a_subscr(k, k - 1);
t.r = a[i__1].r, t.i = a[i__1].i;
c_div(&q__1, &a_ref(k - 1, k - 1), &t);
ak.r = q__1.r, ak.i = q__1.i;
c_div(&q__1, &a_ref(k, k), &t);
akp1.r = q__1.r, akp1.i = q__1.i;
c_div(&q__1, &a_ref(k, k - 1), &t);
akkp1.r = q__1.r, akkp1.i = q__1.i;
q__3.r = ak.r * akp1.r - ak.i * akp1.i, q__3.i = ak.r * akp1.i +
ak.i * akp1.r;
q__2.r = q__3.r - 1.f, q__2.i = q__3.i + 0.f;
q__1.r = t.r * q__2.r - t.i * q__2.i, q__1.i = t.r * q__2.i + t.i
* q__2.r;
d__.r = q__1.r, d__.i = q__1.i;
i__1 = a_subscr(k - 1, k - 1);
c_div(&q__1, &akp1, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k, k);
c_div(&q__1, &ak, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k, k - 1);
q__2.r = -akkp1.r, q__2.i = -akkp1.i;
c_div(&q__1, &q__2, &d__);
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
/* Compute columns K-1 and K of the inverse. */
if (k < *n) {
i__1 = *n - k;
ccopy_(&i__1, &a_ref(k + 1, k), &c__1, &work[1], &c__1);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a_ref(k + 1, k + 1), lda, &work[1]
, &c__1, &c_b2, &a_ref(k + 1, k), &c__1);
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
i__3 = *n - k;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(k + 1, k), &c__1)
;
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(k, k - 1);
i__2 = a_subscr(k, k - 1);
i__3 = *n - k;
cdotu_(&q__2, &i__3, &a_ref(k + 1, k), &c__1, &a_ref(k + 1, k
- 1), &c__1);
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = *n - k;
ccopy_(&i__1, &a_ref(k + 1, k - 1), &c__1, &work[1], &c__1);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
csymv_(uplo, &i__1, &q__1, &a_ref(k + 1, k + 1), lda, &work[1]
, &c__1, &c_b2, &a_ref(k + 1, k - 1), &c__1);
i__1 = a_subscr(k - 1, k - 1);
i__2 = a_subscr(k - 1, k - 1);
i__3 = *n - k;
cdotu_(&q__2, &i__3, &work[1], &c__1, &a_ref(k + 1, k - 1), &
c__1);
q__1.r = a[i__2].r - q__2.r, q__1.i = a[i__2].i - q__2.i;
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
}
kstep = 2;
}
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k) {
/* Interchange rows and columns K and KP in the trailing
submatrix A(k-1:n,k-1:n) */
if (kp < *n) {
i__1 = *n - kp;
cswap_(&i__1, &a_ref(kp + 1, k), &c__1, &a_ref(kp + 1, kp), &
c__1);
}
i__1 = kp - k - 1;
cswap_(&i__1, &a_ref(k + 1, k), &c__1, &a_ref(kp, k + 1), lda);
i__1 = a_subscr(k, k);
temp.r = a[i__1].r, temp.i = a[i__1].i;
i__1 = a_subscr(k, k);
i__2 = a_subscr(kp, kp);
a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i;
i__1 = a_subscr(kp, kp);
a[i__1].r = temp.r, a[i__1].i = temp.i;
if (kstep == 2) {
i__1 = a_subscr(k, k - 1);
temp.r = a[i__1].r, temp.i = a[i__1].i;
i__1 = a_subscr(k, k - 1);
i__2 = a_subscr(kp, k - 1);
a[i__1].r = a[i__2].r, a[i__1].i = a[i__2].i;
i__1 = a_subscr(kp, k - 1);
a[i__1].r = temp.r, a[i__1].i = temp.i;
}
}
k -= kstep;
goto L50;
L60:
;
}
return 0;
/* End of CSYTRI */
} /* csytri_ */
/* Subroutine */ int zunghr_(integer *n, integer *ilo, integer *ihi,
doublecomplex *a, integer *lda, doublecomplex *tau, doublecomplex *
work, integer *lwork, integer *info)
{
/* -- 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
Purpose
=======
ZUNGHR generates a complex unitary matrix Q which is defined as the
product of IHI-ILO elementary reflectors of order N, as returned by
ZGEHRD:
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Arguments
=========
N (input) INTEGER
The order of the matrix Q. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER
ILO and IHI must have the same values as in the previous call
of ZGEHRD. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the vectors which define the elementary reflectors,
as returned by ZGEHRD.
On exit, the N-by-N unitary matrix Q.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU (input) COMPLEX*16 array, dimension (N-1)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by ZGEHRD.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= IHI-ILO.
For optimum performance LWORK >= (IHI-ILO)*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 had an illegal value
=====================================================================
Test the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
/* Local variables */
static integer i__, j, iinfo, nb, nh;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer lwkopt;
static logical lquery;
extern /* Subroutine */ int zungqr_(integer *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *, integer *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
nh = *ihi - *ilo;
lquery = *lwork == -1;
if (*n < 0) {
*info = -1;
} else if (*ilo < 1 || *ilo > max(1,*n)) {
*info = -2;
} else if (*ihi < min(*ilo,*n) || *ihi > *n) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*lwork < max(1,nh) && ! lquery) {
*info = -8;
}
if (*info == 0) {
nb = ilaenv_(&c__1, "ZUNGQR", " ", &nh, &nh, &nh, &c_n1, (ftnlen)6, (
ftnlen)1);
lwkopt = max(1,nh) * nb;
work[1].r = (doublereal) lwkopt, work[1].i = 0.;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZUNGHR", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
work[1].r = 1., work[1].i = 0.;
return 0;
}
/* Shift the vectors which define the elementary reflectors one
column to the right, and set the first ilo and the last n-ihi
rows and columns to those of the unit matrix */
i__1 = *ilo + 1;
for (j = *ihi; j >= i__1; --j) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* L10: */
}
i__2 = *ihi;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = a_subscr(i__, j - 1);
a[i__3].r = a[i__4].r, a[i__3].i = a[i__4].i;
/* L20: */
}
i__2 = *n;
for (i__ = *ihi + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* L30: */
}
/* L40: */
}
i__1 = *ilo;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* L50: */
}
i__2 = a_subscr(j, j);
a[i__2].r = 1., a[i__2].i = 0.;
/* L60: */
}
i__1 = *n;
for (j = *ihi + 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* L70: */
}
i__2 = a_subscr(j, j);
a[i__2].r = 1., a[i__2].i = 0.;
/* L80: */
}
if (nh > 0) {
/* Generate Q(ilo+1:ihi,ilo+1:ihi) */
zungqr_(&nh, &nh, &nh, &a_ref(*ilo + 1, *ilo + 1), lda, &tau[*ilo], &
work[1], lwork, &iinfo);
}
work[1].r = (doublereal) lwkopt, work[1].i = 0.;
return 0;
/* End of ZUNGHR */
} /* zunghr_ */
/* Subroutine */ int cdrgev_(integer *nsizes, integer *nn, integer *ntypes,
logical *dotype, integer *iseed, real *thresh, integer *nounit,
complex *a, integer *lda, complex *b, complex *s, complex *t, complex
*q, integer *ldq, complex *z__, complex *qe, integer *ldqe, complex *
alpha, complex *beta, complex *alpha1, complex *beta1, complex *work,
integer *lwork, real *rwork, real *result, integer *info)
{
/* Initialized data */
static integer kclass[26] = { 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,
2,2,2,3 };
static integer kbmagn[26] = { 1,1,1,1,1,1,1,1,3,2,3,2,2,3,1,1,1,1,1,1,1,3,
2,3,2,1 };
static integer ktrian[26] = { 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,
1,1,1,1 };
static logical lasign[26] = { FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,
TRUE_,FALSE_,TRUE_,TRUE_,FALSE_,FALSE_,TRUE_,TRUE_,TRUE_,FALSE_,
TRUE_,FALSE_,FALSE_,FALSE_,TRUE_,TRUE_,TRUE_,TRUE_,TRUE_,FALSE_ };
static logical lbsign[26] = { FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,
FALSE_,TRUE_,FALSE_,FALSE_,TRUE_,TRUE_,FALSE_,FALSE_,TRUE_,FALSE_,
TRUE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,FALSE_,
FALSE_ };
static integer kz1[6] = { 0,1,2,1,3,3 };
static integer kz2[6] = { 0,0,1,2,1,1 };
static integer kadd[6] = { 0,0,0,0,3,2 };
static integer katype[26] = { 0,1,0,1,2,3,4,1,4,4,1,1,4,4,4,2,4,5,8,7,9,4,
4,4,4,0 };
static integer kbtype[26] = { 0,0,1,1,2,-3,1,4,1,1,4,4,1,1,-4,2,-4,8,8,8,
8,8,8,8,8,0 };
static integer kazero[26] = { 1,1,1,1,1,1,2,1,2,2,1,1,2,2,3,1,3,5,5,5,5,3,
3,3,3,1 };
static integer kbzero[26] = { 1,1,1,1,1,1,1,2,1,1,2,2,1,1,4,1,4,6,6,6,6,4,
4,4,4,1 };
static integer kamagn[26] = { 1,1,1,1,1,1,1,1,2,3,2,3,2,3,1,1,1,1,1,1,1,2,
3,3,2,1 };
/* Format strings */
static char fmt_9999[] = "(\002 CDRGEV: \002,a,\002 returned INFO=\002,i"
"6,\002.\002,/3x,\002N=\002,i6,\002, JTYPE=\002,i6,\002, ISEED="
"(\002,3(i5,\002,\002),i5,\002)\002)";
static char fmt_9998[] = "(\002 CDRGEV: \002,a,\002 Eigenvectors from"
" \002,a,\002 incorrectly \002,\002normalized.\002,/\002 Bits of "
"error=\002,0p,g10.3,\002,\002,3x,\002N=\002,i4,\002, JTYPE=\002,"
"i3,\002, ISEED=(\002,3(i4,\002,\002),i5,\002)\002)";
static char fmt_9997[] = "(/1x,a3,\002 -- Complex Generalized eigenvalue"
" problem \002,\002driver\002)";
static char fmt_9996[] = "(\002 Matrix types (see CDRGEV for details):"
" \002)";
static char fmt_9995[] = "(\002 Special Matrices:\002,23x,\002(J'=transp"
"osed Jordan block)\002,/\002 1=(0,0) 2=(I,0) 3=(0,I) 4=(I,I"
") 5=(J',J') \002,\0026=(diag(J',I), diag(I,J'))\002,/\002 Diag"
"onal Matrices: ( \002,\002D=diag(0,1,2,...) )\002,/\002 7=(D,"
"I) 9=(large*D, small*I\002,\002) 11=(large*I, small*D) 13=(l"
"arge*D, large*I)\002,/\002 8=(I,D) 10=(small*D, large*I) 12="
"(small*I, large*D) \002,\002 14=(small*D, small*I)\002,/\002 15"
"=(D, reversed D)\002)";
static char fmt_9994[] = "(\002 Matrices Rotated by Random \002,a,\002 M"
"atrices U, V:\002,/\002 16=Transposed Jordan Blocks "
" 19=geometric \002,\002alpha, beta=0,1\002,/\002 17=arithm. alp"
"ha&beta \002,\002 20=arithmetic alpha, beta=0,"
"1\002,/\002 18=clustered \002,\002alpha, beta=0,1 21"
"=random alpha, beta=0,1\002,/\002 Large & Small Matrices:\002,"
"/\002 22=(large, small) \002,\00223=(small,large) 24=(smal"
"l,small) 25=(large,large)\002,/\002 26=random O(1) matrices"
".\002)";
static char fmt_9993[] = "(/\002 Tests performed: \002,/\002 1 = max "
"| ( b A - a B )'*l | / const.,\002,/\002 2 = | |VR(i)| - 1 | / u"
"lp,\002,/\002 3 = max | ( b A - a B )*r | / const.\002,/\002 4 ="
" | |VL(i)| - 1 | / ulp,\002,/\002 5 = 0 if W same no matter if r"
" or l computed,\002,/\002 6 = 0 if l same no matter if l compute"
"d,\002,/\002 7 = 0 if r same no matter if r computed,\002,/1x)";
static char fmt_9992[] = "(\002 Matrix order=\002,i5,\002, type=\002,i2"
",\002, seed=\002,4(i4,\002,\002),\002 result \002,i2,\002 is\002"
",0p,f8.2)";
static char fmt_9991[] = "(\002 Matrix order=\002,i5,\002, type=\002,i2"
",\002, seed=\002,4(i4,\002,\002),\002 result \002,i2,\002 is\002"
",1p,e10.3)";
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, qe_dim1,
qe_offset, s_dim1, s_offset, t_dim1, t_offset, z_dim1, z_offset,
i__1, i__2, i__3, i__4, i__5, i__6, i__7;
real r__1, r__2;
complex q__1, q__2, q__3;
/* Builtin functions */
double r_sign(real *, real *), c_abs(complex *);
void r_cnjg(complex *, complex *);
integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void);
/* Local variables */
static integer iadd, ierr, nmax, i__, j, n;
static logical badnn;
extern /* Subroutine */ int cget52_(logical *, integer *, complex *,
integer *, complex *, integer *, complex *, integer *, complex *,
complex *, complex *, real *, real *), cggev_(char *, char *,
integer *, complex *, integer *, complex *, integer *, complex *,
complex *, complex *, integer *, complex *, integer *, complex *,
integer *, real *, integer *);
static real rmagn[4];
static complex ctemp;
static integer nmats, jsize, nerrs, jtype, n1;
extern /* Subroutine */ int clatm4_(integer *, integer *, integer *,
integer *, logical *, real *, real *, real *, integer *, integer *
, complex *, integer *), cunm2r_(char *, char *, integer *,
integer *, integer *, complex *, integer *, complex *, complex *,
integer *, complex *, integer *);
static integer jc, nb, in;
extern /* Subroutine */ int slabad_(real *, real *);
static integer jr;
extern /* Subroutine */ int clarfg_(integer *, complex *, complex *,
integer *, complex *);
extern /* Complex */ VOID clarnd_(complex *, integer *, integer *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), claset_(char *,
integer *, integer *, complex *, complex *, complex *, integer *);
static real safmin, safmax;
static integer ioldsd[4];
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int alasvm_(char *, integer *, integer *, integer
*, integer *), xerbla_(char *, integer *);
static integer minwrk, maxwrk;
static real ulpinv;
static integer mtypes, ntestt;
static real ulp;
/* Fortran I/O blocks */
static cilist io___40 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___42 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___43 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___44 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___45 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___46 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___47 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___48 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___49 = { 0, 0, 0, fmt_9996, 0 };
static cilist io___50 = { 0, 0, 0, fmt_9995, 0 };
static cilist io___51 = { 0, 0, 0, fmt_9994, 0 };
static cilist io___52 = { 0, 0, 0, fmt_9993, 0 };
static cilist io___53 = { 0, 0, 0, fmt_9992, 0 };
static cilist io___54 = { 0, 0, 0, fmt_9991, 0 };
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define q_subscr(a_1,a_2) (a_2)*q_dim1 + a_1
#define q_ref(a_1,a_2) q[q_subscr(a_1,a_2)]
#define z___subscr(a_1,a_2) (a_2)*z_dim1 + a_1
#define z___ref(a_1,a_2) z__[z___subscr(a_1,a_2)]
#define qe_subscr(a_1,a_2) (a_2)*qe_dim1 + a_1
#define qe_ref(a_1,a_2) qe[qe_subscr(a_1,a_2)]
/* -- LAPACK test 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
=======
CDRGEV checks the nonsymmetric generalized eigenvalue problem driver
routine CGGEV.
CGGEV computes for a pair of n-by-n nonsymmetric matrices (A,B) the
generalized eigenvalues and, optionally, the left and right
eigenvectors.
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that A - w*B is singular. It is
usually represented as the pair (alpha,beta), as there is reasonalbe
interpretation for beta=0, and even for both being zero.
A right generalized eigenvector corresponding to a generalized
eigenvalue w for a pair of matrices (A,B) is a vector r such that
(A - wB) * r = 0. A left generalized eigenvector is a vector l such
that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.
When CDRGEV is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following tests
will be performed and compared with the threshhold THRESH.
Results from CGGEV:
(1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of
| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )
where VL**H is the conjugate-transpose of VL.
(2) | |VL(i)| - 1 | / ulp and whether largest component real
VL(i) denotes the i-th column of VL.
(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of
| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )
(4) | |VR(i)| - 1 | / ulp and whether largest component real
VR(i) denotes the i-th column of VR.
(5) W(full) = W(partial)
W(full) denotes the eigenvalues computed when both l and r
are also computed, and W(partial) denotes the eigenvalues
computed when only W, only W and r, or only W and l are
computed.
(6) VL(full) = VL(partial)
VL(full) denotes the left eigenvectors computed when both l
and r are computed, and VL(partial) denotes the result
when only l is computed.
(7) VR(full) = VR(partial)
VR(full) denotes the right eigenvectors computed when both l
and r are also computed, and VR(partial) denotes the result
when only l is computed.
Test Matrices
---- --------
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Arguments
=========
NSIZES (input) INTEGER
The number of sizes of matrices to use. If it is zero,
CDRGES does nothing. NSIZES >= 0.
NN (input) INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0.
NTYPES (input) INTEGER
The number of elements in DOTYPE. If it is zero, CDRGEV
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE (input) LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED (input/output) INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRGES to continue the same random number
sequence.
THRESH (input) REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100. In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix. It must be at least zero.
NOUNIT (input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IERR not equal to 0.)
A (input/workspace) COMPLEX array, dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA (input) INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).
B (input/workspace) COMPLEX array, dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
S (workspace) COMPLEX array, dimension (LDA, max(NN))
The Schur form matrix computed from A by CGGEV. On exit, S
contains the Schur form matrix corresponding to the matrix
in A.
T (workspace) COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGEV.
Q (workspace) COMPLEX array, dimension (LDQ, max(NN))
The (left) eigenvectors matrix computed by CGGEV.
LDQ (input) INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).
Z (workspace) COMPLEX array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by CGGEV.
QE (workspace) COMPLEX array, dimension( LDQ, max(NN) )
QE holds the computed right or left eigenvectors.
LDQE (input) INTEGER
The leading dimension of QE. LDQE >= max(1,max(NN)).
ALPHA (workspace) COMPLEX array, dimension (max(NN))
BETA (workspace) COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CGGEV.
( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
generalized eigenvalue of A and B.
ALPHA1 (workspace) COMPLEX array, dimension (max(NN))
BETA1 (workspace) COMPLEX array, dimension (max(NN))
Like ALPHAR, ALPHAI, BETA, these arrays contain the
eigenvalues of A and B, but those computed when CGGEV only
computes a partial eigendecomposition, i.e. not the
eigenvalues and left and right eigenvectors.
WORK (workspace) COMPLEX array, dimension (LWORK)
LWORK (input) INTEGER
The number of entries in WORK. LWORK >= N*(N+1)
RWORK (workspace) REAL array, dimension (8*N)
Real workspace.
RESULT (output) REAL array, dimension (2)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
=====================================================================
Parameter adjustments */
--nn;
--dotype;
--iseed;
t_dim1 = *lda;
t_offset = 1 + t_dim1 * 1;
t -= t_offset;
s_dim1 = *lda;
s_offset = 1 + s_dim1 * 1;
s -= s_offset;
b_dim1 = *lda;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
z_dim1 = *ldq;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
qe_dim1 = *ldqe;
qe_offset = 1 + qe_dim1 * 1;
qe -= qe_offset;
--alpha;
--beta;
--alpha1;
--beta1;
--work;
--rwork;
--result;
/* Function Body
Check for errors */
*info = 0;
badnn = FALSE_;
nmax = 1;
i__1 = *nsizes;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
i__2 = nmax, i__3 = nn[j];
nmax = max(i__2,i__3);
if (nn[j] < 0) {
badnn = TRUE_;
}
/* L10: */
}
if (*nsizes < 0) {
*info = -1;
} else if (badnn) {
*info = -2;
} else if (*ntypes < 0) {
*info = -3;
} else if (*thresh < 0.f) {
*info = -6;
} else if (*lda <= 1 || *lda < nmax) {
*info = -9;
} else if (*ldq <= 1 || *ldq < nmax) {
*info = -14;
} else if (*ldqe <= 1 || *ldqe < nmax) {
*info = -17;
}
/* 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. */
minwrk = 1;
if (*info == 0 && *lwork >= 1) {
minwrk = nmax * (nmax + 1);
/* Computing MAX */
i__1 = 1, i__2 = ilaenv_(&c__1, "CGEQRF", " ", &nmax, &nmax, &c_n1, &
c_n1, (ftnlen)6, (ftnlen)1), i__1 = max(i__1,i__2), i__2 =
ilaenv_(&c__1, "CUNMQR", "LC", &nmax, &nmax, &nmax, &c_n1, (
ftnlen)6, (ftnlen)2), i__1 = max(i__1,i__2), i__2 = ilaenv_(&
c__1, "CUNGQR", " ", &nmax, &nmax, &nmax, &c_n1, (ftnlen)6, (
ftnlen)1);
nb = max(i__1,i__2);
/* Computing MAX */
i__1 = nmax << 1, i__2 = nmax * (nb + 1), i__1 = max(i__1,i__2), i__2
= nmax * (nmax + 1);
maxwrk = max(i__1,i__2);
work[1].r = (real) maxwrk, work[1].i = 0.f;
}
if (*lwork < minwrk) {
*info = -23;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CDRGEV", &i__1);
return 0;
}
/* Quick return if possible */
if (*nsizes == 0 || *ntypes == 0) {
return 0;
}
ulp = slamch_("Precision");
safmin = slamch_("Safe minimum");
safmin /= ulp;
safmax = 1.f / safmin;
slabad_(&safmin, &safmax);
ulpinv = 1.f / ulp;
/* The values RMAGN(2:3) depend on N, see below. */
rmagn[0] = 0.f;
rmagn[1] = 1.f;
/* Loop over sizes, types */
ntestt = 0;
nerrs = 0;
nmats = 0;
i__1 = *nsizes;
for (jsize = 1; jsize <= i__1; ++jsize) {
n = nn[jsize];
n1 = max(1,n);
rmagn[2] = safmax * ulp / (real) n1;
rmagn[3] = safmin * ulpinv * n1;
if (*nsizes != 1) {
mtypes = min(26,*ntypes);
} else {
mtypes = min(27,*ntypes);
}
i__2 = mtypes;
for (jtype = 1; jtype <= i__2; ++jtype) {
if (! dotype[jtype]) {
goto L210;
}
++nmats;
/* Save ISEED in case of an error. */
for (j = 1; j <= 4; ++j) {
ioldsd[j - 1] = iseed[j];
/* L20: */
}
/* Generate test matrices A and B
Description of control parameters:
KCLASS: =1 means w/o rotation, =2 means w/ rotation,
=3 means random.
KATYPE: the "type" to be passed to CLATM4 for computing A.
KAZERO: the pattern of zeros on the diagonal for A:
=1: ( xxx ), =2: (0, xxx ) =3: ( 0, 0, xxx, 0 ),
=4: ( 0, xxx, 0, 0 ), =5: ( 0, 0, 1, xxx, 0 ),
=6: ( 0, 1, 0, xxx, 0 ). (xxx means a string of
non-zero entries.)
KAMAGN: the magnitude of the matrix: =0: zero, =1: O(1),
=2: large, =3: small.
LASIGN: .TRUE. if the diagonal elements of A are to be
multiplied by a random magnitude 1 number.
KBTYPE, KBZERO, KBMAGN, LBSIGN: the same, but for B.
KTRIAN: =0: don't fill in the upper triangle, =1: do.
KZ1, KZ2, KADD: used to implement KAZERO and KBZERO.
RMAGN: used to implement KAMAGN and KBMAGN. */
if (mtypes > 26) {
goto L100;
}
ierr = 0;
if (kclass[jtype - 1] < 3) {
/* Generate A (w/o rotation) */
if ((i__3 = katype[jtype - 1], abs(i__3)) == 3) {
in = ((n - 1) / 2 << 1) + 1;
if (in != n) {
claset_("Full", &n, &n, &c_b1, &c_b1, &a[a_offset],
lda);
}
} else {
in = n;
}
clatm4_(&katype[jtype - 1], &in, &kz1[kazero[jtype - 1] - 1],
&kz2[kazero[jtype - 1] - 1], &lasign[jtype - 1], &
rmagn[kamagn[jtype - 1]], &ulp, &rmagn[ktrian[jtype -
1] * kamagn[jtype - 1]], &c__2, &iseed[1], &a[
a_offset], lda);
iadd = kadd[kazero[jtype - 1] - 1];
if (iadd > 0 && iadd <= n) {
i__3 = a_subscr(iadd, iadd);
i__4 = kamagn[jtype - 1];
a[i__3].r = rmagn[i__4], a[i__3].i = 0.f;
}
/* Generate B (w/o rotation) */
if ((i__3 = kbtype[jtype - 1], abs(i__3)) == 3) {
in = ((n - 1) / 2 << 1) + 1;
if (in != n) {
claset_("Full", &n, &n, &c_b1, &c_b1, &b[b_offset],
lda);
}
} else {
in = n;
}
clatm4_(&kbtype[jtype - 1], &in, &kz1[kbzero[jtype - 1] - 1],
&kz2[kbzero[jtype - 1] - 1], &lbsign[jtype - 1], &
rmagn[kbmagn[jtype - 1]], &c_b28, &rmagn[ktrian[jtype
- 1] * kbmagn[jtype - 1]], &c__2, &iseed[1], &b[
b_offset], lda);
iadd = kadd[kbzero[jtype - 1] - 1];
if (iadd != 0 && iadd <= n) {
i__3 = b_subscr(iadd, iadd);
i__4 = kbmagn[jtype - 1];
b[i__3].r = rmagn[i__4], b[i__3].i = 0.f;
}
if (kclass[jtype - 1] == 2 && n > 0) {
/* Include rotations
Generate Q, Z as Householder transformations times
a diagonal matrix. */
i__3 = n - 1;
for (jc = 1; jc <= i__3; ++jc) {
i__4 = n;
for (jr = jc; jr <= i__4; ++jr) {
i__5 = q_subscr(jr, jc);
clarnd_(&q__1, &c__3, &iseed[1]);
q[i__5].r = q__1.r, q[i__5].i = q__1.i;
i__5 = z___subscr(jr, jc);
clarnd_(&q__1, &c__3, &iseed[1]);
z__[i__5].r = q__1.r, z__[i__5].i = q__1.i;
/* L30: */
}
i__4 = n + 1 - jc;
clarfg_(&i__4, &q_ref(jc, jc), &q_ref(jc + 1, jc), &
c__1, &work[jc]);
i__4 = (n << 1) + jc;
i__5 = q_subscr(jc, jc);
r__2 = q[i__5].r;
r__1 = r_sign(&c_b28, &r__2);
work[i__4].r = r__1, work[i__4].i = 0.f;
i__4 = q_subscr(jc, jc);
q[i__4].r = 1.f, q[i__4].i = 0.f;
i__4 = n + 1 - jc;
clarfg_(&i__4, &z___ref(jc, jc), &z___ref(jc + 1, jc),
&c__1, &work[n + jc]);
i__4 = n * 3 + jc;
i__5 = z___subscr(jc, jc);
r__2 = z__[i__5].r;
r__1 = r_sign(&c_b28, &r__2);
work[i__4].r = r__1, work[i__4].i = 0.f;
i__4 = z___subscr(jc, jc);
z__[i__4].r = 1.f, z__[i__4].i = 0.f;
/* L40: */
}
clarnd_(&q__1, &c__3, &iseed[1]);
ctemp.r = q__1.r, ctemp.i = q__1.i;
i__3 = q_subscr(n, n);
q[i__3].r = 1.f, q[i__3].i = 0.f;
i__3 = n;
work[i__3].r = 0.f, work[i__3].i = 0.f;
i__3 = n * 3;
r__1 = c_abs(&ctemp);
q__1.r = ctemp.r / r__1, q__1.i = ctemp.i / r__1;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
clarnd_(&q__1, &c__3, &iseed[1]);
ctemp.r = q__1.r, ctemp.i = q__1.i;
i__3 = z___subscr(n, n);
z__[i__3].r = 1.f, z__[i__3].i = 0.f;
i__3 = n << 1;
work[i__3].r = 0.f, work[i__3].i = 0.f;
i__3 = n << 2;
r__1 = c_abs(&ctemp);
q__1.r = ctemp.r / r__1, q__1.i = ctemp.i / r__1;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* Apply the diagonal matrices */
i__3 = n;
for (jc = 1; jc <= i__3; ++jc) {
i__4 = n;
for (jr = 1; jr <= i__4; ++jr) {
i__5 = a_subscr(jr, jc);
i__6 = (n << 1) + jr;
r_cnjg(&q__3, &work[n * 3 + jc]);
q__2.r = work[i__6].r * q__3.r - work[i__6].i *
q__3.i, q__2.i = work[i__6].r * q__3.i +
work[i__6].i * q__3.r;
i__7 = a_subscr(jr, jc);
q__1.r = q__2.r * a[i__7].r - q__2.i * a[i__7].i,
q__1.i = q__2.r * a[i__7].i + q__2.i * a[
i__7].r;
a[i__5].r = q__1.r, a[i__5].i = q__1.i;
i__5 = b_subscr(jr, jc);
i__6 = (n << 1) + jr;
r_cnjg(&q__3, &work[n * 3 + jc]);
q__2.r = work[i__6].r * q__3.r - work[i__6].i *
q__3.i, q__2.i = work[i__6].r * q__3.i +
work[i__6].i * q__3.r;
i__7 = b_subscr(jr, jc);
q__1.r = q__2.r * b[i__7].r - q__2.i * b[i__7].i,
q__1.i = q__2.r * b[i__7].i + q__2.i * b[
i__7].r;
b[i__5].r = q__1.r, b[i__5].i = q__1.i;
/* L50: */
}
/* L60: */
}
i__3 = n - 1;
cunm2r_("L", "N", &n, &n, &i__3, &q[q_offset], ldq, &work[
1], &a[a_offset], lda, &work[(n << 1) + 1], &ierr);
if (ierr != 0) {
goto L90;
}
i__3 = n - 1;
cunm2r_("R", "C", &n, &n, &i__3, &z__[z_offset], ldq, &
work[n + 1], &a[a_offset], lda, &work[(n << 1) +
1], &ierr);
if (ierr != 0) {
goto L90;
}
i__3 = n - 1;
cunm2r_("L", "N", &n, &n, &i__3, &q[q_offset], ldq, &work[
1], &b[b_offset], lda, &work[(n << 1) + 1], &ierr);
if (ierr != 0) {
goto L90;
}
i__3 = n - 1;
cunm2r_("R", "C", &n, &n, &i__3, &z__[z_offset], ldq, &
work[n + 1], &b[b_offset], lda, &work[(n << 1) +
1], &ierr);
if (ierr != 0) {
goto L90;
}
}
} else {
/* Random matrices */
i__3 = n;
for (jc = 1; jc <= i__3; ++jc) {
i__4 = n;
for (jr = 1; jr <= i__4; ++jr) {
i__5 = a_subscr(jr, jc);
i__6 = kamagn[jtype - 1];
clarnd_(&q__2, &c__4, &iseed[1]);
q__1.r = rmagn[i__6] * q__2.r, q__1.i = rmagn[i__6] *
q__2.i;
a[i__5].r = q__1.r, a[i__5].i = q__1.i;
i__5 = b_subscr(jr, jc);
i__6 = kbmagn[jtype - 1];
clarnd_(&q__2, &c__4, &iseed[1]);
q__1.r = rmagn[i__6] * q__2.r, q__1.i = rmagn[i__6] *
q__2.i;
b[i__5].r = q__1.r, b[i__5].i = q__1.i;
/* L70: */
}
/* L80: */
}
}
L90:
if (ierr != 0) {
io___40.ciunit = *nounit;
s_wsfe(&io___40);
do_fio(&c__1, "Generator", (ftnlen)9);
do_fio(&c__1, (char *)&ierr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(ierr);
return 0;
}
L100:
for (i__ = 1; i__ <= 7; ++i__) {
result[i__] = -1.f;
/* L110: */
}
/* Call CGGEV to compute eigenvalues and eigenvectors. */
clacpy_(" ", &n, &n, &a[a_offset], lda, &s[s_offset], lda);
clacpy_(" ", &n, &n, &b[b_offset], lda, &t[t_offset], lda);
cggev_("V", "V", &n, &s[s_offset], lda, &t[t_offset], lda, &alpha[
1], &beta[1], &q[q_offset], ldq, &z__[z_offset], ldq, &
work[1], lwork, &rwork[1], &ierr);
if (ierr != 0 && ierr != n + 1) {
result[1] = ulpinv;
io___42.ciunit = *nounit;
s_wsfe(&io___42);
do_fio(&c__1, "CGGEV1", (ftnlen)6);
do_fio(&c__1, (char *)&ierr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(ierr);
goto L190;
}
/* Do the tests (1) and (2) */
cget52_(&c_true, &n, &a[a_offset], lda, &b[b_offset], lda, &q[
q_offset], ldq, &alpha[1], &beta[1], &work[1], &rwork[1],
&result[1]);
if (result[2] > *thresh) {
io___43.ciunit = *nounit;
s_wsfe(&io___43);
do_fio(&c__1, "Left", (ftnlen)4);
do_fio(&c__1, "CGGEV1", (ftnlen)6);
do_fio(&c__1, (char *)&result[2], (ftnlen)sizeof(real));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
}
/* Do the tests (3) and (4) */
cget52_(&c_false, &n, &a[a_offset], lda, &b[b_offset], lda, &z__[
z_offset], ldq, &alpha[1], &beta[1], &work[1], &rwork[1],
&result[3]);
if (result[4] > *thresh) {
io___44.ciunit = *nounit;
s_wsfe(&io___44);
do_fio(&c__1, "Right", (ftnlen)5);
do_fio(&c__1, "CGGEV1", (ftnlen)6);
do_fio(&c__1, (char *)&result[4], (ftnlen)sizeof(real));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
}
/* Do test (5) */
clacpy_(" ", &n, &n, &a[a_offset], lda, &s[s_offset], lda);
clacpy_(" ", &n, &n, &b[b_offset], lda, &t[t_offset], lda);
cggev_("N", "N", &n, &s[s_offset], lda, &t[t_offset], lda, &
alpha1[1], &beta1[1], &q[q_offset], ldq, &z__[z_offset],
ldq, &work[1], lwork, &rwork[1], &ierr);
if (ierr != 0 && ierr != n + 1) {
result[1] = ulpinv;
io___45.ciunit = *nounit;
s_wsfe(&io___45);
do_fio(&c__1, "CGGEV2", (ftnlen)6);
do_fio(&c__1, (char *)&ierr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(ierr);
goto L190;
}
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i__4 = j;
i__5 = j;
i__6 = j;
i__7 = j;
if (alpha[i__4].r != alpha1[i__5].r || alpha[i__4].i !=
alpha1[i__5].i || (beta[i__6].r != beta1[i__7].r ||
beta[i__6].i != beta1[i__7].i)) {
result[5] = ulpinv;
}
/* L120: */
}
/* Do test (6): Compute eigenvalues and left eigenvectors,
and test them */
clacpy_(" ", &n, &n, &a[a_offset], lda, &s[s_offset], lda);
clacpy_(" ", &n, &n, &b[b_offset], lda, &t[t_offset], lda);
cggev_("V", "N", &n, &s[s_offset], lda, &t[t_offset], lda, &
alpha1[1], &beta1[1], &qe[qe_offset], ldqe, &z__[z_offset]
, ldq, &work[1], lwork, &rwork[1], &ierr);
if (ierr != 0 && ierr != n + 1) {
result[1] = ulpinv;
io___46.ciunit = *nounit;
s_wsfe(&io___46);
do_fio(&c__1, "CGGEV3", (ftnlen)6);
do_fio(&c__1, (char *)&ierr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(ierr);
goto L190;
}
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i__4 = j;
i__5 = j;
i__6 = j;
i__7 = j;
if (alpha[i__4].r != alpha1[i__5].r || alpha[i__4].i !=
alpha1[i__5].i || (beta[i__6].r != beta1[i__7].r ||
beta[i__6].i != beta1[i__7].i)) {
result[6] = ulpinv;
}
/* L130: */
}
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i__4 = n;
for (jc = 1; jc <= i__4; ++jc) {
i__5 = q_subscr(j, jc);
i__6 = qe_subscr(j, jc);
if (q[i__5].r != qe[i__6].r || q[i__5].i != qe[i__6].i) {
result[6] = ulpinv;
}
/* L140: */
}
/* L150: */
}
/* Do test (7): Compute eigenvalues and right eigenvectors,
and test them */
clacpy_(" ", &n, &n, &a[a_offset], lda, &s[s_offset], lda);
clacpy_(" ", &n, &n, &b[b_offset], lda, &t[t_offset], lda);
cggev_("N", "V", &n, &s[s_offset], lda, &t[t_offset], lda, &
alpha1[1], &beta1[1], &q[q_offset], ldq, &qe[qe_offset],
ldqe, &work[1], lwork, &rwork[1], &ierr);
if (ierr != 0 && ierr != n + 1) {
result[1] = ulpinv;
io___47.ciunit = *nounit;
s_wsfe(&io___47);
do_fio(&c__1, "CGGEV4", (ftnlen)6);
do_fio(&c__1, (char *)&ierr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(ierr);
goto L190;
}
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i__4 = j;
i__5 = j;
i__6 = j;
i__7 = j;
if (alpha[i__4].r != alpha1[i__5].r || alpha[i__4].i !=
alpha1[i__5].i || (beta[i__6].r != beta1[i__7].r ||
beta[i__6].i != beta1[i__7].i)) {
result[7] = ulpinv;
}
/* L160: */
}
i__3 = n;
for (j = 1; j <= i__3; ++j) {
i__4 = n;
for (jc = 1; jc <= i__4; ++jc) {
i__5 = z___subscr(j, jc);
i__6 = qe_subscr(j, jc);
if (z__[i__5].r != qe[i__6].r || z__[i__5].i != qe[i__6]
.i) {
result[7] = ulpinv;
}
/* L170: */
}
/* L180: */
}
/* End of Loop -- Check for RESULT(j) > THRESH */
L190:
ntestt += 7;
/* Print out tests which fail. */
for (jr = 1; jr <= 9; ++jr) {
if (result[jr] >= *thresh) {
/* If this is the first test to fail,
print a header to the data file. */
if (nerrs == 0) {
io___48.ciunit = *nounit;
s_wsfe(&io___48);
do_fio(&c__1, "CGV", (ftnlen)3);
e_wsfe();
/* Matrix types */
io___49.ciunit = *nounit;
s_wsfe(&io___49);
e_wsfe();
io___50.ciunit = *nounit;
s_wsfe(&io___50);
e_wsfe();
io___51.ciunit = *nounit;
s_wsfe(&io___51);
do_fio(&c__1, "Orthogonal", (ftnlen)10);
e_wsfe();
/* Tests performed */
io___52.ciunit = *nounit;
s_wsfe(&io___52);
e_wsfe();
}
++nerrs;
if (result[jr] < 1e4f) {
io___53.ciunit = *nounit;
s_wsfe(&io___53);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer))
;
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&jr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&result[jr], (ftnlen)sizeof(
real));
e_wsfe();
} else {
io___54.ciunit = *nounit;
s_wsfe(&io___54);
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer))
;
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(
integer));
do_fio(&c__1, (char *)&jr, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&result[jr], (ftnlen)sizeof(
real));
e_wsfe();
}
}
/* L200: */
}
L210:
;
}
/* L220: */
}
/* Summary */
alasvm_("CGV", nounit, &nerrs, &ntestt, &c__0);
work[1].r = (real) maxwrk, work[1].i = 0.f;
return 0;
/* End of CDRGEV */
} /* cdrgev_ */
/* Subroutine */ int zhgeqz_(char *job, char *compq, char *compz, integer *n,
integer *ilo, integer *ihi, doublecomplex *a, integer *lda,
doublecomplex *b, integer *ldb, doublecomplex *alpha, doublecomplex *
beta, doublecomplex *q, integer *ldq, doublecomplex *z__, integer *
ldz, doublecomplex *work, integer *lwork, doublereal *rwork, 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, i__4, i__5, i__6;
doublereal d__1, d__2, d__3, d__4, d__5, d__6;
doublecomplex z__1, z__2, z__3, z__4, z__5, z__6;
/* Builtin functions */
double z_abs(doublecomplex *);
void d_cnjg(doublecomplex *, doublecomplex *);
double d_imag(doublecomplex *);
void z_div(doublecomplex *, doublecomplex *, doublecomplex *), pow_zi(
doublecomplex *, doublecomplex *, integer *), z_sqrt(
doublecomplex *, doublecomplex *);
/* Local variables */
static doublereal absb, atol, btol, temp, opst;
extern /* Subroutine */ int zrot_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublereal *, doublecomplex *);
static doublereal temp2, c__;
static integer j;
static doublecomplex s, t;
extern logical lsame_(char *, char *);
static doublecomplex ctemp;
static integer iiter, ilast, jiter;
static doublereal anorm;
static integer maxit;
static doublereal bnorm;
static doublecomplex shift;
extern /* Subroutine */ int zscal_(integer *, doublecomplex *,
doublecomplex *, integer *);
static doublereal tempr;
static doublecomplex ctemp2, ctemp3;
static logical ilazr2;
static integer jc, in;
static doublereal ascale, bscale;
static doublecomplex u12;
extern doublereal dlamch_(char *);
static integer jr, nq;
static doublecomplex signbc;
static integer nz;
static doublereal safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static doublecomplex eshift;
static logical ilschr;
static integer icompq, ilastm;
static doublecomplex rtdisc;
static integer ischur;
extern doublereal zlanhs_(char *, integer *, doublecomplex *, integer *,
doublereal *);
static logical ilazro;
static integer icompz, ifirst;
extern /* Subroutine */ int zlartg_(doublecomplex *, doublecomplex *,
doublereal *, doublecomplex *, doublecomplex *);
static integer ifrstm;
extern /* Subroutine */ int zlaset_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, doublecomplex *, integer *);
static integer istart;
static logical lquery;
static doublecomplex ad11, ad12, ad21, ad22;
static integer jch;
static logical ilq, ilz;
static doublereal ulp;
static doublecomplex abi22;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define q_subscr(a_1,a_2) (a_2)*q_dim1 + a_1
#define q_ref(a_1,a_2) q[q_subscr(a_1,a_2)]
#define z___subscr(a_1,a_2) (a_2)*z_dim1 + a_1
#define z___ref(a_1,a_2) z__[z___subscr(a_1,a_2)]
/* -- 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
June 30, 1999
----------------------- 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
=======
ZHGEQZ implements a single-shift version of the QZ
method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i)
of the equation
det( A - w(i) B ) = 0
If JOB='S', then the pair (A,B) is simultaneously
reduced to Schur form (i.e., A and B are both upper triangular) by
applying one unitary tranformation (usually called Q) on the left and
another (usually called Z) on the right. The diagonal elements of
A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).
If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary
transformations used to reduce (A,B) are accumulated into the arrays
Q and Z s.t.:
Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
pp. 241--256.
Arguments
=========
JOB (input) CHARACTER*1
= 'E': compute only ALPHA and BETA. A and B will not
necessarily be put into generalized Schur form.
= 'S': put A and B into generalized Schur form, as well
as computing ALPHA and BETA.
COMPQ (input) CHARACTER*1
= 'N': do not modify Q.
= 'V': multiply the array Q on the right by the conjugate
transpose of the unitary tranformation that is
applied to the left side of A and B to reduce them
to Schur form.
= 'I': like COMPQ='V', except that Q will be initialized to
the identity first.
COMPZ (input) CHARACTER*1
= 'N': do not modify Z.
= 'V': multiply the array Z on the right by the unitary
tranformation that is applied to the right side of
A and B to reduce them to Schur form.
= 'I': like COMPZ='V', except that Z will be initialized to
the identity first.
N (input) INTEGER
The order of the matrices A, B, Q, and Z. 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.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the N-by-N upper Hessenberg matrix A. Elements
below the subdiagonal must be zero.
If JOB='S', then on exit A and B will have been
simultaneously reduced to upper triangular form.
If JOB='E', then on exit A will have been destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max( 1, N ).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. Elements
below the diagonal must be zero.
If JOB='S', then on exit A and B will have been
simultaneously reduced to upper triangular form.
If JOB='E', then on exit B will have been destroyed.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max( 1, N ).
ALPHA (output) COMPLEX*16 array, dimension (N)
The diagonal elements of A when the pair (A,B) has been
reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N
are the generalized eigenvalues.
BETA (output) COMPLEX*16 array, dimension (N)
The diagonal elements of B when the pair (A,B) has been
reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N
are the generalized eigenvalues. A and B are normalized
so that BETA(1),...,BETA(N) are non-negative real numbers.
Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
If COMPQ='N', then Q will not be referenced.
If COMPQ='V' or 'I', then the conjugate transpose of the
unitary transformations which are applied to A and B on
the left will be applied to the array Q on the right.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1.
If COMPQ='V' or 'I', then LDQ >= N.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
If COMPZ='N', then Z will not be referenced.
If COMPZ='V' or 'I', then the unitary transformations which
are applied to A and B on the right will be applied to the
array Z on the right.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1.
If COMPZ='V' or 'I', then LDZ >= N.
WORK (workspace/output) COMPLEX*16 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,N).
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.
RWORK (workspace) DOUBLE PRECISION array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1,...,N: the QZ iteration did not converge. (A,B) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO+1,...,N should be correct.
= N+1,...,2*N: the shift calculation failed. (A,B) is not
in Schur form, but ALPHA(i) and BETA(i),
i=INFO-N+1,...,N should be correct.
> 2*N: various "impossible" errors.
Further Details
===============
We assume that complex ABS works as long as its value is less than
overflow.
=====================================================================
----------------------- Begin Timing Code ------------------------
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;
--alpha;
--beta;
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;
--rwork;
/* Function Body */
latime_1.itcnt = 0.;
/* ------------------------ End Timing Code -------------------------
Decode JOB, COMPQ, COMPZ */
if (lsame_(job, "E")) {
ilschr = FALSE_;
ischur = 1;
} else if (lsame_(job, "S")) {
ilschr = TRUE_;
ischur = 2;
} else {
ischur = 0;
}
if (lsame_(compq, "N")) {
ilq = FALSE_;
icompq = 1;
nq = 0;
} else if (lsame_(compq, "V")) {
ilq = TRUE_;
icompq = 2;
nq = *n;
} else if (lsame_(compq, "I")) {
ilq = TRUE_;
icompq = 3;
nq = *n;
} else {
icompq = 0;
}
if (lsame_(compz, "N")) {
ilz = FALSE_;
icompz = 1;
nz = 0;
} else if (lsame_(compz, "V")) {
ilz = TRUE_;
icompz = 2;
nz = *n;
} else if (lsame_(compz, "I")) {
ilz = TRUE_;
icompz = 3;
nz = *n;
} else {
icompz = 0;
}
/* Check Argument Values */
*info = 0;
i__1 = max(1,*n);
work[1].r = (doublereal) i__1, work[1].i = 0.;
lquery = *lwork == -1;
if (ischur == 0) {
*info = -1;
} else if (icompq == 0) {
*info = -2;
} else if (icompz == 0) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*ilo < 1) {
*info = -5;
} else if (*ihi > *n || *ihi < *ilo - 1) {
*info = -6;
} else if (*lda < *n) {
*info = -8;
} else if (*ldb < *n) {
*info = -10;
} else if (*ldq < 1 || ilq && *ldq < *n) {
*info = -14;
} else if (*ldz < 1 || ilz && *ldz < *n) {
*info = -16;
} else if (*lwork < max(1,*n) && ! lquery) {
*info = -18;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZHGEQZ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible
WORK( 1 ) = CMPLX( 1 ) */
if (*n <= 0) {
work[1].r = 1., work[1].i = 0.;
return 0;
}
/* Initialize Q and Z */
if (icompq == 3) {
zlaset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq);
}
if (icompz == 3) {
zlaset_("Full", n, n, &c_b1, &c_b2, &z__[z_offset], ldz);
}
/* Machine Constants */
in = *ihi + 1 - *ilo;
safmin = dlamch_("S");
ulp = dlamch_("E") * dlamch_("B");
anorm = zlanhs_("F", &in, &a_ref(*ilo, *ilo), lda, &rwork[1]);
bnorm = zlanhs_("F", &in, &b_ref(*ilo, *ilo), ldb, &rwork[1]);
/* Computing MAX */
d__1 = safmin, d__2 = ulp * anorm;
atol = max(d__1,d__2);
/* Computing MAX */
d__1 = safmin, d__2 = ulp * bnorm;
btol = max(d__1,d__2);
ascale = 1. / max(safmin,anorm);
bscale = 1. / max(safmin,bnorm);
/* ---------------------- Begin Timing Code -------------------------
Count ops for norms, etc. */
opst = 0.;
/* Computing 2nd power */
i__1 = *n;
latime_1.ops += (doublereal) ((i__1 * i__1 << 2) + *n * 12 - 5);
/* ----------------------- End Timing Code --------------------------
Set Eigenvalues IHI+1:N */
i__1 = *n;
for (j = *ihi + 1; j <= i__1; ++j) {
absb = z_abs(&b_ref(j, j));
if (absb > safmin) {
i__2 = b_subscr(j, j);
z__2.r = b[i__2].r / absb, z__2.i = b[i__2].i / absb;
d_cnjg(&z__1, &z__2);
signbc.r = z__1.r, signbc.i = z__1.i;
i__2 = b_subscr(j, j);
b[i__2].r = absb, b[i__2].i = 0.;
if (ilschr) {
i__2 = j - 1;
zscal_(&i__2, &signbc, &b_ref(1, j), &c__1);
zscal_(&j, &signbc, &a_ref(1, j), &c__1);
/* ----------------- Begin Timing Code --------------------- */
opst += (doublereal) ((j - 1) * 12);
/* ------------------ End Timing Code ---------------------- */
} else {
i__2 = a_subscr(j, j);
i__3 = a_subscr(j, j);
z__1.r = a[i__3].r * signbc.r - a[i__3].i * signbc.i, z__1.i =
a[i__3].r * signbc.i + a[i__3].i * signbc.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
}
if (ilz) {
zscal_(n, &signbc, &z___ref(1, j), &c__1);
}
/* ------------------- Begin Timing Code ---------------------- */
opst += (doublereal) (nz * 6 + 13);
/* -------------------- End Timing Code ----------------------- */
} else {
i__2 = b_subscr(j, j);
b[i__2].r = 0., b[i__2].i = 0.;
}
i__2 = j;
i__3 = a_subscr(j, j);
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = j;
i__3 = b_subscr(j, j);
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L10: */
}
/* If IHI < ILO, skip QZ steps */
if (*ihi < *ilo) {
goto L190;
}
/* MAIN QZ ITERATION LOOP
Initialize dynamic indices
Eigenvalues ILAST+1:N have been found.
Column operations modify rows IFRSTM:whatever
Row operations modify columns whatever:ILASTM
If only eigenvalues are being computed, then
IFRSTM is the row of the last splitting row above row ILAST;
this is always at least ILO.
IITER counts iterations since the last eigenvalue was found,
to tell when to use an extraordinary shift.
MAXIT is the maximum number of QZ sweeps allowed. */
ilast = *ihi;
if (ilschr) {
ifrstm = 1;
ilastm = *n;
} else {
ifrstm = *ilo;
ilastm = *ihi;
}
iiter = 0;
eshift.r = 0., eshift.i = 0.;
maxit = (*ihi - *ilo + 1) * 30;
i__1 = maxit;
for (jiter = 1; jiter <= i__1; ++jiter) {
/* Check for too many iterations. */
if (jiter > maxit) {
goto L180;
}
/* Split the matrix if possible.
Two tests:
1: A(j,j-1)=0 or j=ILO
2: B(j,j)=0
Special case: j=ILAST */
if (ilast == *ilo) {
goto L60;
} else {
i__2 = a_subscr(ilast, ilast - 1);
if ((d__1 = a[i__2].r, abs(d__1)) + (d__2 = d_imag(&a_ref(ilast,
ilast - 1)), abs(d__2)) <= atol) {
i__2 = a_subscr(ilast, ilast - 1);
a[i__2].r = 0., a[i__2].i = 0.;
goto L60;
}
}
if (z_abs(&b_ref(ilast, ilast)) <= btol) {
i__2 = b_subscr(ilast, ilast);
b[i__2].r = 0., b[i__2].i = 0.;
goto L50;
}
/* General case: j<ILAST */
i__2 = *ilo;
for (j = ilast - 1; j >= i__2; --j) {
/* Test 1: for A(j,j-1)=0 or j=ILO */
if (j == *ilo) {
ilazro = TRUE_;
} else {
i__3 = a_subscr(j, j - 1);
if ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a_ref(j,
j - 1)), abs(d__2)) <= atol) {
i__3 = a_subscr(j, j - 1);
a[i__3].r = 0., a[i__3].i = 0.;
ilazro = TRUE_;
} else {
ilazro = FALSE_;
}
}
/* Test 2: for B(j,j)=0 */
if (z_abs(&b_ref(j, j)) < btol) {
i__3 = b_subscr(j, j);
b[i__3].r = 0., b[i__3].i = 0.;
/* Test 1a: Check for 2 consecutive small subdiagonals in A */
ilazr2 = FALSE_;
if (! ilazro) {
i__3 = a_subscr(j, j - 1);
i__4 = a_subscr(j + 1, j);
i__5 = a_subscr(j, j);
if (((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&
a_ref(j, j - 1)), abs(d__2))) * (ascale * ((d__3 =
a[i__4].r, abs(d__3)) + (d__4 = d_imag(&a_ref(j
+ 1, j)), abs(d__4)))) <= ((d__5 = a[i__5].r, abs(
d__5)) + (d__6 = d_imag(&a_ref(j, j)), abs(d__6)))
* (ascale * atol)) {
ilazr2 = TRUE_;
}
}
/* If both tests pass (1 & 2), i.e., the leading diagonal
element of B in the block is zero, split a 1x1 block off
at the top. (I.e., at the J-th row/column) The leading
diagonal element of the remainder can also be zero, so
this may have to be done repeatedly. */
if (ilazro || ilazr2) {
i__3 = ilast - 1;
for (jch = j; jch <= i__3; ++jch) {
i__4 = a_subscr(jch, jch);
ctemp.r = a[i__4].r, ctemp.i = a[i__4].i;
zlartg_(&ctemp, &a_ref(jch + 1, jch), &c__, &s, &
a_ref(jch, jch));
i__4 = a_subscr(jch + 1, jch);
a[i__4].r = 0., a[i__4].i = 0.;
i__4 = ilastm - jch;
zrot_(&i__4, &a_ref(jch, jch + 1), lda, &a_ref(jch +
1, jch + 1), lda, &c__, &s);
i__4 = ilastm - jch;
zrot_(&i__4, &b_ref(jch, jch + 1), ldb, &b_ref(jch +
1, jch + 1), ldb, &c__, &s);
if (ilq) {
d_cnjg(&z__1, &s);
zrot_(n, &q_ref(1, jch), &c__1, &q_ref(1, jch + 1)
, &c__1, &c__, &z__1);
}
if (ilazr2) {
i__4 = a_subscr(jch, jch - 1);
i__5 = a_subscr(jch, jch - 1);
z__1.r = c__ * a[i__5].r, z__1.i = c__ * a[i__5]
.i;
a[i__4].r = z__1.r, a[i__4].i = z__1.i;
}
ilazr2 = FALSE_;
/* --------------- Begin Timing Code ----------------- */
opst += (doublereal) ((ilastm - jch) * 40 + 32 + nq *
20);
/* ---------------- End Timing Code ------------------ */
i__4 = b_subscr(jch + 1, jch + 1);
if ((d__1 = b[i__4].r, abs(d__1)) + (d__2 = d_imag(&
b_ref(jch + 1, jch + 1)), abs(d__2)) >= btol)
{
if (jch + 1 >= ilast) {
goto L60;
} else {
ifirst = jch + 1;
goto L70;
}
}
i__4 = b_subscr(jch + 1, jch + 1);
b[i__4].r = 0., b[i__4].i = 0.;
/* L20: */
}
goto L50;
} else {
/* Only test 2 passed -- chase the zero to B(ILAST,ILAST)
Then process as in the case B(ILAST,ILAST)=0 */
i__3 = ilast - 1;
for (jch = j; jch <= i__3; ++jch) {
i__4 = b_subscr(jch, jch + 1);
ctemp.r = b[i__4].r, ctemp.i = b[i__4].i;
zlartg_(&ctemp, &b_ref(jch + 1, jch + 1), &c__, &s, &
b_ref(jch, jch + 1));
i__4 = b_subscr(jch + 1, jch + 1);
b[i__4].r = 0., b[i__4].i = 0.;
if (jch < ilastm - 1) {
i__4 = ilastm - jch - 1;
zrot_(&i__4, &b_ref(jch, jch + 2), ldb, &b_ref(
jch + 1, jch + 2), ldb, &c__, &s);
}
i__4 = ilastm - jch + 2;
zrot_(&i__4, &a_ref(jch, jch - 1), lda, &a_ref(jch +
1, jch - 1), lda, &c__, &s);
if (ilq) {
d_cnjg(&z__1, &s);
zrot_(n, &q_ref(1, jch), &c__1, &q_ref(1, jch + 1)
, &c__1, &c__, &z__1);
}
i__4 = a_subscr(jch + 1, jch);
ctemp.r = a[i__4].r, ctemp.i = a[i__4].i;
zlartg_(&ctemp, &a_ref(jch + 1, jch - 1), &c__, &s, &
a_ref(jch + 1, jch));
i__4 = a_subscr(jch + 1, jch - 1);
a[i__4].r = 0., a[i__4].i = 0.;
i__4 = jch + 1 - ifrstm;
zrot_(&i__4, &a_ref(ifrstm, jch), &c__1, &a_ref(
ifrstm, jch - 1), &c__1, &c__, &s);
i__4 = jch - ifrstm;
zrot_(&i__4, &b_ref(ifrstm, jch), &c__1, &b_ref(
ifrstm, jch - 1), &c__1, &c__, &s);
if (ilz) {
zrot_(n, &z___ref(1, jch), &c__1, &z___ref(1, jch
- 1), &c__1, &c__, &s);
}
/* L30: */
}
/* ---------------- Begin Timing Code ------------------- */
opst += (doublereal) ((ilastm + 1 - ifrstm) * 40 + 64 + (
nq + nz) * 20) * (doublereal) (ilast - j);
/* ----------------- End Timing Code -------------------- */
goto L50;
}
} else if (ilazro) {
/* Only test 1 passed -- work on J:ILAST */
ifirst = j;
goto L70;
}
/* Neither test passed -- try next J
L40: */
}
/* (Drop-through is "impossible") */
*info = (*n << 1) + 1;
goto L210;
/* B(ILAST,ILAST)=0 -- clear A(ILAST,ILAST-1) to split off a
1x1 block. */
L50:
i__2 = a_subscr(ilast, ilast);
ctemp.r = a[i__2].r, ctemp.i = a[i__2].i;
zlartg_(&ctemp, &a_ref(ilast, ilast - 1), &c__, &s, &a_ref(ilast,
ilast));
i__2 = a_subscr(ilast, ilast - 1);
a[i__2].r = 0., a[i__2].i = 0.;
i__2 = ilast - ifrstm;
zrot_(&i__2, &a_ref(ifrstm, ilast), &c__1, &a_ref(ifrstm, ilast - 1),
&c__1, &c__, &s);
i__2 = ilast - ifrstm;
zrot_(&i__2, &b_ref(ifrstm, ilast), &c__1, &b_ref(ifrstm, ilast - 1),
&c__1, &c__, &s);
if (ilz) {
zrot_(n, &z___ref(1, ilast), &c__1, &z___ref(1, ilast - 1), &c__1,
&c__, &s);
}
/* --------------------- Begin Timing Code ----------------------- */
opst += (doublereal) ((ilast - ifrstm) * 40 + 32 + nz * 20);
/* ---------------------- End Timing Code ------------------------
A(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA */
L60:
absb = z_abs(&b_ref(ilast, ilast));
if (absb > safmin) {
i__2 = b_subscr(ilast, ilast);
z__2.r = b[i__2].r / absb, z__2.i = b[i__2].i / absb;
d_cnjg(&z__1, &z__2);
signbc.r = z__1.r, signbc.i = z__1.i;
i__2 = b_subscr(ilast, ilast);
b[i__2].r = absb, b[i__2].i = 0.;
if (ilschr) {
i__2 = ilast - ifrstm;
zscal_(&i__2, &signbc, &b_ref(ifrstm, ilast), &c__1);
i__2 = ilast + 1 - ifrstm;
zscal_(&i__2, &signbc, &a_ref(ifrstm, ilast), &c__1);
/* ----------------- Begin Timing Code --------------------- */
opst += (doublereal) ((ilast - ifrstm) * 12);
/* ------------------ End Timing Code ---------------------- */
} else {
i__2 = a_subscr(ilast, ilast);
i__3 = a_subscr(ilast, ilast);
z__1.r = a[i__3].r * signbc.r - a[i__3].i * signbc.i, z__1.i =
a[i__3].r * signbc.i + a[i__3].i * signbc.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
}
if (ilz) {
zscal_(n, &signbc, &z___ref(1, ilast), &c__1);
}
/* ------------------- Begin Timing Code ---------------------- */
opst += (doublereal) (nz * 6 + 13);
/* -------------------- End Timing Code ----------------------- */
} else {
i__2 = b_subscr(ilast, ilast);
b[i__2].r = 0., b[i__2].i = 0.;
}
i__2 = ilast;
i__3 = a_subscr(ilast, ilast);
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = ilast;
i__3 = b_subscr(ilast, ilast);
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* Go to next block -- exit if finished. */
--ilast;
if (ilast < *ilo) {
goto L190;
}
/* Reset counters */
iiter = 0;
eshift.r = 0., eshift.i = 0.;
if (! ilschr) {
ilastm = ilast;
if (ifrstm > ilast) {
ifrstm = *ilo;
}
}
goto L160;
/* QZ step
This iteration only involves rows/columns IFIRST:ILAST. We
assume IFIRST < ILAST, and that the diagonal of B is non-zero. */
L70:
++iiter;
if (! ilschr) {
ifrstm = ifirst;
}
/* Compute the Shift.
At this point, IFIRST < ILAST, and the diagonal elements of
B(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
magnitude) */
if (iiter / 10 * 10 != iiter) {
/* The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
the bottom-right 2x2 block of A inv(B) which is nearest to
the bottom-right element.
We factor B as U*D, where U has unit diagonals, and
compute (A*inv(D))*inv(U). */
i__2 = b_subscr(ilast - 1, ilast);
z__2.r = bscale * b[i__2].r, z__2.i = bscale * b[i__2].i;
i__3 = b_subscr(ilast, ilast);
z__3.r = bscale * b[i__3].r, z__3.i = bscale * b[i__3].i;
z_div(&z__1, &z__2, &z__3);
u12.r = z__1.r, u12.i = z__1.i;
i__2 = a_subscr(ilast - 1, ilast - 1);
z__2.r = ascale * a[i__2].r, z__2.i = ascale * a[i__2].i;
i__3 = b_subscr(ilast - 1, ilast - 1);
z__3.r = bscale * b[i__3].r, z__3.i = bscale * b[i__3].i;
z_div(&z__1, &z__2, &z__3);
ad11.r = z__1.r, ad11.i = z__1.i;
i__2 = a_subscr(ilast, ilast - 1);
z__2.r = ascale * a[i__2].r, z__2.i = ascale * a[i__2].i;
i__3 = b_subscr(ilast - 1, ilast - 1);
z__3.r = bscale * b[i__3].r, z__3.i = bscale * b[i__3].i;
z_div(&z__1, &z__2, &z__3);
ad21.r = z__1.r, ad21.i = z__1.i;
i__2 = a_subscr(ilast - 1, ilast);
z__2.r = ascale * a[i__2].r, z__2.i = ascale * a[i__2].i;
i__3 = b_subscr(ilast, ilast);
z__3.r = bscale * b[i__3].r, z__3.i = bscale * b[i__3].i;
z_div(&z__1, &z__2, &z__3);
ad12.r = z__1.r, ad12.i = z__1.i;
i__2 = a_subscr(ilast, ilast);
z__2.r = ascale * a[i__2].r, z__2.i = ascale * a[i__2].i;
i__3 = b_subscr(ilast, ilast);
z__3.r = bscale * b[i__3].r, z__3.i = bscale * b[i__3].i;
z_div(&z__1, &z__2, &z__3);
ad22.r = z__1.r, ad22.i = z__1.i;
z__2.r = u12.r * ad21.r - u12.i * ad21.i, z__2.i = u12.r * ad21.i
+ u12.i * ad21.r;
z__1.r = ad22.r - z__2.r, z__1.i = ad22.i - z__2.i;
abi22.r = z__1.r, abi22.i = z__1.i;
z__2.r = ad11.r + abi22.r, z__2.i = ad11.i + abi22.i;
z__1.r = z__2.r * .5, z__1.i = z__2.i * .5;
t.r = z__1.r, t.i = z__1.i;
pow_zi(&z__4, &t, &c__2);
z__5.r = ad12.r * ad21.r - ad12.i * ad21.i, z__5.i = ad12.r *
ad21.i + ad12.i * ad21.r;
z__3.r = z__4.r + z__5.r, z__3.i = z__4.i + z__5.i;
z__6.r = ad11.r * ad22.r - ad11.i * ad22.i, z__6.i = ad11.r *
ad22.i + ad11.i * ad22.r;
z__2.r = z__3.r - z__6.r, z__2.i = z__3.i - z__6.i;
z_sqrt(&z__1, &z__2);
rtdisc.r = z__1.r, rtdisc.i = z__1.i;
z__1.r = t.r - abi22.r, z__1.i = t.i - abi22.i;
z__2.r = t.r - abi22.r, z__2.i = t.i - abi22.i;
temp = z__1.r * rtdisc.r + d_imag(&z__2) * d_imag(&rtdisc);
if (temp <= 0.) {
z__1.r = t.r + rtdisc.r, z__1.i = t.i + rtdisc.i;
shift.r = z__1.r, shift.i = z__1.i;
} else {
z__1.r = t.r - rtdisc.r, z__1.i = t.i - rtdisc.i;
shift.r = z__1.r, shift.i = z__1.i;
}
/* ------------------- Begin Timing Code ---------------------- */
opst += 116.;
/* -------------------- End Timing Code ----------------------- */
} else {
/* Exceptional shift. Chosen for no particularly good reason. */
i__2 = a_subscr(ilast - 1, ilast);
z__4.r = ascale * a[i__2].r, z__4.i = ascale * a[i__2].i;
i__3 = b_subscr(ilast - 1, ilast - 1);
z__5.r = bscale * b[i__3].r, z__5.i = bscale * b[i__3].i;
z_div(&z__3, &z__4, &z__5);
d_cnjg(&z__2, &z__3);
z__1.r = eshift.r + z__2.r, z__1.i = eshift.i + z__2.i;
eshift.r = z__1.r, eshift.i = z__1.i;
shift.r = eshift.r, shift.i = eshift.i;
/* ------------------- Begin Timing Code ---------------------- */
opst += 15.;
/* -------------------- End Timing Code ----------------------- */
}
/* Now check for two consecutive small subdiagonals. */
i__2 = ifirst + 1;
for (j = ilast - 1; j >= i__2; --j) {
istart = j;
i__3 = a_subscr(j, j);
z__2.r = ascale * a[i__3].r, z__2.i = ascale * a[i__3].i;
i__4 = b_subscr(j, j);
z__4.r = bscale * b[i__4].r, z__4.i = bscale * b[i__4].i;
z__3.r = shift.r * z__4.r - shift.i * z__4.i, z__3.i = shift.r *
z__4.i + shift.i * z__4.r;
z__1.r = z__2.r - z__3.r, z__1.i = z__2.i - z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
temp = (d__1 = ctemp.r, abs(d__1)) + (d__2 = d_imag(&ctemp), abs(
d__2));
i__3 = a_subscr(j + 1, j);
temp2 = ascale * ((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&
a_ref(j + 1, j)), abs(d__2)));
tempr = max(temp,temp2);
if (tempr < 1. && tempr != 0.) {
temp /= tempr;
temp2 /= tempr;
}
i__3 = a_subscr(j, j - 1);
if (((d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a_ref(j, j -
1)), abs(d__2))) * temp2 <= temp * atol) {
goto L90;
}
/* L80: */
}
istart = ifirst;
i__2 = a_subscr(ifirst, ifirst);
z__2.r = ascale * a[i__2].r, z__2.i = ascale * a[i__2].i;
i__3 = b_subscr(ifirst, ifirst);
z__4.r = bscale * b[i__3].r, z__4.i = bscale * b[i__3].i;
z__3.r = shift.r * z__4.r - shift.i * z__4.i, z__3.i = shift.r *
z__4.i + shift.i * z__4.r;
z__1.r = z__2.r - z__3.r, z__1.i = z__2.i - z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
/* --------------------- Begin Timing Code ----------------------- */
opst += -6.;
/* ---------------------- End Timing Code ------------------------ */
L90:
/* Do an implicit-shift QZ sweep.
Initial Q */
i__2 = a_subscr(istart + 1, istart);
z__1.r = ascale * a[i__2].r, z__1.i = ascale * a[i__2].i;
ctemp2.r = z__1.r, ctemp2.i = z__1.i;
/* --------------------- Begin Timing Code ----------------------- */
opst += (doublereal) ((ilast - istart) * 18 + 2);
/* ---------------------- End Timing Code ------------------------ */
zlartg_(&ctemp, &ctemp2, &c__, &s, &ctemp3);
/* Sweep */
i__2 = ilast - 1;
for (j = istart; j <= i__2; ++j) {
if (j > istart) {
i__3 = a_subscr(j, j - 1);
ctemp.r = a[i__3].r, ctemp.i = a[i__3].i;
zlartg_(&ctemp, &a_ref(j + 1, j - 1), &c__, &s, &a_ref(j, j -
1));
i__3 = a_subscr(j + 1, j - 1);
a[i__3].r = 0., a[i__3].i = 0.;
}
i__3 = ilastm;
for (jc = j; jc <= i__3; ++jc) {
i__4 = a_subscr(j, jc);
z__2.r = c__ * a[i__4].r, z__2.i = c__ * a[i__4].i;
i__5 = a_subscr(j + 1, jc);
z__3.r = s.r * a[i__5].r - s.i * a[i__5].i, z__3.i = s.r * a[
i__5].i + s.i * a[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
i__4 = a_subscr(j + 1, jc);
d_cnjg(&z__4, &s);
z__3.r = -z__4.r, z__3.i = -z__4.i;
i__5 = a_subscr(j, jc);
z__2.r = z__3.r * a[i__5].r - z__3.i * a[i__5].i, z__2.i =
z__3.r * a[i__5].i + z__3.i * a[i__5].r;
i__6 = a_subscr(j + 1, jc);
z__5.r = c__ * a[i__6].r, z__5.i = c__ * a[i__6].i;
z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i;
a[i__4].r = z__1.r, a[i__4].i = z__1.i;
i__4 = a_subscr(j, jc);
a[i__4].r = ctemp.r, a[i__4].i = ctemp.i;
i__4 = b_subscr(j, jc);
z__2.r = c__ * b[i__4].r, z__2.i = c__ * b[i__4].i;
i__5 = b_subscr(j + 1, jc);
z__3.r = s.r * b[i__5].r - s.i * b[i__5].i, z__3.i = s.r * b[
i__5].i + s.i * b[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp2.r = z__1.r, ctemp2.i = z__1.i;
i__4 = b_subscr(j + 1, jc);
d_cnjg(&z__4, &s);
z__3.r = -z__4.r, z__3.i = -z__4.i;
i__5 = b_subscr(j, jc);
z__2.r = z__3.r * b[i__5].r - z__3.i * b[i__5].i, z__2.i =
z__3.r * b[i__5].i + z__3.i * b[i__5].r;
i__6 = b_subscr(j + 1, jc);
z__5.r = c__ * b[i__6].r, z__5.i = c__ * b[i__6].i;
z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i;
b[i__4].r = z__1.r, b[i__4].i = z__1.i;
i__4 = b_subscr(j, jc);
b[i__4].r = ctemp2.r, b[i__4].i = ctemp2.i;
/* L100: */
}
if (ilq) {
i__3 = *n;
for (jr = 1; jr <= i__3; ++jr) {
i__4 = q_subscr(jr, j);
z__2.r = c__ * q[i__4].r, z__2.i = c__ * q[i__4].i;
d_cnjg(&z__4, &s);
i__5 = q_subscr(jr, j + 1);
z__3.r = z__4.r * q[i__5].r - z__4.i * q[i__5].i, z__3.i =
z__4.r * q[i__5].i + z__4.i * q[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
i__4 = q_subscr(jr, j + 1);
z__3.r = -s.r, z__3.i = -s.i;
i__5 = q_subscr(jr, j);
z__2.r = z__3.r * q[i__5].r - z__3.i * q[i__5].i, z__2.i =
z__3.r * q[i__5].i + z__3.i * q[i__5].r;
i__6 = q_subscr(jr, j + 1);
z__4.r = c__ * q[i__6].r, z__4.i = c__ * q[i__6].i;
z__1.r = z__2.r + z__4.r, z__1.i = z__2.i + z__4.i;
q[i__4].r = z__1.r, q[i__4].i = z__1.i;
i__4 = q_subscr(jr, j);
q[i__4].r = ctemp.r, q[i__4].i = ctemp.i;
/* L110: */
}
}
i__3 = b_subscr(j + 1, j + 1);
ctemp.r = b[i__3].r, ctemp.i = b[i__3].i;
zlartg_(&ctemp, &b_ref(j + 1, j), &c__, &s, &b_ref(j + 1, j + 1));
i__3 = b_subscr(j + 1, j);
b[i__3].r = 0., b[i__3].i = 0.;
/* Computing MIN */
i__4 = j + 2;
i__3 = min(i__4,ilast);
for (jr = ifrstm; jr <= i__3; ++jr) {
i__4 = a_subscr(jr, j + 1);
z__2.r = c__ * a[i__4].r, z__2.i = c__ * a[i__4].i;
i__5 = a_subscr(jr, j);
z__3.r = s.r * a[i__5].r - s.i * a[i__5].i, z__3.i = s.r * a[
i__5].i + s.i * a[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
i__4 = a_subscr(jr, j);
d_cnjg(&z__4, &s);
z__3.r = -z__4.r, z__3.i = -z__4.i;
i__5 = a_subscr(jr, j + 1);
z__2.r = z__3.r * a[i__5].r - z__3.i * a[i__5].i, z__2.i =
z__3.r * a[i__5].i + z__3.i * a[i__5].r;
i__6 = a_subscr(jr, j);
z__5.r = c__ * a[i__6].r, z__5.i = c__ * a[i__6].i;
z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i;
a[i__4].r = z__1.r, a[i__4].i = z__1.i;
i__4 = a_subscr(jr, j + 1);
a[i__4].r = ctemp.r, a[i__4].i = ctemp.i;
/* L120: */
}
i__3 = j;
for (jr = ifrstm; jr <= i__3; ++jr) {
i__4 = b_subscr(jr, j + 1);
z__2.r = c__ * b[i__4].r, z__2.i = c__ * b[i__4].i;
i__5 = b_subscr(jr, j);
z__3.r = s.r * b[i__5].r - s.i * b[i__5].i, z__3.i = s.r * b[
i__5].i + s.i * b[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
i__4 = b_subscr(jr, j);
d_cnjg(&z__4, &s);
z__3.r = -z__4.r, z__3.i = -z__4.i;
i__5 = b_subscr(jr, j + 1);
z__2.r = z__3.r * b[i__5].r - z__3.i * b[i__5].i, z__2.i =
z__3.r * b[i__5].i + z__3.i * b[i__5].r;
i__6 = b_subscr(jr, j);
z__5.r = c__ * b[i__6].r, z__5.i = c__ * b[i__6].i;
z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i;
b[i__4].r = z__1.r, b[i__4].i = z__1.i;
i__4 = b_subscr(jr, j + 1);
b[i__4].r = ctemp.r, b[i__4].i = ctemp.i;
/* L130: */
}
if (ilz) {
i__3 = *n;
for (jr = 1; jr <= i__3; ++jr) {
i__4 = z___subscr(jr, j + 1);
z__2.r = c__ * z__[i__4].r, z__2.i = c__ * z__[i__4].i;
i__5 = z___subscr(jr, j);
z__3.r = s.r * z__[i__5].r - s.i * z__[i__5].i, z__3.i =
s.r * z__[i__5].i + s.i * z__[i__5].r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
ctemp.r = z__1.r, ctemp.i = z__1.i;
i__4 = z___subscr(jr, j);
d_cnjg(&z__4, &s);
z__3.r = -z__4.r, z__3.i = -z__4.i;
i__5 = z___subscr(jr, j + 1);
z__2.r = z__3.r * z__[i__5].r - z__3.i * z__[i__5].i,
z__2.i = z__3.r * z__[i__5].i + z__3.i * z__[i__5]
.r;
i__6 = z___subscr(jr, j);
z__5.r = c__ * z__[i__6].r, z__5.i = c__ * z__[i__6].i;
z__1.r = z__2.r + z__5.r, z__1.i = z__2.i + z__5.i;
z__[i__4].r = z__1.r, z__[i__4].i = z__1.i;
i__4 = z___subscr(jr, j + 1);
z__[i__4].r = ctemp.r, z__[i__4].i = ctemp.i;
/* L140: */
}
}
/* L150: */
}
/* --------------------- Begin Timing Code ----------------------- */
opst += (doublereal) (ilast - istart) * (doublereal) ((ilastm -
ifrstm) * 40 + 184 + (nq + nz) * 20) - 20;
/* ---------------------- End Timing Code ------------------------ */
L160:
/* --------------------- Begin Timing Code -----------------------
End of iteration -- add in "small" contributions. */
latime_1.ops += opst;
opst = 0.;
/* ---------------------- End Timing Code ------------------------
L170: */
}
/* Drop-through = non-convergence */
L180:
*info = ilast;
/* ---------------------- Begin Timing Code ------------------------- */
latime_1.ops += opst;
opst = 0.;
/* ----------------------- End Timing Code -------------------------- */
goto L210;
/* Successful completion of all QZ steps */
L190:
/* Set Eigenvalues 1:ILO-1 */
i__1 = *ilo - 1;
for (j = 1; j <= i__1; ++j) {
absb = z_abs(&b_ref(j, j));
if (absb > safmin) {
i__2 = b_subscr(j, j);
z__2.r = b[i__2].r / absb, z__2.i = b[i__2].i / absb;
d_cnjg(&z__1, &z__2);
signbc.r = z__1.r, signbc.i = z__1.i;
i__2 = b_subscr(j, j);
b[i__2].r = absb, b[i__2].i = 0.;
if (ilschr) {
i__2 = j - 1;
zscal_(&i__2, &signbc, &b_ref(1, j), &c__1);
zscal_(&j, &signbc, &a_ref(1, j), &c__1);
/* ----------------- Begin Timing Code --------------------- */
opst += (doublereal) ((j - 1) * 12);
/* ------------------ End Timing Code ---------------------- */
} else {
i__2 = a_subscr(j, j);
i__3 = a_subscr(j, j);
z__1.r = a[i__3].r * signbc.r - a[i__3].i * signbc.i, z__1.i =
a[i__3].r * signbc.i + a[i__3].i * signbc.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
}
if (ilz) {
zscal_(n, &signbc, &z___ref(1, j), &c__1);
}
/* ------------------- Begin Timing Code ---------------------- */
opst += (doublereal) (nz * 6 + 13);
/* -------------------- End Timing Code ----------------------- */
} else {
i__2 = b_subscr(j, j);
b[i__2].r = 0., b[i__2].i = 0.;
}
i__2 = j;
i__3 = a_subscr(j, j);
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = j;
i__3 = b_subscr(j, j);
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L200: */
}
/* Normal Termination */
*info = 0;
/* Exit (other than argument error) -- return optimal workspace size */
L210:
/* ---------------------- Begin Timing Code ------------------------- */
latime_1.ops += opst;
opst = 0.;
latime_1.itcnt = (doublereal) jiter;
/* ----------------------- End Timing Code -------------------------- */
z__1.r = (doublereal) (*n), z__1.i = 0.;
work[1].r = z__1.r, work[1].i = z__1.i;
return 0;
/* End of ZHGEQZ */
} /* zhgeqz_ */
/* Subroutine */ int ctgsen_(integer *ijob, logical *wantq, logical *wantz,
logical *select, integer *n, complex *a, integer *lda, complex *b,
integer *ldb, complex *alpha, complex *beta, complex *q, integer *ldq,
complex *z__, integer *ldz, integer *m, real *pl, real *pr, real *
dif, complex *work, integer *lwork, integer *iwork, integer *liwork,
integer *info)
{
/* -- 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
Purpose
=======
CTGSEN reorders the generalized Schur decomposition of a complex
matrix pair (A, B) (in terms of an unitary equivalence trans-
formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues
appears in the leading diagonal blocks of the pair (A,B). The leading
columns of Q and Z form unitary bases of the corresponding left and
right eigenspaces (deflating subspaces). (A, B) must be in
generalized Schur canonical form, that is, A and B are both upper
triangular.
CTGSEN also computes the generalized eigenvalues
w(j)= ALPHA(j) / BETA(j)
of the reordered matrix pair (A, B).
Optionally, the routine computes estimates of reciprocal condition
numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
(A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
between the matrix pairs (A11, B11) and (A22,B22) that correspond to
the selected cluster and the eigenvalues outside the cluster, resp.,
and norms of "projections" onto left and right eigenspaces w.r.t.
the selected cluster in the (1,1)-block.
Arguments
=========
IJOB (input) integer
Specifies whether condition numbers are required for the
cluster of eigenvalues (PL and PR) or the deflating subspaces
(Difu and Difl):
=0: Only reorder w.r.t. SELECT. No extras.
=1: Reciprocal of norms of "projections" onto left and right
eigenspaces w.r.t. the selected cluster (PL and PR).
=2: Upper bounds on Difu and Difl. F-norm-based estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1-norm-based estimate
(DIF(1:2)).
About 5 times as expensive as IJOB = 2.
=4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
version to get it all.
=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
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.
SELECT (input) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select an eigenvalue w(j), SELECT(j) must be set to
.TRUE..
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) COMPLEX array, dimension(LDA,N)
On entry, the upper triangular matrix A, in generalized
Schur canonical form.
On exit, A is overwritten by the reordered matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) COMPLEX array, dimension(LDB,N)
On entry, the upper triangular matrix B, in generalized
Schur canonical form.
On exit, B is overwritten by the reordered matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHA (output) COMPLEX array, dimension (N)
BETA (output) COMPLEX array, dimension (N)
The diagonal elements of A and B, respectively,
when the pair (A,B) has been reduced to generalized Schur
form. ALPHA(i)/BETA(i) i=1,...,N are the generalized
eigenvalues.
Q (input/output) COMPLEX array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
On exit, Q has been postmultiplied by the left unitary
transformation matrix which reorder (A, B); The leading M
columns of Q form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTQ = .FALSE., Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1.
If WANTQ = .TRUE., LDQ >= N.
Z (input/output) COMPLEX array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
On exit, Z has been postmultiplied by the left unitary
transformation matrix which reorder (A, B); The leading M
columns of Z form orthonormal bases for the specified pair of
left eigenspaces (deflating subspaces).
If WANTZ = .FALSE., Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1.
If WANTZ = .TRUE., LDZ >= N.
M (output) INTEGER
The dimension of the specified pair of left and right
eigenspaces, (deflating subspaces) 0 <= M <= N.
PL, PR (output) REAL
If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
reciprocal of the norm of "projections" onto left and right
eigenspace with respect to the selected cluster.
0 < PL, PR <= 1.
If M = 0 or M = N, PL = PR = 1.
If IJOB = 0, 2 or 3 PL, PR are not referenced.
DIF (output) REAL array, dimension (2).
If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
estimates of Difu and Difl, computed using reversed
communication with CLACON.
If M = 0 or N, DIF(1:2) = F-norm([A, B]).
If IJOB = 0 or 1, DIF is not referenced.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
IF IJOB = 0, WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1
If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M)
If IJOB = 3 or 5, LWORK >= 4*M*(N-M)
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/output) INTEGER, dimension (LIWORK)
IF IJOB = 0, IWORK is not referenced. Otherwise,
on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1.
If IJOB = 1, 2 or 4, LIWORK >= N+2;
If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
=0: Successful exit.
<0: If INFO = -i, the i-th argument had an illegal value.
=1: Reordering of (A, B) failed because the transformed
matrix pair (A, B) would be too far from generalized
Schur form; the problem is very ill-conditioned.
(A, B) may have been partially reordered.
If requested, 0 is returned in DIF(*), PL and PR.
Further Details
===============
CTGSEN first collects the selected eigenvalues by computing unitary
U and W that move them to the top left corner of (A, B). In other
words, the selected eigenvalues are the eigenvalues of (A11, B11) in
U'*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2
where N = n1+n2 and U' means the conjugate transpose of U. The first
n1 columns of U and W span the specified pair of left and right
eigenspaces (deflating subspaces) of (A, B).
If (A, B) has been obtained from the generalized real Schur
decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the
reordered generalized Schur form of (C, D) is given by
(C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
and the first n1 columns of Q*U and Z*W span the corresponding
deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
Note that if the selected eigenvalue is sufficiently ill-conditioned,
then its value may differ significantly from its value before
reordering.
The reciprocal condition numbers of the left and right eigenspaces
spanned by the first n1 columns of U and W (or Q*U and Z*W) may
be returned in DIF(1:2), corresponding to Difu and Difl, resp.
The Difu and Difl are defined as:
Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
and
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
where sigma-min(Zu) is the smallest singular value of the
(2*n1*n2)-by-(2*n1*n2) matrix
Zu = [ kron(In2, A11) -kron(A22', In1) ]
[ kron(In2, B11) -kron(B22', In1) ].
Here, Inx is the identity matrix of size nx and A22' is the
transpose of A22. kron(X, Y) is the Kronecker product between
the matrices X and Y.
When DIF(2) is small, small changes in (A, B) can cause large changes
in the deflating subspace. An approximate (asymptotic) bound on the
maximum angular error in the computed deflating subspaces is
EPS * norm((A, B)) / DIF(2),
where EPS is the machine precision.
The reciprocal norm of the projectors on the left and right
eigenspaces associated with (A11, B11) may be returned in PL and PR.
They are computed as follows. First we compute L and R so that
P*(A, B)*Q is block diagonal, where
P = ( I -L ) n1 Q = ( I R ) n1
( 0 I ) n2 and ( 0 I ) n2
n1 n2 n1 n2
and (L, R) is the solution to the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / PL.
There are also global error bounds which valid for perturbations up
to a certain restriction: A lower bound (x) on the smallest
F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
(i.e. (A + E, B + F), is
x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
An approximate bound on x can be computed from DIF(1:2), PL and PR.
If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
(L', R') and unperturbed (L, R) left and right deflating subspaces
associated with the selected cluster in the (1,1)-blocks can be
bounded as
max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
See LAPACK User's Guide section 4.11 or the following references
for more information.
Note that if the default method for computing the Frobenius-norm-
based estimate DIF is not wanted (see CLATDF), then the parameter
IDIFJB (see below) should be changed from 3 to 4 (routine CLATDF
(IJOB = 2 will be used)). See CTGSYL for more details.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
References
==========
[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.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
1996.
=====================================================================
Decode and test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* 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;
complex q__1, q__2;
/* Builtin functions */
double sqrt(doublereal), c_abs(complex *);
void r_cnjg(complex *, complex *);
/* Local variables */
static integer kase, ierr;
static real dsum;
static logical swap;
static integer i__, k;
extern /* Subroutine */ int cscal_(integer *, complex *, complex *,
integer *);
static logical wantd;
static integer lwmin;
static logical wantp;
static integer n1, n2;
static logical wantd1, wantd2;
static real dscale;
static integer ks;
extern /* Subroutine */ int clacon_(integer *, complex *, complex *, real
*, integer *);
extern doublereal slamch_(char *);
static real rdscal;
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *);
static real safmin;
extern /* Subroutine */ int ctgexc_(logical *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, integer *,
complex *, integer *, integer *, integer *, integer *), xerbla_(
char *, integer *), classq_(integer *, complex *, integer
*, real *, real *);
static integer liwmin;
extern /* Subroutine */ int ctgsyl_(char *, integer *, integer *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, real *, real *, complex *, integer *, integer *, integer *);
static integer mn2;
static logical lquery;
static integer ijb;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define q_subscr(a_1,a_2) (a_2)*q_dim1 + a_1
#define q_ref(a_1,a_2) q[q_subscr(a_1,a_2)]
--select;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--alpha;
--beta;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
--dif;
--work;
--iwork;
/* Function Body */
*info = 0;
lquery = *lwork == -1 || *liwork == -1;
if (*ijob < 0 || *ijob > 5) {
*info = -1;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldq < 1 || *wantq && *ldq < *n) {
*info = -13;
} else if (*ldz < 1 || *wantz && *ldz < *n) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTGSEN", &i__1);
return 0;
}
ierr = 0;
wantp = *ijob == 1 || *ijob >= 4;
wantd1 = *ijob == 2 || *ijob == 4;
wantd2 = *ijob == 3 || *ijob == 5;
wantd = wantd1 || wantd2;
/* Set M to the dimension of the specified pair of deflating
subspaces. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = a_subscr(k, k);
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = b_subscr(k, k);
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
if (k < *n) {
if (select[k]) {
++(*m);
}
} else {
if (select[*n]) {
++(*m);
}
}
/* L10: */
}
if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m);
lwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + 2;
liwmin = max(i__1,i__2);
} else if (*ijob == 3 || *ijob == 5) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 2) * (*n - *m);
lwmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = max(i__1,i__2), i__2 =
*n + 2;
liwmin = max(i__1,i__2);
} else {
lwmin = 1;
liwmin = 1;
}
work[1].r = (real) lwmin, work[1].i = 0.f;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -21;
} else if (*liwork < liwmin && ! lquery) {
*info = -23;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTGSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible. */
if (*m == *n || *m == 0) {
if (wantp) {
*pl = 1.f;
*pr = 1.f;
}
if (wantd) {
dscale = 0.f;
dsum = 1.f;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
classq_(n, &a_ref(1, i__), &c__1, &dscale, &dsum);
classq_(n, &b_ref(1, i__), &c__1, &dscale, &dsum);
/* L20: */
}
dif[1] = dscale * sqrt(dsum);
dif[2] = dif[1];
}
goto L70;
}
/* Get machine constant */
safmin = slamch_("S");
/* Collect the selected blocks at the top-left corner of (A, B). */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
swap = select[k];
if (swap) {
++ks;
/* Swap the K-th block to position KS. Compute unitary Q
and Z that will swap adjacent diagonal blocks in (A, B). */
if (k != ks) {
ctgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb,
&q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, &
ierr);
}
if (ierr > 0) {
/* Swap is rejected: exit. */
*info = 1;
if (wantp) {
*pl = 0.f;
*pr = 0.f;
}
if (wantd) {
dif[1] = 0.f;
dif[2] = 0.f;
}
goto L70;
}
}
/* L30: */
}
if (wantp) {
/* Solve generalized Sylvester equation for R and L:
A11 * R - L * A22 = A12
B11 * R - L * B22 = B12 */
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
clacpy_("Full", &n1, &n2, &a_ref(1, i__), lda, &work[1], &n1);
clacpy_("Full", &n1, &n2, &b_ref(1, i__), ldb, &work[n1 * n2 + 1], &
n1);
ijb = 0;
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a_ref(i__, i__), lda,
&work[1], &n1, &b[b_offset], ldb, &b_ref(i__, i__), ldb, &
work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1)
+ 1], &i__1, &iwork[1], &ierr);
/* Estimate the reciprocal of norms of "projections" onto
left and right eigenspaces */
rdscal = 0.f;
dsum = 1.f;
i__1 = n1 * n2;
classq_(&i__1, &work[1], &c__1, &rdscal, &dsum);
*pl = rdscal * sqrt(dsum);
if (*pl == 0.f) {
*pl = 1.f;
} else {
*pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl));
}
rdscal = 0.f;
dsum = 1.f;
i__1 = n1 * n2;
classq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum);
*pr = rdscal * sqrt(dsum);
if (*pr == 0.f) {
*pr = 1.f;
} else {
*pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr));
}
}
if (wantd) {
/* Compute estimates Difu and Difl. */
if (wantd1) {
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 3;
/* Frobenius norm-based Difu estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a_ref(i__, i__),
lda, &work[1], &n1, &b[b_offset], ldb, &b_ref(i__, i__),
ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &work[(n1
* n2 << 1) + 1], &i__1, &iwork[1], &ierr);
/* Frobenius norm-based Difl estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n2, &n1, &a_ref(i__, i__), lda, &a[a_offset],
lda, &work[1], &n2, &b_ref(i__, i__), ldb, &b[b_offset],
ldb, &work[n1 * n2 + 1], &n2, &dscale, &dif[2], &work[(n1
* n2 << 1) + 1], &i__1, &iwork[1], &ierr);
} else {
/* Compute 1-norm-based estimates of Difu and Difl using
reversed communication with CLACON. In each step a
generalized Sylvester equation or a transposed variant
is solved. */
kase = 0;
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 0;
mn2 = (n1 << 1) * n2;
/* 1-norm-based estimate of Difu. */
L40:
clacon_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a_ref(
i__, i__), lda, &work[1], &n1, &b[b_offset], ldb,
&b_ref(i__, i__), ldb, &work[n1 * n2 + 1], &n1, &
dscale, &dif[1], &work[(n1 * n2 << 1) + 1], &i__1,
&iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a_ref(
i__, i__), lda, &work[1], &n1, &b[b_offset], ldb,
&b_ref(i__, i__), ldb, &work[n1 * n2 + 1], &n1, &
dscale, &dif[1], &work[(n1 * n2 << 1) + 1], &i__1,
&iwork[1], &ierr);
}
goto L40;
}
dif[1] = dscale / dif[1];
/* 1-norm-based estimate of Difl. */
L50:
clacon_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("N", &ijb, &n2, &n1, &a_ref(i__, i__), lda, &a[
a_offset], lda, &work[1], &n2, &b_ref(i__, i__),
ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &
dscale, &dif[2], &work[(n1 * n2 << 1) + 1], &i__1,
&iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ctgsyl_("C", &ijb, &n2, &n1, &a_ref(i__, i__), lda, &a[
a_offset], lda, &work[1], &n2, &b[b_offset], ldb,
&b_ref(i__, i__), ldb, &work[n1 * n2 + 1], &n2, &
dscale, &dif[2], &work[(n1 * n2 << 1) + 1], &i__1,
&iwork[1], &ierr);
}
goto L50;
}
dif[2] = dscale / dif[2];
}
}
/* If B(K,K) is complex, make it real and positive (normalization
of the generalized Schur form) and Store the generalized
eigenvalues of reordered pair (A, B) */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
dscale = c_abs(&b_ref(k, k));
if (dscale > safmin) {
i__2 = b_subscr(k, k);
q__2.r = b[i__2].r / dscale, q__2.i = b[i__2].i / dscale;
r_cnjg(&q__1, &q__2);
work[1].r = q__1.r, work[1].i = q__1.i;
i__2 = b_subscr(k, k);
q__1.r = b[i__2].r / dscale, q__1.i = b[i__2].i / dscale;
work[2].r = q__1.r, work[2].i = q__1.i;
i__2 = b_subscr(k, k);
b[i__2].r = dscale, b[i__2].i = 0.f;
i__2 = *n - k;
cscal_(&i__2, &work[1], &b_ref(k, k + 1), ldb);
i__2 = *n - k + 1;
cscal_(&i__2, &work[1], &a_ref(k, k), lda);
if (*wantq) {
cscal_(n, &work[2], &q_ref(1, k), &c__1);
}
} else {
i__2 = b_subscr(k, k);
b[i__2].r = 0.f, b[i__2].i = 0.f;
}
i__2 = k;
i__3 = a_subscr(k, k);
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = b_subscr(k, k);
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L60: */
}
L70:
work[1].r = (real) lwmin, work[1].i = 0.f;
iwork[1] = liwmin;
return 0;
/* End of CTGSEN */
} /* ctgsen_ */
/* Subroutine */ int cchkbd_(integer *nsizes, integer *mval, integer *nval,
integer *ntypes, logical *dotype, integer *nrhs, integer *iseed, real
*thresh, complex *a, integer *lda, real *bd, real *be, real *s1, real
*s2, complex *x, integer *ldx, complex *y, complex *z__, complex *q,
integer *ldq, complex *pt, integer *ldpt, complex *u, complex *vt,
complex *work, integer *lwork, real *rwork, integer *nout, integer *
info)
{
/* Initialized data */
static integer ktype[16] = { 1,2,4,4,4,4,4,6,6,6,6,6,9,9,9,10 };
static integer kmagn[16] = { 1,1,1,1,1,2,3,1,1,1,2,3,1,2,3,0 };
static integer kmode[16] = { 0,0,4,3,1,4,4,4,3,1,4,4,0,0,0,0 };
/* Format strings */
static char fmt_9998[] = "(\002 CCHKBD: \002,a,\002 returned INFO=\002,i"
"6,\002.\002,/9x,\002M=\002,i6,\002, N=\002,i6,\002, JTYPE=\002,i"
"6,\002, ISEED=(\002,3(i5,\002,\002),i5,\002)\002)";
static char fmt_9999[] = "(\002 M=\002,i5,\002, N=\002,i5,\002, type "
"\002,i2,\002, seed=\002,4(i4,\002,\002),\002 test(\002,i2,\002)"
"=\002,g11.4)";
/* System generated locals */
integer a_dim1, a_offset, pt_dim1, pt_offset, q_dim1, q_offset, u_dim1,
u_offset, vt_dim1, vt_offset, x_dim1, x_offset, y_dim1, y_offset,
z_dim1, z_offset, i__1, i__2, i__3, i__4, i__5, i__6, i__7;
real r__1, r__2, r__3, r__4, r__5, r__6, r__7;
/* Builtin functions
Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen);
double log(doublereal), sqrt(doublereal), exp(doublereal);
integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void);
/* Local variables */
static real cond;
static integer jcol;
static char path[3];
static integer mmax, nmax;
static real unfl, ovfl;
static char uplo[1];
static real temp1, temp2;
static integer i__, j, m, n;
extern /* Subroutine */ int cbdt01_(integer *, integer *, integer *,
complex *, integer *, complex *, integer *, real *, real *,
complex *, integer *, complex *, real *, real *);
static logical badmm, badnn;
extern /* Subroutine */ int cbdt02_(integer *, integer *, complex *,
integer *, complex *, integer *, complex *, integer *, complex *,
real *, real *), cbdt03_(char *, integer *, integer *, real *,
real *, complex *, integer *, real *, complex *, integer *,
complex *, real *);
static integer nfail, imode;
extern /* Subroutine */ int cgemm_(char *, char *, integer *, integer *,
integer *, complex *, complex *, integer *, complex *, integer *,
complex *, complex *, integer *);
static real dumma[1];
static integer iinfo;
extern /* Subroutine */ int cunt01_(char *, integer *, integer *, complex
*, integer *, complex *, integer *, real *, real *);
static real anorm;
static integer mnmin, mnmax, jsize, itype, jtype, iwork[1], ntest;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *), slahd2_(integer *, char *);
static integer log2ui;
static logical bidiag;
extern /* Subroutine */ int cgebrd_(integer *, integer *, complex *,
integer *, real *, real *, complex *, complex *, complex *,
integer *, integer *), slabad_(real *, real *);
static integer mq;
extern doublereal slamch_(char *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), claset_(char *,
integer *, integer *, complex *, complex *, complex *, integer *), xerbla_(char *, integer *);
static integer ioldsd[4];
extern /* Subroutine */ int cbdsqr_(char *, integer *, integer *, integer
*, integer *, real *, real *, complex *, integer *, complex *,
integer *, complex *, integer *, real *, integer *),
cungbr_(char *, integer *, integer *, integer *, complex *,
integer *, complex *, complex *, integer *, integer *),
alasum_(char *, integer *, integer *, integer *, integer *);
extern doublereal slarnd_(integer *, integer *);
extern /* Subroutine */ int clatmr_(integer *, integer *, char *, integer
*, char *, complex *, integer *, real *, complex *, char *, char *
, complex *, integer *, real *, complex *, integer *, real *,
char *, integer *, integer *, integer *, real *, real *, char *,
complex *, integer *, integer *, integer *), clatms_(integer *, integer *,
char *, integer *, char *, real *, integer *, real *, real *,
integer *, integer *, char *, complex *, integer *, complex *,
integer *);
static real amninv;
extern /* Subroutine */ int ssvdch_(integer *, real *, real *, real *,
real *, integer *);
static integer minwrk;
static real rtunfl, rtovfl, ulpinv, result[14];
static integer mtypes;
static real ulp;
/* Fortran I/O blocks */
static cilist io___40 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___41 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___43 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___44 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___45 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___46 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___50 = { 0, 0, 0, fmt_9999, 0 };
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
/* -- LAPACK test 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
=======
CCHKBD checks the singular value decomposition (SVD) routines.
CGEBRD reduces a complex general m by n matrix A to real upper or
lower bidiagonal form by an orthogonal transformation: Q' * A * P = B
(or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n
and lower bidiagonal if m < n.
CUNGBR generates the orthogonal matrices Q and P' from CGEBRD.
Note that Q and P are not necessarily square.
CBDSQR computes the singular value decomposition of the bidiagonal
matrix B as B = U S V'. It is called three times to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
3) A = (UQ) S (P'V'), the SVD of the original matrix A.
In addition, CBDSQR has an option to apply the left orthogonal matrix
U to a matrix X, useful in least squares applications.
For each pair of matrix dimensions (M,N) and each selected matrix
type, an M by N matrix A and an M by NRHS matrix X are generated.
The problem dimensions are as follows
A: M x N
Q: M x min(M,N) (but M x M if NRHS > 0)
P: min(M,N) x N
B: min(M,N) x min(M,N)
U, V: min(M,N) x min(M,N)
S1, S2 diagonal, order min(M,N)
X: M x NRHS
For each generated matrix, 14 tests are performed:
Test CGEBRD and CUNGBR
(1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
(2) | I - Q' Q | / ( M ulp )
(3) | I - PT PT' | / ( N ulp )
Test CBDSQR on bidiagonal matrix B
(4) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(5) | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
and Z = U' Y.
(6) | I - U' U | / ( min(M,N) ulp )
(7) | I - VT VT' | / ( min(M,N) ulp )
(8) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(9) 0 if the true singular values of B are within THRESH of
those in S1. 2*THRESH if they are not. (Tested using
SSVDCH)
(10) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
Test CBDSQR on matrix A
(11) | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
(12) | X - (QU) Z | / ( |X| max(M,k) ulp )
(13) | I - (QU)'(QU) | / ( M ulp )
(14) | I - (VT PT) (PT'VT') | / ( N ulp )
The possible matrix types are
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (3), but multiplied by SQRT( overflow threshold )
(7) Same as (3), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U D V, where U and V are orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U D V, where U and V are orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U D V, where U and V are orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Rectangular matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Special case:
(16) A bidiagonal matrix with random entries chosen from a
logarithmic distribution on [ulp^2,ulp^(-2)] (I.e., each
entry is e^x, where x is chosen uniformly on
[ 2 log(ulp), -2 log(ulp) ] .) For *this* type:
(a) CGEBRD is not called to reduce it to bidiagonal form.
(b) the bidiagonal is min(M,N) x min(M,N); if M<N, the
matrix will be lower bidiagonal, otherwise upper.
(c) only tests 5--8 and 14 are performed.
A subset of the full set of matrix types may be selected through
the logical array DOTYPE.
Arguments
==========
NSIZES (input) INTEGER
The number of values of M and N contained in the vectors
MVAL and NVAL. The matrix sizes are used in pairs (M,N).
MVAL (input) INTEGER array, dimension (NM)
The values of the matrix row dimension M.
NVAL (input) INTEGER array, dimension (NM)
The values of the matrix column dimension N.
NTYPES (input) INTEGER
The number of elements in DOTYPE. If it is zero, CCHKBD
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrices are in A and B.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE (input) LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix
of type j will be generated. If NTYPES is smaller than the
maximum number of types defined (PARAMETER MAXTYP), then
types NTYPES+1 through MAXTYP will not be generated. If
NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through
DOTYPE(NTYPES) will be ignored.
NRHS (input) INTEGER
The number of columns in the "right-hand side" matrices X, Y,
and Z, used in testing CBDSQR. If NRHS = 0, then the
operations on the right-hand side will not be tested.
NRHS must be at least 0.
ISEED (input/output) INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The values of ISEED are changed on exit, and can be
used in the next call to CCHKBD to continue the same random
number sequence.
THRESH (input) REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0. Note that the
expected value of the test ratios is O(1), so THRESH should
be a reasonably small multiple of 1, e.g., 10 or 100.
A (workspace) COMPLEX array, dimension (LDA,NMAX)
where NMAX is the maximum value of N in NVAL.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,MMAX),
where MMAX is the maximum value of M in MVAL.
BD (workspace) REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
BE (workspace) REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
S1 (workspace) REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
S2 (workspace) REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
X (workspace) COMPLEX array, dimension (LDX,NRHS)
LDX (input) INTEGER
The leading dimension of the arrays X, Y, and Z.
LDX >= max(1,MMAX).
Y (workspace) COMPLEX array, dimension (LDX,NRHS)
Z (workspace) COMPLEX array, dimension (LDX,NRHS)
Q (workspace) COMPLEX array, dimension (LDQ,MMAX)
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,MMAX).
PT (workspace) COMPLEX array, dimension (LDPT,NMAX)
LDPT (input) INTEGER
The leading dimension of the arrays PT, U, and V.
LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
U (workspace) COMPLEX array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
V (workspace) COMPLEX array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
WORK (workspace) COMPLEX array, dimension (LWORK)
LWORK (input) INTEGER
The number of entries in WORK. This must be at least
3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all
pairs (M,N)=(MM(j),NN(j))
RWORK (workspace) REAL array, dimension
(5*max(min(M,N)))
NOUT (input) INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
INFO (output) INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some MM(j) < 0
-3: Some NN(j) < 0
-4: NTYPES < 0
-6: NRHS < 0
-8: THRESH < 0
-11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).
-17: LDB < 1 or LDB < MMAX.
-21: LDQ < 1 or LDQ < MMAX.
-23: LDP < 1 or LDP < MNMAX.
-27: LWORK too small.
If CLATMR, CLATMS, CGEBRD, CUNGBR, or CBDSQR,
returns an error code, the
absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
MMAX Largest value in NN.
NMAX Largest value in NN.
MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal
matrix.)
MNMAX The maximum value of MNMIN for j=1,...,NSIZES.
NFAIL The number of tests which have exceeded THRESH
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
RTOVFL, RTUNFL Square roots of the previous 2 values.
ULP, ULPINV Finest relative precision and its inverse.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
======================================================================
Parameter adjustments */
--mval;
--nval;
--dotype;
--iseed;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--bd;
--be;
--s1;
--s2;
z_dim1 = *ldx;
z_offset = 1 + z_dim1 * 1;
z__ -= z_offset;
y_dim1 = *ldx;
y_offset = 1 + y_dim1 * 1;
y -= y_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
vt_dim1 = *ldpt;
vt_offset = 1 + vt_dim1 * 1;
vt -= vt_offset;
u_dim1 = *ldpt;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
pt_dim1 = *ldpt;
pt_offset = 1 + pt_dim1 * 1;
pt -= pt_offset;
--work;
--rwork;
/* Function Body
Check for errors */
*info = 0;
badmm = FALSE_;
badnn = FALSE_;
mmax = 1;
nmax = 1;
mnmax = 1;
minwrk = 1;
i__1 = *nsizes;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
i__2 = mmax, i__3 = mval[j];
mmax = max(i__2,i__3);
if (mval[j] < 0) {
badmm = TRUE_;
}
/* Computing MAX */
i__2 = nmax, i__3 = nval[j];
nmax = max(i__2,i__3);
if (nval[j] < 0) {
badnn = TRUE_;
}
/* Computing MAX
Computing MIN */
i__4 = mval[j], i__5 = nval[j];
i__2 = mnmax, i__3 = min(i__4,i__5);
mnmax = max(i__2,i__3);
/* Computing MAX
Computing MAX */
i__4 = mval[j], i__5 = nval[j], i__4 = max(i__4,i__5);
/* Computing MIN */
i__6 = nval[j], i__7 = mval[j];
i__2 = minwrk, i__3 = (mval[j] + nval[j]) * 3, i__2 = max(i__2,i__3),
i__3 = mval[j] * (mval[j] + max(i__4,*nrhs) + 1) + nval[j] *
min(i__6,i__7);
minwrk = max(i__2,i__3);
/* L10: */
}
/* Check for errors */
if (*nsizes < 0) {
*info = -1;
} else if (badmm) {
*info = -2;
} else if (badnn) {
*info = -3;
} else if (*ntypes < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -6;
} else if (*lda < mmax) {
*info = -11;
} else if (*ldx < mmax) {
*info = -17;
} else if (*ldq < mmax) {
*info = -21;
} else if (*ldpt < mnmax) {
*info = -23;
} else if (minwrk > *lwork) {
*info = -27;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CCHKBD", &i__1);
return 0;
}
/* Initialize constants */
s_copy(path, "Complex precision", (ftnlen)1, (ftnlen)17);
s_copy(path + 1, "BD", (ftnlen)2, (ftnlen)2);
nfail = 0;
ntest = 0;
unfl = slamch_("Safe minimum");
ovfl = slamch_("Overflow");
slabad_(&unfl, &ovfl);
ulp = slamch_("Precision");
ulpinv = 1.f / ulp;
log2ui = (integer) (log(ulpinv) / log(2.f));
rtunfl = sqrt(unfl);
rtovfl = sqrt(ovfl);
infoc_1.infot = 0;
/* Loop over sizes, types */
i__1 = *nsizes;
for (jsize = 1; jsize <= i__1; ++jsize) {
m = mval[jsize];
n = nval[jsize];
mnmin = min(m,n);
/* Computing MAX */
i__2 = max(m,n);
amninv = 1.f / max(i__2,1);
if (*nsizes != 1) {
mtypes = min(16,*ntypes);
} else {
mtypes = min(17,*ntypes);
}
i__2 = mtypes;
for (jtype = 1; jtype <= i__2; ++jtype) {
if (! dotype[jtype]) {
goto L170;
}
for (j = 1; j <= 4; ++j) {
ioldsd[j - 1] = iseed[j];
/* L20: */
}
for (j = 1; j <= 14; ++j) {
result[j - 1] = -1.f;
/* L30: */
}
*(unsigned char *)uplo = ' ';
/* Compute "A"
Control parameters:
KMAGN KMODE KTYPE
=1 O(1) clustered 1 zero
=2 large clustered 2 identity
=3 small exponential (none)
=4 arithmetic diagonal, (w/ eigenvalues)
=5 random symmetric, w/ eigenvalues
=6 nonsymmetric, w/ singular values
=7 random diagonal
=8 random symmetric
=9 random nonsymmetric
=10 random bidiagonal (log. distrib.) */
if (mtypes > 16) {
goto L100;
}
itype = ktype[jtype - 1];
imode = kmode[jtype - 1];
/* Compute norm */
switch (kmagn[jtype - 1]) {
case 1: goto L40;
case 2: goto L50;
case 3: goto L60;
}
L40:
anorm = 1.f;
goto L70;
L50:
anorm = rtovfl * ulp * amninv;
goto L70;
L60:
anorm = rtunfl * max(m,n) * ulpinv;
goto L70;
L70:
claset_("Full", lda, &n, &c_b1, &c_b1, &a[a_offset], lda);
iinfo = 0;
cond = ulpinv;
bidiag = FALSE_;
if (itype == 1) {
/* Zero matrix */
iinfo = 0;
} else if (itype == 2) {
/* Identity */
i__3 = mnmin;
for (jcol = 1; jcol <= i__3; ++jcol) {
i__4 = a_subscr(jcol, jcol);
a[i__4].r = anorm, a[i__4].i = 0.f;
/* L80: */
}
} else if (itype == 4) {
/* Diagonal Matrix, [Eigen]values Specified */
clatms_(&mnmin, &mnmin, "S", &iseed[1], "N", &rwork[1], &
imode, &cond, &anorm, &c__0, &c__0, "N", &a[a_offset],
lda, &work[1], &iinfo);
} else if (itype == 5) {
/* Symmetric, eigenvalues specified */
clatms_(&mnmin, &mnmin, "S", &iseed[1], "S", &rwork[1], &
imode, &cond, &anorm, &m, &n, "N", &a[a_offset], lda,
&work[1], &iinfo);
} else if (itype == 6) {
/* Nonsymmetric, singular values specified */
clatms_(&m, &n, "S", &iseed[1], "N", &rwork[1], &imode, &cond,
&anorm, &m, &n, "N", &a[a_offset], lda, &work[1], &
iinfo);
} else if (itype == 7) {
/* Diagonal, random entries */
clatmr_(&mnmin, &mnmin, "S", &iseed[1], "N", &work[1], &c__6,
&c_b37, &c_b2, "T", "N", &work[mnmin + 1], &c__1, &
c_b37, &work[(mnmin << 1) + 1], &c__1, &c_b37, "N",
iwork, &c__0, &c__0, &c_b47, &anorm, "NO", &a[
a_offset], lda, iwork, &iinfo);
} else if (itype == 8) {
/* Symmetric, random entries */
clatmr_(&mnmin, &mnmin, "S", &iseed[1], "S", &work[1], &c__6,
&c_b37, &c_b2, "T", "N", &work[mnmin + 1], &c__1, &
c_b37, &work[m + mnmin + 1], &c__1, &c_b37, "N",
iwork, &m, &n, &c_b47, &anorm, "NO", &a[a_offset],
lda, iwork, &iinfo);
} else if (itype == 9) {
/* Nonsymmetric, random entries */
clatmr_(&m, &n, "S", &iseed[1], "N", &work[1], &c__6, &c_b37,
&c_b2, "T", "N", &work[mnmin + 1], &c__1, &c_b37, &
work[m + mnmin + 1], &c__1, &c_b37, "N", iwork, &m, &
n, &c_b47, &anorm, "NO", &a[a_offset], lda, iwork, &
iinfo);
} else if (itype == 10) {
/* Bidiagonal, random entries */
temp1 = log(ulp) * -2.f;
i__3 = mnmin;
for (j = 1; j <= i__3; ++j) {
bd[j] = exp(temp1 * slarnd_(&c__2, &iseed[1]));
if (j < mnmin) {
be[j] = exp(temp1 * slarnd_(&c__2, &iseed[1]));
}
/* L90: */
}
iinfo = 0;
bidiag = TRUE_;
if (m >= n) {
*(unsigned char *)uplo = 'U';
} else {
*(unsigned char *)uplo = 'L';
}
} else {
iinfo = 1;
}
if (iinfo == 0) {
/* Generate Right-Hand Side */
if (bidiag) {
clatmr_(&mnmin, nrhs, "S", &iseed[1], "N", &work[1], &
c__6, &c_b37, &c_b2, "T", "N", &work[mnmin + 1], &
c__1, &c_b37, &work[(mnmin << 1) + 1], &c__1, &
c_b37, "N", iwork, &mnmin, nrhs, &c_b47, &c_b37,
"NO", &y[y_offset], ldx, iwork, &iinfo);
} else {
clatmr_(&m, nrhs, "S", &iseed[1], "N", &work[1], &c__6, &
c_b37, &c_b2, "T", "N", &work[m + 1], &c__1, &
c_b37, &work[(m << 1) + 1], &c__1, &c_b37, "N",
iwork, &m, nrhs, &c_b47, &c_b37, "NO", &x[
x_offset], ldx, iwork, &iinfo);
}
}
/* Error Exit */
if (iinfo != 0) {
io___40.ciunit = *nout;
s_wsfe(&io___40);
do_fio(&c__1, "Generator", (ftnlen)9);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(iinfo);
return 0;
}
L100:
/* Call CGEBRD and CUNGBR to compute B, Q, and P, do tests. */
if (! bidiag) {
/* Compute transformations to reduce A to bidiagonal form:
B := Q' * A * P. */
clacpy_(" ", &m, &n, &a[a_offset], lda, &q[q_offset], ldq);
i__3 = *lwork - (mnmin << 1);
cgebrd_(&m, &n, &q[q_offset], ldq, &bd[1], &be[1], &work[1], &
work[mnmin + 1], &work[(mnmin << 1) + 1], &i__3, &
iinfo);
/* Check error code from CGEBRD. */
if (iinfo != 0) {
io___41.ciunit = *nout;
s_wsfe(&io___41);
do_fio(&c__1, "CGEBRD", (ftnlen)6);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer))
;
e_wsfe();
*info = abs(iinfo);
return 0;
}
clacpy_(" ", &m, &n, &q[q_offset], ldq, &pt[pt_offset], ldpt);
if (m >= n) {
*(unsigned char *)uplo = 'U';
} else {
*(unsigned char *)uplo = 'L';
}
/* Generate Q */
mq = m;
if (*nrhs <= 0) {
mq = mnmin;
}
i__3 = *lwork - (mnmin << 1);
cungbr_("Q", &m, &mq, &n, &q[q_offset], ldq, &work[1], &work[(
mnmin << 1) + 1], &i__3, &iinfo);
/* Check error code from CUNGBR. */
if (iinfo != 0) {
io___43.ciunit = *nout;
s_wsfe(&io___43);
do_fio(&c__1, "CUNGBR(Q)", (ftnlen)9);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer))
;
e_wsfe();
*info = abs(iinfo);
return 0;
}
/* Generate P' */
i__3 = *lwork - (mnmin << 1);
cungbr_("P", &mnmin, &n, &m, &pt[pt_offset], ldpt, &work[
mnmin + 1], &work[(mnmin << 1) + 1], &i__3, &iinfo);
/* Check error code from CUNGBR. */
if (iinfo != 0) {
io___44.ciunit = *nout;
s_wsfe(&io___44);
do_fio(&c__1, "CUNGBR(P)", (ftnlen)9);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer))
;
e_wsfe();
*info = abs(iinfo);
return 0;
}
/* Apply Q' to an M by NRHS matrix X: Y := Q' * X. */
cgemm_("Conjugate transpose", "No transpose", &m, nrhs, &m, &
c_b2, &q[q_offset], ldq, &x[x_offset], ldx, &c_b1, &y[
y_offset], ldx);
/* Test 1: Check the decomposition A := Q * B * PT
2: Check the orthogonality of Q
3: Check the orthogonality of PT */
cbdt01_(&m, &n, &c__1, &a[a_offset], lda, &q[q_offset], ldq, &
bd[1], &be[1], &pt[pt_offset], ldpt, &work[1], &rwork[
1], result);
cunt01_("Columns", &m, &mq, &q[q_offset], ldq, &work[1],
lwork, &rwork[1], &result[1]);
cunt01_("Rows", &mnmin, &n, &pt[pt_offset], ldpt, &work[1],
lwork, &rwork[1], &result[2]);
}
/* Use CBDSQR to form the SVD of the bidiagonal matrix B:
B := U * S1 * VT, and compute Z = U' * Y. */
scopy_(&mnmin, &bd[1], &c__1, &s1[1], &c__1);
if (mnmin > 0) {
i__3 = mnmin - 1;
scopy_(&i__3, &be[1], &c__1, &rwork[1], &c__1);
}
clacpy_(" ", &m, nrhs, &y[y_offset], ldx, &z__[z_offset], ldx);
claset_("Full", &mnmin, &mnmin, &c_b1, &c_b2, &u[u_offset], ldpt);
claset_("Full", &mnmin, &mnmin, &c_b1, &c_b2, &vt[vt_offset],
ldpt);
cbdsqr_(uplo, &mnmin, &mnmin, &mnmin, nrhs, &s1[1], &rwork[1], &
vt[vt_offset], ldpt, &u[u_offset], ldpt, &z__[z_offset],
ldx, &rwork[mnmin + 1], &iinfo);
/* Check error code from CBDSQR. */
if (iinfo != 0) {
io___45.ciunit = *nout;
s_wsfe(&io___45);
do_fio(&c__1, "CBDSQR(vects)", (ftnlen)13);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(iinfo);
if (iinfo < 0) {
return 0;
} else {
result[3] = ulpinv;
goto L150;
}
}
/* Use CBDSQR to compute only the singular values of the
bidiagonal matrix B; U, VT, and Z should not be modified. */
scopy_(&mnmin, &bd[1], &c__1, &s2[1], &c__1);
if (mnmin > 0) {
i__3 = mnmin - 1;
scopy_(&i__3, &be[1], &c__1, &rwork[1], &c__1);
}
cbdsqr_(uplo, &mnmin, &c__0, &c__0, &c__0, &s2[1], &rwork[1], &vt[
vt_offset], ldpt, &u[u_offset], ldpt, &z__[z_offset], ldx,
&rwork[mnmin + 1], &iinfo);
/* Check error code from CBDSQR. */
if (iinfo != 0) {
io___46.ciunit = *nout;
s_wsfe(&io___46);
do_fio(&c__1, "CBDSQR(values)", (ftnlen)14);
do_fio(&c__1, (char *)&iinfo, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer));
e_wsfe();
*info = abs(iinfo);
if (iinfo < 0) {
return 0;
} else {
result[8] = ulpinv;
goto L150;
}
}
/* Test 4: Check the decomposition B := U * S1 * VT
5: Check the computation Z := U' * Y
6: Check the orthogonality of U
7: Check the orthogonality of VT */
cbdt03_(uplo, &mnmin, &c__1, &bd[1], &be[1], &u[u_offset], ldpt, &
s1[1], &vt[vt_offset], ldpt, &work[1], &result[3]);
cbdt02_(&mnmin, nrhs, &y[y_offset], ldx, &z__[z_offset], ldx, &u[
u_offset], ldpt, &work[1], &rwork[1], &result[4]);
cunt01_("Columns", &mnmin, &mnmin, &u[u_offset], ldpt, &work[1],
lwork, &rwork[1], &result[5]);
cunt01_("Rows", &mnmin, &mnmin, &vt[vt_offset], ldpt, &work[1],
lwork, &rwork[1], &result[6]);
/* Test 8: Check that the singular values are sorted in
non-increasing order and are non-negative */
result[7] = 0.f;
i__3 = mnmin - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
if (s1[i__] < s1[i__ + 1]) {
result[7] = ulpinv;
}
if (s1[i__] < 0.f) {
result[7] = ulpinv;
}
/* L110: */
}
if (mnmin >= 1) {
if (s1[mnmin] < 0.f) {
result[7] = ulpinv;
}
}
/* Test 9: Compare CBDSQR with and without singular vectors */
temp2 = 0.f;
i__3 = mnmin;
for (j = 1; j <= i__3; ++j) {
/* Computing MAX
Computing MAX */
r__6 = (r__1 = s1[j], dabs(r__1)), r__7 = (r__2 = s2[j], dabs(
r__2));
r__4 = sqrt(unfl) * dmax(s1[1],1.f), r__5 = ulp * dmax(r__6,
r__7);
temp1 = (r__3 = s1[j] - s2[j], dabs(r__3)) / dmax(r__4,r__5);
temp2 = dmax(temp1,temp2);
/* L120: */
}
result[8] = temp2;
/* Test 10: Sturm sequence test of singular values
Go up by factors of two until it succeeds */
temp1 = *thresh * (.5f - ulp);
i__3 = log2ui;
for (j = 0; j <= i__3; ++j) {
ssvdch_(&mnmin, &bd[1], &be[1], &s1[1], &temp1, &iinfo);
if (iinfo == 0) {
goto L140;
}
temp1 *= 2.f;
/* L130: */
}
L140:
result[9] = temp1;
/* Use CBDSQR to form the decomposition A := (QU) S (VT PT)
from the bidiagonal form A := Q B PT. */
if (! bidiag) {
scopy_(&mnmin, &bd[1], &c__1, &s2[1], &c__1);
if (mnmin > 0) {
i__3 = mnmin - 1;
scopy_(&i__3, &be[1], &c__1, &rwork[1], &c__1);
}
cbdsqr_(uplo, &mnmin, &n, &m, nrhs, &s2[1], &rwork[1], &pt[
pt_offset], ldpt, &q[q_offset], ldq, &y[y_offset],
ldx, &rwork[mnmin + 1], &iinfo);
/* Test 11: Check the decomposition A := Q*U * S2 * VT*PT
12: Check the computation Z := U' * Q' * X
13: Check the orthogonality of Q*U
14: Check the orthogonality of VT*PT */
cbdt01_(&m, &n, &c__0, &a[a_offset], lda, &q[q_offset], ldq, &
s2[1], dumma, &pt[pt_offset], ldpt, &work[1], &rwork[
1], &result[10]);
cbdt02_(&m, nrhs, &x[x_offset], ldx, &y[y_offset], ldx, &q[
q_offset], ldq, &work[1], &rwork[1], &result[11]);
cunt01_("Columns", &m, &mq, &q[q_offset], ldq, &work[1],
lwork, &rwork[1], &result[12]);
cunt01_("Rows", &mnmin, &n, &pt[pt_offset], ldpt, &work[1],
lwork, &rwork[1], &result[13]);
}
/* End of Loop -- Check for RESULT(j) > THRESH */
L150:
for (j = 1; j <= 14; ++j) {
if (result[j - 1] >= *thresh) {
if (nfail == 0) {
slahd2_(nout, path);
}
io___50.ciunit = *nout;
s_wsfe(&io___50);
do_fio(&c__1, (char *)&m, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&jtype, (ftnlen)sizeof(integer));
do_fio(&c__4, (char *)&ioldsd[0], (ftnlen)sizeof(integer))
;
do_fio(&c__1, (char *)&j, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&result[j - 1], (ftnlen)sizeof(real)
);
e_wsfe();
++nfail;
}
/* L160: */
}
if (! bidiag) {
ntest += 14;
} else {
ntest += 5;
}
L170:
;
}
/* L180: */
}
/* Summary */
alasum_(path, nout, &nfail, &ntest, &c__0);
return 0;
/* End of CCHKBD */
} /* cchkbd_ */
/* Subroutine */ int zgeqr2_(integer *m, integer *n, doublecomplex *a,
integer *lda, doublecomplex *tau, doublecomplex *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
=======
ZGEQR2 computes a QR factorization of a complex m by n matrix A:
A = Q * R.
Arguments
=========
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.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, the elements on and above the diagonal of the array
contain the min(m,n) by n upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors (see Further Details).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU (output) COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK (workspace) COMPLEX*16 array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Further Details
===============
The matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
=====================================================================
Test the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublecomplex z__1;
/* Builtin functions */
void d_cnjg(doublecomplex *, doublecomplex *);
/* Local variables */
static integer i__, k;
static doublecomplex alpha;
extern /* Subroutine */ int zlarf_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *, doublecomplex *), xerbla_(char *, integer *), zlarfg_(integer *, doublecomplex *, doublecomplex *,
integer *, doublecomplex *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGEQR2", &i__1);
return 0;
}
k = min(*m,*n);
i__1 = k;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Generate elementary reflector H(i) to annihilate A(i+1:m,i)
Computing MIN */
i__2 = i__ + 1;
i__3 = *m - i__ + 1;
zlarfg_(&i__3, &a_ref(i__, i__), &a_ref(min(i__2,*m), i__), &c__1, &
tau[i__]);
if (i__ < *n) {
/* Apply H(i)' to A(i:m,i+1:n) from the left */
i__2 = a_subscr(i__, i__);
alpha.r = a[i__2].r, alpha.i = a[i__2].i;
i__2 = a_subscr(i__, i__);
a[i__2].r = 1., a[i__2].i = 0.;
i__2 = *m - i__ + 1;
i__3 = *n - i__;
d_cnjg(&z__1, &tau[i__]);
zlarf_("Left", &i__2, &i__3, &a_ref(i__, i__), &c__1, &z__1, &
a_ref(i__, i__ + 1), lda, &work[1]);
i__2 = a_subscr(i__, i__);
a[i__2].r = alpha.r, a[i__2].i = alpha.i;
}
/* L10: */
}
return 0;
/* End of ZGEQR2 */
} /* zgeqr2_ */
/* Subroutine */ int zlacp2_(char *uplo, integer *m, integer *n, doublereal *
a, integer *lda, doublecomplex *b, integer *ldb)
{
/* -- 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
=======
ZLACP2 copies all or part of a real two-dimensional matrix A to a
complex matrix B.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies the part of the matrix A to be copied to B.
= 'U': Upper triangular part
= 'L': Lower triangular part
Otherwise: All of the matrix A
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.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The m by n matrix A. If UPLO = 'U', only the upper trapezium
is accessed; if UPLO = 'L', only the lower trapezium is
accessed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (output) COMPLEX*16 array, dimension (LDB,N)
On exit, B = A in the locations specified by UPLO.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,M).
=====================================================================
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
/* Local variables */
static integer i__, j;
extern logical lsame_(char *, char *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
/* Function Body */
if (lsame_(uplo, "U")) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = min(j,*m);
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
i__4 = a_subscr(i__, j);
b[i__3].r = a[i__4], b[i__3].i = 0.;
/* L10: */
}
/* L20: */
}
} else if (lsame_(uplo, "L")) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
i__4 = a_subscr(i__, j);
b[i__3].r = a[i__4], b[i__3].i = 0.;
/* L30: */
}
/* L40: */
}
} else {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
i__4 = a_subscr(i__, j);
b[i__3].r = a[i__4], b[i__3].i = 0.;
/* L50: */
}
/* L60: */
}
}
return 0;
/* End of ZLACP2 */
} /* zlacp2_ */
/* Subroutine */ int ctrsyl_(char *trana, char *tranb, integer *isgn, integer
*m, integer *n, complex *a, integer *lda, complex *b, integer *ldb,
complex *c__, integer *ldc, real *scale, integer *info)
{
/* -- 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
Purpose
=======
CTRSYL solves the complex Sylvester matrix equation:
op(A)*X + X*op(B) = scale*C or
op(A)*X - X*op(B) = scale*C,
where op(A) = A or A**H, and A and B are both upper triangular. A is
M-by-M and B is N-by-N; the right hand side C and the solution X are
M-by-N; and scale is an output scale factor, set <= 1 to avoid
overflow in X.
Arguments
=========
TRANA (input) CHARACTER*1
Specifies the option op(A):
= 'N': op(A) = A (No transpose)
= 'C': op(A) = A**H (Conjugate transpose)
TRANB (input) CHARACTER*1
Specifies the option op(B):
= 'N': op(B) = B (No transpose)
= 'C': op(B) = B**H (Conjugate transpose)
ISGN (input) INTEGER
Specifies the sign in the equation:
= +1: solve op(A)*X + X*op(B) = scale*C
= -1: solve op(A)*X - X*op(B) = scale*C
M (input) INTEGER
The order of the matrix A, and the number of rows in the
matrices X and C. M >= 0.
N (input) INTEGER
The order of the matrix B, and the number of columns in the
matrices X and C. N >= 0.
A (input) COMPLEX array, dimension (LDA,M)
The upper triangular matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input) COMPLEX array, dimension (LDB,N)
The upper triangular matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input/output) COMPLEX array, dimension (LDC,N)
On entry, the M-by-N right hand side matrix C.
On exit, C is overwritten by the solution matrix X.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,M)
SCALE (output) REAL
The scale factor, scale, set <= 1 to avoid overflow in X.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: A and B have common or very close eigenvalues; perturbed
values were used to solve the equation (but the matrices
A and B are unchanged).
=====================================================================
Decode and Test input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, c_dim1, c_offset, i__1, i__2,
i__3, i__4;
real r__1, r__2;
complex q__1, q__2, q__3, q__4;
/* Builtin functions */
double r_imag(complex *);
void r_cnjg(complex *, complex *);
/* Local variables */
static real smin;
static complex suml, sumr;
static integer j, k, l;
extern /* Complex */ VOID cdotc_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern logical lsame_(char *, char *);
extern /* Complex */ VOID cdotu_(complex *, integer *, complex *, integer
*, complex *, integer *);
static complex a11;
static real db;
extern /* Subroutine */ int slabad_(real *, real *);
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
static complex x11;
extern /* Complex */ VOID cladiv_(complex *, complex *, complex *);
static real scaloc;
extern doublereal slamch_(char *);
extern /* Subroutine */ int csscal_(integer *, real *, complex *, integer
*), xerbla_(char *, integer *);
static real bignum;
static logical notrna, notrnb;
static real smlnum, da11;
static complex vec;
static real dum[1], eps, sgn;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define c___subscr(a_1,a_2) (a_2)*c_dim1 + a_1
#define c___ref(a_1,a_2) c__[c___subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
c_dim1 = *ldc;
c_offset = 1 + c_dim1 * 1;
c__ -= c_offset;
/* Function Body */
notrna = lsame_(trana, "N");
notrnb = lsame_(tranb, "N");
*info = 0;
if (! notrna && ! lsame_(trana, "T") && ! lsame_(
trana, "C")) {
*info = -1;
} else if (! notrnb && ! lsame_(tranb, "T") && !
lsame_(tranb, "C")) {
*info = -2;
} else if (*isgn != 1 && *isgn != -1) {
*info = -3;
} else if (*m < 0) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*m)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldc < max(1,*m)) {
*info = -11;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTRSYL", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0) {
return 0;
}
/* Set constants to control overflow */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = smlnum * (real) (*m * *n) / eps;
bignum = 1.f / smlnum;
/* Computing MAX */
r__1 = smlnum, r__2 = eps * clange_("M", m, m, &a[a_offset], lda, dum), r__1 = max(r__1,r__2), r__2 = eps * clange_("M", n, n,
&b[b_offset], ldb, dum);
smin = dmax(r__1,r__2);
*scale = 1.f;
sgn = (real) (*isgn);
if (notrna && notrnb) {
/* Solve A*X + ISGN*X*B = scale*C.
The (K,L)th block of X is determined starting from
bottom-left corner column by column by
A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
Where
M L-1
R(K,L) = SUM [A(K,I)*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)].
I=K+1 J=1 */
i__1 = *n;
for (l = 1; l <= i__1; ++l) {
for (k = *m; k >= 1; --k) {
/* Computing MIN */
i__2 = k + 1;
/* Computing MIN */
i__3 = k + 1;
i__4 = *m - k;
cdotu_(&q__1, &i__4, &a_ref(k, min(i__2,*m)), lda, &c___ref(
min(i__3,*m), l), &c__1);
suml.r = q__1.r, suml.i = q__1.i;
i__2 = l - 1;
cdotu_(&q__1, &i__2, &c___ref(k, 1), ldc, &b_ref(1, l), &c__1)
;
sumr.r = q__1.r, sumr.i = q__1.i;
i__2 = c___subscr(k, l);
q__3.r = sgn * sumr.r, q__3.i = sgn * sumr.i;
q__2.r = suml.r + q__3.r, q__2.i = suml.i + q__3.i;
q__1.r = c__[i__2].r - q__2.r, q__1.i = c__[i__2].i - q__2.i;
vec.r = q__1.r, vec.i = q__1.i;
scaloc = 1.f;
i__2 = a_subscr(k, k);
i__3 = b_subscr(l, l);
q__2.r = sgn * b[i__3].r, q__2.i = sgn * b[i__3].i;
q__1.r = a[i__2].r + q__2.r, q__1.i = a[i__2].i + q__2.i;
a11.r = q__1.r, a11.i = q__1.i;
da11 = (r__1 = a11.r, dabs(r__1)) + (r__2 = r_imag(&a11),
dabs(r__2));
if (da11 <= smin) {
a11.r = smin, a11.i = 0.f;
da11 = smin;
*info = 1;
}
db = (r__1 = vec.r, dabs(r__1)) + (r__2 = r_imag(&vec), dabs(
r__2));
if (da11 < 1.f && db > 1.f) {
if (db > bignum * da11) {
scaloc = 1.f / db;
}
}
q__3.r = scaloc, q__3.i = 0.f;
q__2.r = vec.r * q__3.r - vec.i * q__3.i, q__2.i = vec.r *
q__3.i + vec.i * q__3.r;
cladiv_(&q__1, &q__2, &a11);
x11.r = q__1.r, x11.i = q__1.i;
if (scaloc != 1.f) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
csscal_(m, &scaloc, &c___ref(1, j), &c__1);
/* L10: */
}
*scale *= scaloc;
}
i__2 = c___subscr(k, l);
c__[i__2].r = x11.r, c__[i__2].i = x11.i;
/* L20: */
}
/* L30: */
}
} else if (! notrna && notrnb) {
/* Solve A' *X + ISGN*X*B = scale*C.
The (K,L)th block of X is determined starting from
upper-left corner column by column by
A'(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
Where
K-1 L-1
R(K,L) = SUM [A'(I,K)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)]
I=1 J=1 */
i__1 = *n;
for (l = 1; l <= i__1; ++l) {
i__2 = *m;
for (k = 1; k <= i__2; ++k) {
i__3 = k - 1;
cdotc_(&q__1, &i__3, &a_ref(1, k), &c__1, &c___ref(1, l), &
c__1);
suml.r = q__1.r, suml.i = q__1.i;
i__3 = l - 1;
cdotu_(&q__1, &i__3, &c___ref(k, 1), ldc, &b_ref(1, l), &c__1)
;
sumr.r = q__1.r, sumr.i = q__1.i;
i__3 = c___subscr(k, l);
q__3.r = sgn * sumr.r, q__3.i = sgn * sumr.i;
q__2.r = suml.r + q__3.r, q__2.i = suml.i + q__3.i;
q__1.r = c__[i__3].r - q__2.r, q__1.i = c__[i__3].i - q__2.i;
vec.r = q__1.r, vec.i = q__1.i;
scaloc = 1.f;
r_cnjg(&q__2, &a_ref(k, k));
i__3 = b_subscr(l, l);
q__3.r = sgn * b[i__3].r, q__3.i = sgn * b[i__3].i;
q__1.r = q__2.r + q__3.r, q__1.i = q__2.i + q__3.i;
a11.r = q__1.r, a11.i = q__1.i;
da11 = (r__1 = a11.r, dabs(r__1)) + (r__2 = r_imag(&a11),
dabs(r__2));
if (da11 <= smin) {
a11.r = smin, a11.i = 0.f;
da11 = smin;
*info = 1;
}
db = (r__1 = vec.r, dabs(r__1)) + (r__2 = r_imag(&vec), dabs(
r__2));
if (da11 < 1.f && db > 1.f) {
if (db > bignum * da11) {
scaloc = 1.f / db;
}
}
q__3.r = scaloc, q__3.i = 0.f;
q__2.r = vec.r * q__3.r - vec.i * q__3.i, q__2.i = vec.r *
q__3.i + vec.i * q__3.r;
cladiv_(&q__1, &q__2, &a11);
x11.r = q__1.r, x11.i = q__1.i;
if (scaloc != 1.f) {
i__3 = *n;
for (j = 1; j <= i__3; ++j) {
csscal_(m, &scaloc, &c___ref(1, j), &c__1);
/* L40: */
}
*scale *= scaloc;
}
i__3 = c___subscr(k, l);
c__[i__3].r = x11.r, c__[i__3].i = x11.i;
/* L50: */
}
/* L60: */
}
} else if (! notrna && ! notrnb) {
/* Solve A'*X + ISGN*X*B' = C.
The (K,L)th block of X is determined starting from
upper-right corner column by column by
A'(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
Where
K-1
R(K,L) = SUM [A'(I,K)*X(I,L)] +
I=1
N
ISGN*SUM [X(K,J)*B'(L,J)].
J=L+1 */
for (l = *n; l >= 1; --l) {
i__1 = *m;
for (k = 1; k <= i__1; ++k) {
i__2 = k - 1;
cdotc_(&q__1, &i__2, &a_ref(1, k), &c__1, &c___ref(1, l), &
c__1);
suml.r = q__1.r, suml.i = q__1.i;
/* Computing MIN */
i__2 = l + 1;
/* Computing MIN */
i__3 = l + 1;
i__4 = *n - l;
cdotc_(&q__1, &i__4, &c___ref(k, min(i__2,*n)), ldc, &b_ref(l,
min(i__3,*n)), ldb);
sumr.r = q__1.r, sumr.i = q__1.i;
i__2 = c___subscr(k, l);
r_cnjg(&q__4, &sumr);
q__3.r = sgn * q__4.r, q__3.i = sgn * q__4.i;
q__2.r = suml.r + q__3.r, q__2.i = suml.i + q__3.i;
q__1.r = c__[i__2].r - q__2.r, q__1.i = c__[i__2].i - q__2.i;
vec.r = q__1.r, vec.i = q__1.i;
scaloc = 1.f;
i__2 = a_subscr(k, k);
i__3 = b_subscr(l, l);
q__3.r = sgn * b[i__3].r, q__3.i = sgn * b[i__3].i;
q__2.r = a[i__2].r + q__3.r, q__2.i = a[i__2].i + q__3.i;
r_cnjg(&q__1, &q__2);
a11.r = q__1.r, a11.i = q__1.i;
da11 = (r__1 = a11.r, dabs(r__1)) + (r__2 = r_imag(&a11),
dabs(r__2));
if (da11 <= smin) {
a11.r = smin, a11.i = 0.f;
da11 = smin;
*info = 1;
}
db = (r__1 = vec.r, dabs(r__1)) + (r__2 = r_imag(&vec), dabs(
r__2));
if (da11 < 1.f && db > 1.f) {
if (db > bignum * da11) {
scaloc = 1.f / db;
}
}
q__3.r = scaloc, q__3.i = 0.f;
q__2.r = vec.r * q__3.r - vec.i * q__3.i, q__2.i = vec.r *
q__3.i + vec.i * q__3.r;
cladiv_(&q__1, &q__2, &a11);
x11.r = q__1.r, x11.i = q__1.i;
if (scaloc != 1.f) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
csscal_(m, &scaloc, &c___ref(1, j), &c__1);
/* L70: */
}
*scale *= scaloc;
}
i__2 = c___subscr(k, l);
c__[i__2].r = x11.r, c__[i__2].i = x11.i;
/* L80: */
}
/* L90: */
}
} else if (notrna && ! notrnb) {
/* Solve A*X + ISGN*X*B' = C.
The (K,L)th block of X is determined starting from
bottom-left corner column by column by
A(K,K)*X(K,L) + ISGN*X(K,L)*B'(L,L) = C(K,L) - R(K,L)
Where
M N
R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B'(L,J)]
I=K+1 J=L+1 */
for (l = *n; l >= 1; --l) {
for (k = *m; k >= 1; --k) {
/* Computing MIN */
i__1 = k + 1;
/* Computing MIN */
i__2 = k + 1;
i__3 = *m - k;
cdotu_(&q__1, &i__3, &a_ref(k, min(i__1,*m)), lda, &c___ref(
min(i__2,*m), l), &c__1);
suml.r = q__1.r, suml.i = q__1.i;
/* Computing MIN */
i__1 = l + 1;
/* Computing MIN */
i__2 = l + 1;
i__3 = *n - l;
cdotc_(&q__1, &i__3, &c___ref(k, min(i__1,*n)), ldc, &b_ref(l,
min(i__2,*n)), ldb);
sumr.r = q__1.r, sumr.i = q__1.i;
i__1 = c___subscr(k, l);
r_cnjg(&q__4, &sumr);
q__3.r = sgn * q__4.r, q__3.i = sgn * q__4.i;
q__2.r = suml.r + q__3.r, q__2.i = suml.i + q__3.i;
q__1.r = c__[i__1].r - q__2.r, q__1.i = c__[i__1].i - q__2.i;
vec.r = q__1.r, vec.i = q__1.i;
scaloc = 1.f;
i__1 = a_subscr(k, k);
r_cnjg(&q__3, &b_ref(l, l));
q__2.r = sgn * q__3.r, q__2.i = sgn * q__3.i;
q__1.r = a[i__1].r + q__2.r, q__1.i = a[i__1].i + q__2.i;
a11.r = q__1.r, a11.i = q__1.i;
da11 = (r__1 = a11.r, dabs(r__1)) + (r__2 = r_imag(&a11),
dabs(r__2));
if (da11 <= smin) {
a11.r = smin, a11.i = 0.f;
da11 = smin;
*info = 1;
}
db = (r__1 = vec.r, dabs(r__1)) + (r__2 = r_imag(&vec), dabs(
r__2));
if (da11 < 1.f && db > 1.f) {
if (db > bignum * da11) {
scaloc = 1.f / db;
}
}
q__3.r = scaloc, q__3.i = 0.f;
q__2.r = vec.r * q__3.r - vec.i * q__3.i, q__2.i = vec.r *
q__3.i + vec.i * q__3.r;
cladiv_(&q__1, &q__2, &a11);
x11.r = q__1.r, x11.i = q__1.i;
if (scaloc != 1.f) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
csscal_(m, &scaloc, &c___ref(1, j), &c__1);
/* L100: */
}
*scale *= scaloc;
}
i__1 = c___subscr(k, l);
c__[i__1].r = x11.r, c__[i__1].i = x11.i;
/* L110: */
}
/* L120: */
}
}
return 0;
/* End of CTRSYL */
} /* ctrsyl_ */
/* Subroutine */ int zgeru_(integer *m, integer *n, doublecomplex *alpha,
doublecomplex *x, integer *incx, doublecomplex *y, integer *incy,
doublecomplex *a, integer *lda)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4, i__5;
doublecomplex z__1, z__2;
/* Local variables */
static integer info;
static doublecomplex temp;
static integer i__, j, ix, jy, kx;
extern /* Subroutine */ int xerbla_(char *, integer *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
/* Purpose
=======
ZGERU performs the rank 1 operation
A := alpha*x*y' + A,
where alpha is a scalar, x is an m element vector, y is an n element
vector and A is an m by n matrix.
Parameters
==========
M - INTEGER.
On entry, M specifies the number of rows of the matrix A.
M must be at least zero.
Unchanged on exit.
N - INTEGER.
On entry, N specifies the number of columns of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA - COMPLEX*16 .
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
X - COMPLEX*16 array of dimension at least
( 1 + ( m - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the m
element vector x.
Unchanged on exit.
INCX - INTEGER.
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
Y - COMPLEX*16 array of dimension at least
( 1 + ( n - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the n
element vector y.
Unchanged on exit.
INCY - INTEGER.
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
A - COMPLEX*16 array of DIMENSION ( LDA, n ).
Before entry, the leading m by n part of the array A must
contain the matrix of coefficients. On exit, A is
overwritten by the updated matrix.
LDA - INTEGER.
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, m ).
Unchanged on exit.
Level 2 Blas routine.
-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
Test the input parameters.
Parameter adjustments */
--x;
--y;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
/* Function Body */
info = 0;
if (*m < 0) {
info = 1;
} else if (*n < 0) {
info = 2;
} else if (*incx == 0) {
info = 5;
} else if (*incy == 0) {
info = 7;
} else if (*lda < max(1,*m)) {
info = 9;
}
if (info != 0) {
xerbla_("ZGERU ", &info);
return 0;
}
/* Quick return if possible. */
if (*m == 0 || *n == 0 || alpha->r == 0. && alpha->i == 0.) {
return 0;
}
/* Start the operations. In this version the elements of A are
accessed sequentially with one pass through A. */
if (*incy > 0) {
jy = 1;
} else {
jy = 1 - (*n - 1) * *incy;
}
if (*incx == 1) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = jy;
if (y[i__2].r != 0. || y[i__2].i != 0.) {
i__2 = jy;
z__1.r = alpha->r * y[i__2].r - alpha->i * y[i__2].i, z__1.i =
alpha->r * y[i__2].i + alpha->i * y[i__2].r;
temp.r = z__1.r, temp.i = z__1.i;
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = a_subscr(i__, j);
i__5 = i__;
z__2.r = x[i__5].r * temp.r - x[i__5].i * temp.i, z__2.i =
x[i__5].r * temp.i + x[i__5].i * temp.r;
z__1.r = a[i__4].r + z__2.r, z__1.i = a[i__4].i + z__2.i;
a[i__3].r = z__1.r, a[i__3].i = z__1.i;
/* L10: */
}
}
jy += *incy;
/* L20: */
}
} else {
if (*incx > 0) {
kx = 1;
} else {
kx = 1 - (*m - 1) * *incx;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = jy;
if (y[i__2].r != 0. || y[i__2].i != 0.) {
i__2 = jy;
z__1.r = alpha->r * y[i__2].r - alpha->i * y[i__2].i, z__1.i =
alpha->r * y[i__2].i + alpha->i * y[i__2].r;
temp.r = z__1.r, temp.i = z__1.i;
ix = kx;
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = a_subscr(i__, j);
i__5 = ix;
z__2.r = x[i__5].r * temp.r - x[i__5].i * temp.i, z__2.i =
x[i__5].r * temp.i + x[i__5].i * temp.r;
z__1.r = a[i__4].r + z__2.r, z__1.i = a[i__4].i + z__2.i;
a[i__3].r = z__1.r, a[i__3].i = z__1.i;
ix += *incx;
/* L30: */
}
}
jy += *incy;
/* L40: */
}
}
return 0;
/* End of ZGERU . */
} /* zgeru_ */
/* Subroutine */ int chetrs_(char *uplo, integer *n, integer *nrhs, complex *
a, integer *lda, integer *ipiv, complex *b, integer *ldb, 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
=======
CHETRS solves a system of linear equations A*X = B with a complex
Hermitian matrix A using the factorization A = U*D*U**H or
A = L*D*L**H computed by CHETRF.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 'U': Upper triangular, form is A = U*D*U**H;
= 'L': Lower triangular, form is A = L*D*L**H.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
A (input) COMPLEX array, dimension (LDA,N)
The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by CHETRF.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
B (input/output) COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side matrix B.
On exit, the solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
=====================================================================
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {1.f,0.f};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
complex q__1, q__2, q__3;
/* Builtin functions */
void c_div(complex *, complex *, complex *), r_cnjg(complex *, complex *);
/* Local variables */
static complex akm1k;
static integer j, k;
static real s;
extern logical lsame_(char *, char *);
static complex denom;
extern /* Subroutine */ int cgemv_(char *, integer *, integer *, complex *
, complex *, integer *, complex *, integer *, complex *, complex *
, integer *), cgeru_(integer *, integer *, complex *,
complex *, integer *, complex *, integer *, complex *, integer *),
cswap_(integer *, complex *, integer *, complex *, integer *);
static logical upper;
static complex ak, bk;
static integer kp;
extern /* Subroutine */ int clacgv_(integer *, complex *, integer *),
csscal_(integer *, real *, complex *, integer *), xerbla_(char *,
integer *);
static complex akm1, bkm1;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHETRS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
return 0;
}
if (upper) {
/* Solve A*X = B, where A = U*D*U'.
First solve U*D*X = B, overwriting B with X.
K is the main loop index, decreasing from N to 1 in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = *n;
L10:
/* If K < 1, exit from loop. */
if (k < 1) {
goto L30;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Interchange rows K and IPIV(K). */
kp = ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Multiply by inv(U(K)), where U(K) is the transformation
stored in column K of A. */
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(1, k), &c__1, &b_ref(k, 1), ldb,
&b_ref(1, 1), ldb);
/* Multiply by the inverse of the diagonal block. */
i__1 = a_subscr(k, k);
s = 1.f / a[i__1].r;
csscal_(nrhs, &s, &b_ref(k, 1), ldb);
--k;
} else {
/* 2 x 2 diagonal block
Interchange rows K-1 and -IPIV(K). */
kp = -ipiv[k];
if (kp != k - 1) {
cswap_(nrhs, &b_ref(k - 1, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Multiply by inv(U(K)), where U(K) is the transformation
stored in columns K-1 and K of A. */
i__1 = k - 2;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(1, k), &c__1, &b_ref(k, 1), ldb,
&b_ref(1, 1), ldb);
i__1 = k - 2;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(1, k - 1), &c__1, &b_ref(k - 1,
1), ldb, &b_ref(1, 1), ldb);
/* Multiply by the inverse of the diagonal block. */
i__1 = a_subscr(k - 1, k);
akm1k.r = a[i__1].r, akm1k.i = a[i__1].i;
c_div(&q__1, &a_ref(k - 1, k - 1), &akm1k);
akm1.r = q__1.r, akm1.i = q__1.i;
r_cnjg(&q__2, &akm1k);
c_div(&q__1, &a_ref(k, k), &q__2);
ak.r = q__1.r, ak.i = q__1.i;
q__2.r = akm1.r * ak.r - akm1.i * ak.i, q__2.i = akm1.r * ak.i +
akm1.i * ak.r;
q__1.r = q__2.r - 1.f, q__1.i = q__2.i + 0.f;
denom.r = q__1.r, denom.i = q__1.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
c_div(&q__1, &b_ref(k - 1, j), &akm1k);
bkm1.r = q__1.r, bkm1.i = q__1.i;
r_cnjg(&q__2, &akm1k);
c_div(&q__1, &b_ref(k, j), &q__2);
bk.r = q__1.r, bk.i = q__1.i;
i__2 = b_subscr(k - 1, j);
q__3.r = ak.r * bkm1.r - ak.i * bkm1.i, q__3.i = ak.r *
bkm1.i + ak.i * bkm1.r;
q__2.r = q__3.r - bk.r, q__2.i = q__3.i - bk.i;
c_div(&q__1, &q__2, &denom);
b[i__2].r = q__1.r, b[i__2].i = q__1.i;
i__2 = b_subscr(k, j);
q__3.r = akm1.r * bk.r - akm1.i * bk.i, q__3.i = akm1.r *
bk.i + akm1.i * bk.r;
q__2.r = q__3.r - bkm1.r, q__2.i = q__3.i - bkm1.i;
c_div(&q__1, &q__2, &denom);
b[i__2].r = q__1.r, b[i__2].i = q__1.i;
/* L20: */
}
k += -2;
}
goto L10;
L30:
/* Next solve U'*X = B, overwriting B with X.
K is the main loop index, increasing from 1 to N in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = 1;
L40:
/* If K > N, exit from loop. */
if (k > *n) {
goto L50;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Multiply by inv(U'(K)), where U(K) is the transformation
stored in column K of A. */
if (k > 1) {
clacgv_(nrhs, &b_ref(k, 1), ldb);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b[b_offset]
, ldb, &a_ref(1, k), &c__1, &c_b1, &b_ref(k, 1), ldb);
clacgv_(nrhs, &b_ref(k, 1), ldb);
}
/* Interchange rows K and IPIV(K). */
kp = ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
++k;
} else {
/* 2 x 2 diagonal block
Multiply by inv(U'(K+1)), where U(K+1) is the transformation
stored in columns K and K+1 of A. */
if (k > 1) {
clacgv_(nrhs, &b_ref(k, 1), ldb);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b[b_offset]
, ldb, &a_ref(1, k), &c__1, &c_b1, &b_ref(k, 1), ldb);
clacgv_(nrhs, &b_ref(k, 1), ldb);
clacgv_(nrhs, &b_ref(k + 1, 1), ldb);
i__1 = k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b[b_offset]
, ldb, &a_ref(1, k + 1), &c__1, &c_b1, &b_ref(k + 1,
1), ldb);
clacgv_(nrhs, &b_ref(k + 1, 1), ldb);
}
/* Interchange rows K and -IPIV(K). */
kp = -ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
k += 2;
}
goto L40;
L50:
;
} else {
/* Solve A*X = B, where A = L*D*L'.
First solve L*D*X = B, overwriting B with X.
K is the main loop index, increasing from 1 to N in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = 1;
L60:
/* If K > N, exit from loop. */
if (k > *n) {
goto L80;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Interchange rows K and IPIV(K). */
kp = ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Multiply by inv(L(K)), where L(K) is the transformation
stored in column K of A. */
if (k < *n) {
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(k + 1, k), &c__1, &b_ref(k,
1), ldb, &b_ref(k + 1, 1), ldb);
}
/* Multiply by the inverse of the diagonal block. */
i__1 = a_subscr(k, k);
s = 1.f / a[i__1].r;
csscal_(nrhs, &s, &b_ref(k, 1), ldb);
++k;
} else {
/* 2 x 2 diagonal block
Interchange rows K+1 and -IPIV(K). */
kp = -ipiv[k];
if (kp != k + 1) {
cswap_(nrhs, &b_ref(k + 1, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Multiply by inv(L(K)), where L(K) is the transformation
stored in columns K and K+1 of A. */
if (k < *n - 1) {
i__1 = *n - k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(k + 2, k), &c__1, &b_ref(k,
1), ldb, &b_ref(k + 2, 1), ldb);
i__1 = *n - k - 1;
q__1.r = -1.f, q__1.i = 0.f;
cgeru_(&i__1, nrhs, &q__1, &a_ref(k + 2, k + 1), &c__1, &
b_ref(k + 1, 1), ldb, &b_ref(k + 2, 1), ldb);
}
/* Multiply by the inverse of the diagonal block. */
i__1 = a_subscr(k + 1, k);
akm1k.r = a[i__1].r, akm1k.i = a[i__1].i;
r_cnjg(&q__2, &akm1k);
c_div(&q__1, &a_ref(k, k), &q__2);
akm1.r = q__1.r, akm1.i = q__1.i;
c_div(&q__1, &a_ref(k + 1, k + 1), &akm1k);
ak.r = q__1.r, ak.i = q__1.i;
q__2.r = akm1.r * ak.r - akm1.i * ak.i, q__2.i = akm1.r * ak.i +
akm1.i * ak.r;
q__1.r = q__2.r - 1.f, q__1.i = q__2.i + 0.f;
denom.r = q__1.r, denom.i = q__1.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
r_cnjg(&q__2, &akm1k);
c_div(&q__1, &b_ref(k, j), &q__2);
bkm1.r = q__1.r, bkm1.i = q__1.i;
c_div(&q__1, &b_ref(k + 1, j), &akm1k);
bk.r = q__1.r, bk.i = q__1.i;
i__2 = b_subscr(k, j);
q__3.r = ak.r * bkm1.r - ak.i * bkm1.i, q__3.i = ak.r *
bkm1.i + ak.i * bkm1.r;
q__2.r = q__3.r - bk.r, q__2.i = q__3.i - bk.i;
c_div(&q__1, &q__2, &denom);
b[i__2].r = q__1.r, b[i__2].i = q__1.i;
i__2 = b_subscr(k + 1, j);
q__3.r = akm1.r * bk.r - akm1.i * bk.i, q__3.i = akm1.r *
bk.i + akm1.i * bk.r;
q__2.r = q__3.r - bkm1.r, q__2.i = q__3.i - bkm1.i;
c_div(&q__1, &q__2, &denom);
b[i__2].r = q__1.r, b[i__2].i = q__1.i;
/* L70: */
}
k += 2;
}
goto L60;
L80:
/* Next solve L'*X = B, overwriting B with X.
K is the main loop index, decreasing from N to 1 in steps of
1 or 2, depending on the size of the diagonal blocks. */
k = *n;
L90:
/* If K < 1, exit from loop. */
if (k < 1) {
goto L100;
}
if (ipiv[k] > 0) {
/* 1 x 1 diagonal block
Multiply by inv(L'(K)), where L(K) is the transformation
stored in column K of A. */
if (k < *n) {
clacgv_(nrhs, &b_ref(k, 1), ldb);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b_ref(k +
1, 1), ldb, &a_ref(k + 1, k), &c__1, &c_b1, &b_ref(k,
1), ldb);
clacgv_(nrhs, &b_ref(k, 1), ldb);
}
/* Interchange rows K and IPIV(K). */
kp = ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
--k;
} else {
/* 2 x 2 diagonal block
Multiply by inv(L'(K-1)), where L(K-1) is the transformation
stored in columns K-1 and K of A. */
if (k < *n) {
clacgv_(nrhs, &b_ref(k, 1), ldb);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b_ref(k +
1, 1), ldb, &a_ref(k + 1, k), &c__1, &c_b1, &b_ref(k,
1), ldb);
clacgv_(nrhs, &b_ref(k, 1), ldb);
clacgv_(nrhs, &b_ref(k - 1, 1), ldb);
i__1 = *n - k;
q__1.r = -1.f, q__1.i = 0.f;
cgemv_("Conjugate transpose", &i__1, nrhs, &q__1, &b_ref(k +
1, 1), ldb, &a_ref(k + 1, k - 1), &c__1, &c_b1, &
b_ref(k - 1, 1), ldb);
clacgv_(nrhs, &b_ref(k - 1, 1), ldb);
}
/* Interchange rows K and -IPIV(K). */
kp = -ipiv[k];
if (kp != k) {
cswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
k += -2;
}
goto L90;
L100:
;
}
return 0;
/* End of CHETRS */
} /* chetrs_ */
/* Subroutine */ int zunm2r_(char *side, char *trans, integer *m, integer *n,
integer *k, doublecomplex *a, integer *lda, doublecomplex *tau,
doublecomplex *c__, integer *ldc, doublecomplex *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
=======
ZUNM2R overwrites the general complex m-by-n matrix C with
Q * C if SIDE = 'L' and TRANS = 'N', or
Q'* C if SIDE = 'L' and TRANS = 'C', or
C * Q if SIDE = 'R' and TRANS = 'N', or
C * Q' if SIDE = 'R' and TRANS = 'C',
where Q is a complex unitary matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by ZGEQRF. Q is of order m if SIDE = 'L' and of order n
if SIDE = 'R'.
Arguments
=========
SIDE (input) CHARACTER*1
= 'L': apply Q or Q' from the Left
= 'R': apply Q or Q' from the Right
TRANS (input) CHARACTER*1
= 'N': apply Q (No transpose)
= 'C': apply Q' (Conjugate transpose)
M (input) INTEGER
The number of rows of the matrix C. M >= 0.
N (input) INTEGER
The number of columns of the matrix C. N >= 0.
K (input) INTEGER
The number of elementary reflectors whose product defines
the matrix Q.
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
A (input) COMPLEX*16 array, dimension (LDA,K)
The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
ZGEQRF in the first k columns of its array argument A.
A is modified by the routine but restored on exit.
LDA (input) INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).
TAU (input) COMPLEX*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by ZGEQRF.
C (input/output) COMPLEX*16 array, dimension (LDC,N)
On entry, the m-by-n matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK (workspace) COMPLEX*16 array, dimension
(N) if SIDE = 'L',
(M) if SIDE = 'R'
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
=====================================================================
Test the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, c_dim1, c_offset, i__1, i__2, i__3;
doublecomplex z__1;
/* Builtin functions */
void d_cnjg(doublecomplex *, doublecomplex *);
/* Local variables */
static logical left;
static doublecomplex taui;
static integer i__;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int zlarf_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *, doublecomplex *);
static integer i1, i2, i3, ic, jc, mi, ni, nq;
extern /* Subroutine */ int xerbla_(char *, integer *);
static logical notran;
static doublecomplex aii;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define c___subscr(a_1,a_2) (a_2)*c_dim1 + a_1
#define c___ref(a_1,a_2) c__[c___subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
c_dim1 = *ldc;
c_offset = 1 + c_dim1 * 1;
c__ -= c_offset;
--work;
/* Function Body */
*info = 0;
left = lsame_(side, "L");
notran = lsame_(trans, "N");
/* NQ is the order of Q */
if (left) {
nq = *m;
} else {
nq = *n;
}
if (! left && ! lsame_(side, "R")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "C")) {
*info = -2;
} else if (*m < 0) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*k < 0 || *k > nq) {
*info = -5;
} else if (*lda < max(1,nq)) {
*info = -7;
} else if (*ldc < max(1,*m)) {
*info = -10;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZUNM2R", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0 || *k == 0) {
return 0;
}
if (left && ! notran || ! left && notran) {
i1 = 1;
i2 = *k;
i3 = 1;
} else {
i1 = *k;
i2 = 1;
i3 = -1;
}
if (left) {
ni = *n;
jc = 1;
} else {
mi = *m;
ic = 1;
}
i__1 = i2;
i__2 = i3;
for (i__ = i1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) {
if (left) {
/* H(i) or H(i)' is applied to C(i:m,1:n) */
mi = *m - i__ + 1;
ic = i__;
} else {
/* H(i) or H(i)' is applied to C(1:m,i:n) */
ni = *n - i__ + 1;
jc = i__;
}
/* Apply H(i) or H(i)' */
if (notran) {
i__3 = i__;
taui.r = tau[i__3].r, taui.i = tau[i__3].i;
} else {
d_cnjg(&z__1, &tau[i__]);
taui.r = z__1.r, taui.i = z__1.i;
}
i__3 = a_subscr(i__, i__);
aii.r = a[i__3].r, aii.i = a[i__3].i;
i__3 = a_subscr(i__, i__);
a[i__3].r = 1., a[i__3].i = 0.;
zlarf_(side, &mi, &ni, &a_ref(i__, i__), &c__1, &taui, &c___ref(ic,
jc), ldc, &work[1]);
i__3 = a_subscr(i__, i__);
a[i__3].r = aii.r, a[i__3].i = aii.i;
/* L10: */
}
return 0;
/* End of ZUNM2R */
} /* zunm2r_ */
/* Subroutine */ int zggbal_(char *job, integer *n, doublecomplex *a, integer
*lda, doublecomplex *b, integer *ldb, integer *ilo, integer *ihi,
doublereal *lscale, doublereal *rscale, doublereal *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
=======
ZGGBAL balances a pair of general complex matrices (A,B). This
involves, first, permuting A and B by similarity transformations to
isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
elements on the diagonal; and second, applying a diagonal similarity
transformation to rows and columns ILO to IHI to make the rows
and columns as close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrices, and improve the
accuracy of the computed eigenvalues and/or eigenvectors in the
generalized eigenvalue problem A*x = lambda*B*x.
Arguments
=========
JOB (input) CHARACTER*1
Specifies the operations to be performed on A and B:
= 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
and RSCALE(I) = 1.0 for i=1,...,N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB,N)
On entry, the input matrix B.
On exit, B is overwritten by the balanced matrix.
If JOB = 'N', B is not referenced.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ILO (output) INTEGER
IHI (output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 and B(i,j) = 0 if i > j and
j = 1,...,ILO-1 or i = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the left side of A and B. If P(j) is the index of the
row interchanged with row j, and D(j) is the scaling factor
applied to row j, then
LSCALE(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.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
to the right side of A and B. If P(j) is the index of the
column interchanged with column j, and D(j) is the scaling
factor applied to column j, then
RSCALE(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.
WORK (workspace) DOUBLE PRECISION array, dimension (6*N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
See R.C. WARD, Balancing the generalized eigenvalue problem,
SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static doublereal c_b35 = 10.;
static doublereal c_b71 = .5;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2, i__3, i__4;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double d_lg10(doublereal *), d_imag(doublecomplex *), z_abs(doublecomplex
*), d_sign(doublereal *, doublereal *), pow_di(doublereal *,
integer *);
/* Local variables */
static integer lcab;
static doublereal beta, coef;
static integer irab, lrab;
static doublereal basl, cmax;
extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *,
integer *);
static doublereal coef2, coef5;
static integer i__, j, k, l, m;
static doublereal gamma, t, alpha;
extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *,
integer *);
extern logical lsame_(char *, char *);
static doublereal sfmin, sfmax;
static integer iflow;
extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *);
static integer kount;
extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *);
static integer jc;
static doublereal ta, tb, tc;
extern doublereal dlamch_(char *);
static integer ir, it;
static doublereal ew;
static integer nr;
static doublereal pgamma;
extern /* Subroutine */ int xerbla_(char *, integer *), zdscal_(
integer *, doublereal *, doublecomplex *, integer *);
static integer lsfmin;
extern integer izamax_(integer *, doublecomplex *, integer *);
static integer lsfmax, ip1, jp1, lm1;
static doublereal cab, rab, ewc, cor, sum;
static integer nrp2, icab;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--lscale;
--rscale;
--work;
/* Function Body */
*info = 0;
if (! lsame_(job, "N") && ! lsame_(job, "P") && ! lsame_(job, "S")
&& ! lsame_(job, "B")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
} else if (*ldb < max(1,*n)) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGGBAL", &i__1);
return 0;
}
k = 1;
l = *n;
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (lsame_(job, "N")) {
*ilo = 1;
*ihi = *n;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
lscale[i__] = 1.;
rscale[i__] = 1.;
/* L10: */
}
return 0;
}
if (k == l) {
*ilo = 1;
*ihi = 1;
lscale[1] = 1.;
rscale[1] = 1.;
return 0;
}
if (lsame_(job, "S")) {
goto L190;
}
goto L30;
/* Permute the matrices A and B to isolate the eigenvalues.
Find row with one nonzero in columns 1 through L */
L20:
l = lm1;
if (l != 1) {
goto L30;
}
rscale[1] = 1.;
lscale[1] = 1.;
goto L190;
L30:
lm1 = l - 1;
for (i__ = l; i__ >= 1; --i__) {
i__1 = lm1;
for (j = 1; j <= i__1; ++j) {
jp1 = j + 1;
i__2 = a_subscr(i__, j);
i__3 = b_subscr(i__, j);
if (a[i__2].r != 0. || a[i__2].i != 0. || (b[i__3].r != 0. || b[
i__3].i != 0.)) {
goto L50;
}
/* L40: */
}
j = l;
goto L70;
L50:
i__1 = l;
for (j = jp1; j <= i__1; ++j) {
i__2 = a_subscr(i__, j);
i__3 = b_subscr(i__, j);
if (a[i__2].r != 0. || a[i__2].i != 0. || (b[i__3].r != 0. || b[
i__3].i != 0.)) {
goto L80;
}
/* L60: */
}
j = jp1 - 1;
L70:
m = l;
iflow = 1;
goto L160;
L80:
;
}
goto L100;
/* Find column with one nonzero in rows K through N */
L90:
++k;
L100:
i__1 = l;
for (j = k; j <= i__1; ++j) {
i__2 = lm1;
for (i__ = k; i__ <= i__2; ++i__) {
ip1 = i__ + 1;
i__3 = a_subscr(i__, j);
i__4 = b_subscr(i__, j);
if (a[i__3].r != 0. || a[i__3].i != 0. || (b[i__4].r != 0. || b[
i__4].i != 0.)) {
goto L120;
}
/* L110: */
}
i__ = l;
goto L140;
L120:
i__2 = l;
for (i__ = ip1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = b_subscr(i__, j);
if (a[i__3].r != 0. || a[i__3].i != 0. || (b[i__4].r != 0. || b[
i__4].i != 0.)) {
goto L150;
}
/* L130: */
}
i__ = ip1 - 1;
L140:
m = k;
iflow = 2;
goto L160;
L150:
;
}
goto L190;
/* Permute rows M and I */
L160:
lscale[m] = (doublereal) i__;
if (i__ == m) {
goto L170;
}
i__1 = *n - k + 1;
zswap_(&i__1, &a_ref(i__, k), lda, &a_ref(m, k), lda);
i__1 = *n - k + 1;
zswap_(&i__1, &b_ref(i__, k), ldb, &b_ref(m, k), ldb);
/* Permute columns M and J */
L170:
rscale[m] = (doublereal) j;
if (j == m) {
goto L180;
}
zswap_(&l, &a_ref(1, j), &c__1, &a_ref(1, m), &c__1);
zswap_(&l, &b_ref(1, j), &c__1, &b_ref(1, m), &c__1);
L180:
switch (iflow) {
case 1: goto L20;
case 2: goto L90;
}
L190:
*ilo = k;
*ihi = l;
if (*ilo == *ihi) {
return 0;
}
if (lsame_(job, "P")) {
return 0;
}
/* Balance the submatrix in rows ILO to IHI. */
nr = *ihi - *ilo + 1;
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
rscale[i__] = 0.;
lscale[i__] = 0.;
work[i__] = 0.;
work[i__ + *n] = 0.;
work[i__ + (*n << 1)] = 0.;
work[i__ + *n * 3] = 0.;
work[i__ + (*n << 2)] = 0.;
work[i__ + *n * 5] = 0.;
/* L200: */
}
/* Compute right side vector in resulting linear equations */
basl = d_lg10(&c_b35);
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
i__2 = *ihi;
for (j = *ilo; j <= i__2; ++j) {
i__3 = a_subscr(i__, j);
if (a[i__3].r == 0. && a[i__3].i == 0.) {
ta = 0.;
goto L210;
}
i__3 = a_subscr(i__, j);
d__3 = (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a_ref(i__,
j)), abs(d__2));
ta = d_lg10(&d__3) / basl;
L210:
i__3 = b_subscr(i__, j);
if (b[i__3].r == 0. && b[i__3].i == 0.) {
tb = 0.;
goto L220;
}
i__3 = b_subscr(i__, j);
d__3 = (d__1 = b[i__3].r, abs(d__1)) + (d__2 = d_imag(&b_ref(i__,
j)), abs(d__2));
tb = d_lg10(&d__3) / basl;
L220:
work[i__ + (*n << 2)] = work[i__ + (*n << 2)] - ta - tb;
work[j + *n * 5] = work[j + *n * 5] - ta - tb;
/* L230: */
}
/* L240: */
}
coef = 1. / (doublereal) (nr << 1);
coef2 = coef * coef;
coef5 = coef2 * .5;
nrp2 = nr + 2;
beta = 0.;
it = 1;
/* Start generalized conjugate gradient iteration */
L250:
gamma = ddot_(&nr, &work[*ilo + (*n << 2)], &c__1, &work[*ilo + (*n << 2)]
, &c__1) + ddot_(&nr, &work[*ilo + *n * 5], &c__1, &work[*ilo + *
n * 5], &c__1);
ew = 0.;
ewc = 0.;
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
ew += work[i__ + (*n << 2)];
ewc += work[i__ + *n * 5];
/* L260: */
}
/* Computing 2nd power */
d__1 = ew;
/* Computing 2nd power */
d__2 = ewc;
/* Computing 2nd power */
d__3 = ew - ewc;
gamma = coef * gamma - coef2 * (d__1 * d__1 + d__2 * d__2) - coef5 * (
d__3 * d__3);
if (gamma == 0.) {
goto L350;
}
if (it != 1) {
beta = gamma / pgamma;
}
t = coef5 * (ewc - ew * 3.);
tc = coef5 * (ew - ewc * 3.);
dscal_(&nr, &beta, &work[*ilo], &c__1);
dscal_(&nr, &beta, &work[*ilo + *n], &c__1);
daxpy_(&nr, &coef, &work[*ilo + (*n << 2)], &c__1, &work[*ilo + *n], &
c__1);
daxpy_(&nr, &coef, &work[*ilo + *n * 5], &c__1, &work[*ilo], &c__1);
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
work[i__] += tc;
work[i__ + *n] += t;
/* L270: */
}
/* Apply matrix to vector */
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
kount = 0;
sum = 0.;
i__2 = *ihi;
for (j = *ilo; j <= i__2; ++j) {
i__3 = a_subscr(i__, j);
if (a[i__3].r == 0. && a[i__3].i == 0.) {
goto L280;
}
++kount;
sum += work[j];
L280:
i__3 = b_subscr(i__, j);
if (b[i__3].r == 0. && b[i__3].i == 0.) {
goto L290;
}
++kount;
sum += work[j];
L290:
;
}
work[i__ + (*n << 1)] = (doublereal) kount * work[i__ + *n] + sum;
/* L300: */
}
i__1 = *ihi;
for (j = *ilo; j <= i__1; ++j) {
kount = 0;
sum = 0.;
i__2 = *ihi;
for (i__ = *ilo; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
if (a[i__3].r == 0. && a[i__3].i == 0.) {
goto L310;
}
++kount;
sum += work[i__ + *n];
L310:
i__3 = b_subscr(i__, j);
if (b[i__3].r == 0. && b[i__3].i == 0.) {
goto L320;
}
++kount;
sum += work[i__ + *n];
L320:
;
}
work[j + *n * 3] = (doublereal) kount * work[j] + sum;
/* L330: */
}
sum = ddot_(&nr, &work[*ilo + *n], &c__1, &work[*ilo + (*n << 1)], &c__1)
+ ddot_(&nr, &work[*ilo], &c__1, &work[*ilo + *n * 3], &c__1);
alpha = gamma / sum;
/* Determine correction to current iteration */
cmax = 0.;
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
cor = alpha * work[i__ + *n];
if (abs(cor) > cmax) {
cmax = abs(cor);
}
lscale[i__] += cor;
cor = alpha * work[i__];
if (abs(cor) > cmax) {
cmax = abs(cor);
}
rscale[i__] += cor;
/* L340: */
}
if (cmax < .5) {
goto L350;
}
d__1 = -alpha;
daxpy_(&nr, &d__1, &work[*ilo + (*n << 1)], &c__1, &work[*ilo + (*n << 2)]
, &c__1);
d__1 = -alpha;
daxpy_(&nr, &d__1, &work[*ilo + *n * 3], &c__1, &work[*ilo + *n * 5], &
c__1);
pgamma = gamma;
++it;
if (it <= nrp2) {
goto L250;
}
/* End generalized conjugate gradient iteration */
L350:
sfmin = dlamch_("S");
sfmax = 1. / sfmin;
lsfmin = (integer) (d_lg10(&sfmin) / basl + 1.);
lsfmax = (integer) (d_lg10(&sfmax) / basl);
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
i__2 = *n - *ilo + 1;
irab = izamax_(&i__2, &a_ref(i__, *ilo), lda);
rab = z_abs(&a_ref(i__, irab + *ilo - 1));
i__2 = *n - *ilo + 1;
irab = izamax_(&i__2, &b_ref(i__, *ilo), lda);
/* Computing MAX */
d__1 = rab, d__2 = z_abs(&b_ref(i__, irab + *ilo - 1));
rab = max(d__1,d__2);
d__1 = rab + sfmin;
lrab = (integer) (d_lg10(&d__1) / basl + 1.);
ir = (integer) (lscale[i__] + d_sign(&c_b71, &lscale[i__]));
/* Computing MIN */
i__2 = max(ir,lsfmin), i__2 = min(i__2,lsfmax), i__3 = lsfmax - lrab;
ir = min(i__2,i__3);
lscale[i__] = pow_di(&c_b35, &ir);
icab = izamax_(ihi, &a_ref(1, i__), &c__1);
cab = z_abs(&a_ref(icab, i__));
icab = izamax_(ihi, &b_ref(1, i__), &c__1);
/* Computing MAX */
d__1 = cab, d__2 = z_abs(&b_ref(icab, i__));
cab = max(d__1,d__2);
d__1 = cab + sfmin;
lcab = (integer) (d_lg10(&d__1) / basl + 1.);
jc = (integer) (rscale[i__] + d_sign(&c_b71, &rscale[i__]));
/* Computing MIN */
i__2 = max(jc,lsfmin), i__2 = min(i__2,lsfmax), i__3 = lsfmax - lcab;
jc = min(i__2,i__3);
rscale[i__] = pow_di(&c_b35, &jc);
/* L360: */
}
/* Row scaling of matrices A and B */
i__1 = *ihi;
for (i__ = *ilo; i__ <= i__1; ++i__) {
i__2 = *n - *ilo + 1;
zdscal_(&i__2, &lscale[i__], &a_ref(i__, *ilo), lda);
i__2 = *n - *ilo + 1;
zdscal_(&i__2, &lscale[i__], &b_ref(i__, *ilo), ldb);
/* L370: */
}
/* Column scaling of matrices A and B */
i__1 = *ihi;
for (j = *ilo; j <= i__1; ++j) {
zdscal_(ihi, &rscale[j], &a_ref(1, j), &c__1);
zdscal_(ihi, &rscale[j], &b_ref(1, j), &c__1);
/* L380: */
}
return 0;
/* End of ZGGBAL */
} /* zggbal_ */
/* Subroutine */ int ctrti2_(char *uplo, char *diag, integer *n, complex *a,
integer *lda, 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
=======
CTRTI2 computes the inverse of a complex upper or lower triangular
matrix.
This is the Level 2 BLAS version of the algorithm.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
DIAG (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the triangular matrix A. If UPLO = 'U', the
leading n by n upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If DIAG = 'U', the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {1.f,0.f};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
complex q__1;
/* Builtin functions */
void c_div(complex *, complex *, complex *);
/* Local variables */
static integer j;
extern /* Subroutine */ int cscal_(integer *, complex *, complex *,
integer *);
extern logical lsame_(char *, char *);
static logical upper;
extern /* Subroutine */ int ctrmv_(char *, char *, char *, integer *,
complex *, integer *, complex *, integer *), xerbla_(char *, integer *);
static logical nounit;
static complex ajj;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTRTI2", &i__1);
return 0;
}
if (upper) {
/* Compute inverse of upper triangular matrix. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (nounit) {
i__2 = a_subscr(j, j);
c_div(&q__1, &c_b1, &a_ref(j, j));
a[i__2].r = q__1.r, a[i__2].i = q__1.i;
i__2 = a_subscr(j, j);
q__1.r = -a[i__2].r, q__1.i = -a[i__2].i;
ajj.r = q__1.r, ajj.i = q__1.i;
} else {
q__1.r = -1.f, q__1.i = 0.f;
ajj.r = q__1.r, ajj.i = q__1.i;
}
/* Compute elements 1:j-1 of j-th column. */
i__2 = j - 1;
ctrmv_("Upper", "No transpose", diag, &i__2, &a[a_offset], lda, &
a_ref(1, j), &c__1);
i__2 = j - 1;
cscal_(&i__2, &ajj, &a_ref(1, j), &c__1);
/* L10: */
}
} else {
/* Compute inverse of lower triangular matrix. */
for (j = *n; j >= 1; --j) {
if (nounit) {
i__1 = a_subscr(j, j);
c_div(&q__1, &c_b1, &a_ref(j, j));
a[i__1].r = q__1.r, a[i__1].i = q__1.i;
i__1 = a_subscr(j, j);
q__1.r = -a[i__1].r, q__1.i = -a[i__1].i;
ajj.r = q__1.r, ajj.i = q__1.i;
} else {
q__1.r = -1.f, q__1.i = 0.f;
ajj.r = q__1.r, ajj.i = q__1.i;
}
if (j < *n) {
/* Compute elements j+1:n of j-th column. */
i__1 = *n - j;
ctrmv_("Lower", "No transpose", diag, &i__1, &a_ref(j + 1, j
+ 1), lda, &a_ref(j + 1, j), &c__1);
i__1 = *n - j;
cscal_(&i__1, &ajj, &a_ref(j + 1, j), &c__1);
}
/* L20: */
}
}
return 0;
/* End of CTRTI2 */
} /* ctrti2_ */
/* Subroutine */ int cpoequ_(integer *n, complex *a, integer *lda, real *s,
real *scond, real *amax, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
CPOEQU computes row and column scalings intended to equilibrate a
Hermitian positive definite matrix A and reduce its condition number
(with respect to the two-norm). S contains the scale factors,
S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
choice of S puts the condition number of B within a factor N of the
smallest possible condition number over all possible diagonal
scalings.
Arguments
=========
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input) COMPLEX array, dimension (LDA,N)
The N-by-N Hermitian positive definite matrix whose scaling
factors are to be computed. Only the diagonal elements of A
are referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
S (output) REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND (output) REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.
AMAX (output) REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2;
real r__1, r__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static real smin;
static integer i__;
extern /* Subroutine */ int xerbla_(char *, integer *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--s;
/* Function Body */
*info = 0;
if (*n < 0) {
*info = -1;
} else if (*lda < max(1,*n)) {
*info = -3;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CPOEQU", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*scond = 1.f;
*amax = 0.f;
return 0;
}
/* Find the minimum and maximum diagonal elements. */
i__1 = a_subscr(1, 1);
s[1] = a[i__1].r;
smin = s[1];
*amax = s[1];
i__1 = *n;
for (i__ = 2; i__ <= i__1; ++i__) {
i__2 = a_subscr(i__, i__);
s[i__] = a[i__2].r;
/* Computing MIN */
r__1 = smin, r__2 = s[i__];
smin = dmin(r__1,r__2);
/* Computing MAX */
r__1 = *amax, r__2 = s[i__];
*amax = dmax(r__1,r__2);
/* L10: */
}
if (smin <= 0.f) {
/* Find the first non-positive diagonal element and return. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (s[i__] <= 0.f) {
*info = i__;
return 0;
}
/* L20: */
}
} else {
/* Set the scale factors to the reciprocals
of the diagonal elements. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
s[i__] = 1.f / sqrt(s[i__]);
/* L30: */
}
/* Compute SCOND = min(S(I)) / max(S(I)) */
*scond = sqrt(smin) / sqrt(*amax);
}
return 0;
/* End of CPOEQU */
} /* cpoequ_ */
/* Subroutine */ int zlavsy_(char *uplo, char *trans, char *diag, integer *n,
integer *nrhs, doublecomplex *a, integer *lda, integer *ipiv,
doublecomplex *b, integer *ldb, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
doublecomplex z__1, z__2, z__3;
/* Local variables */
static integer j, k;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int zscal_(integer *, doublecomplex *,
doublecomplex *, integer *), zgemv_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, integer *, doublecomplex *,
integer *, doublecomplex *, doublecomplex *, integer *),
zgeru_(integer *, integer *, doublecomplex *, doublecomplex *,
integer *, doublecomplex *, integer *, doublecomplex *, integer *)
, zswap_(integer *, doublecomplex *, integer *, doublecomplex *,
integer *);
static doublecomplex t1, t2, d11, d12, d21, d22;
static integer kp;
extern /* Subroutine */ int xerbla_(char *, integer *);
static logical nounit;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
/* -- 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
=======
ZLAVSY performs one of the matrix-vector operations
x := A*x or x := A'*x,
where x is an N element vector and A is one of the factors
from the symmetric factorization computed by ZSYTRF.
ZSYTRF produces a factorization of the form
U * D * U' or L * D * L' ,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, U' (or L') is the transpose of
U (or L), and D is symmetric and block diagonal with 1 x 1 and
2 x 2 diagonal blocks. The multipliers for the transformations
and the upper or lower triangular parts of the diagonal blocks
are stored in the leading upper or lower triangle of the 2-D
array A.
If TRANS = 'N' or 'n', ZLAVSY multiplies either by U or U * D
(or L or L * D).
If TRANS = 'T' or 't', ZLAVSY multiplies either by U' or D * U'
(or L' or D * L' ).
Arguments
==========
UPLO - CHARACTER*1
On entry, UPLO specifies whether the triangular matrix
stored in A is upper or lower triangular.
UPLO = 'U' or 'u' The matrix is upper triangular.
UPLO = 'L' or 'l' The matrix is lower triangular.
Unchanged on exit.
TRANS - CHARACTER*1
On entry, TRANS specifies the operation to be performed as
follows:
TRANS = 'N' or 'n' x := A*x.
TRANS = 'T' or 't' x := A'*x.
Unchanged on exit.
DIAG - CHARACTER*1
On entry, DIAG specifies whether the diagonal blocks are
assumed to be unit matrices:
DIAG = 'U' or 'u' Diagonal blocks are unit matrices.
DIAG = 'N' or 'n' Diagonal blocks are non-unit.
Unchanged on exit.
N - INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
NRHS - INTEGER
On entry, NRHS specifies the number of right hand sides,
i.e., the number of vectors x to be multiplied by A.
NRHS must be at least zero.
Unchanged on exit.
A - COMPLEX*16 array, dimension( LDA, N )
On entry, A contains a block diagonal matrix and the
multipliers of the transformations used to obtain it,
stored as a 2-D triangular matrix.
Unchanged on exit.
LDA - INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling ( sub ) program. LDA must be at least
max( 1, N ).
Unchanged on exit.
IPIV - INTEGER array, dimension( N )
On entry, IPIV contains the vector of pivot indices as
determined by ZSYTRF or ZHETRF.
If IPIV( K ) = K, no interchange was done.
If IPIV( K ) <> K but IPIV( K ) > 0, then row K was inter-
changed with row IPIV( K ) and a 1 x 1 pivot block was used.
If IPIV( K ) < 0 and UPLO = 'U', then row K-1 was exchanged
with row | IPIV( K ) | and a 2 x 2 pivot block was used.
If IPIV( K ) < 0 and UPLO = 'L', then row K+1 was exchanged
with row | IPIV( K ) | and a 2 x 2 pivot block was used.
B - COMPLEX*16 array, dimension( LDB, NRHS )
On entry, B contains NRHS vectors of length N.
On exit, B is overwritten with the product A * B.
LDB - INTEGER
On entry, LDB contains the leading dimension of B as
declared in the calling program. LDB must be at least
max( 1, N ).
Unchanged on exit.
INFO - INTEGER
INFO is the error flag.
On exit, a value of 0 indicates a successful exit.
A negative value, say -K, indicates that the K-th argument
has an illegal value.
=====================================================================
Test the input parameters.
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
/* Function Body */
*info = 0;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! lsame_(trans, "N") && ! lsame_(trans,
"T")) {
*info = -2;
} else if (! lsame_(diag, "U") && ! lsame_(diag,
"N")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldb < max(1,*n)) {
*info = -9;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLAVSY ", &i__1);
return 0;
}
/* Quick return if possible. */
if (*n == 0) {
return 0;
}
nounit = lsame_(diag, "N");
/* ------------------------------------------
Compute B := A * B (No transpose)
------------------------------------------ */
if (lsame_(trans, "N")) {
/* Compute B := U*B
where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1)) */
if (lsame_(uplo, "U")) {
/* Loop forward applying the transformations. */
k = 1;
L10:
if (k > *n) {
goto L30;
}
if (ipiv[k] > 0) {
/* 1 x 1 pivot block
Multiply by the diagonal element if forming U * D. */
if (nounit) {
zscal_(nrhs, &a_ref(k, k), &b_ref(k, 1), ldb);
}
/* Multiply by P(K) * inv(U(K)) if K > 1. */
if (k > 1) {
/* Apply the transformation. */
i__1 = k - 1;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(1, k), &c__1, &b_ref(k,
1), ldb, &b_ref(1, 1), ldb);
/* Interchange if P(K) != I. */
kp = ipiv[k];
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
}
++k;
} else {
/* 2 x 2 pivot block
Multiply by the diagonal block if forming U * D. */
if (nounit) {
i__1 = a_subscr(k, k);
d11.r = a[i__1].r, d11.i = a[i__1].i;
i__1 = a_subscr(k + 1, k + 1);
d22.r = a[i__1].r, d22.i = a[i__1].i;
i__1 = a_subscr(k, k + 1);
d12.r = a[i__1].r, d12.i = a[i__1].i;
d21.r = d12.r, d21.i = d12.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = b_subscr(k, j);
t1.r = b[i__2].r, t1.i = b[i__2].i;
i__2 = b_subscr(k + 1, j);
t2.r = b[i__2].r, t2.i = b[i__2].i;
i__2 = b_subscr(k, j);
z__2.r = d11.r * t1.r - d11.i * t1.i, z__2.i = d11.r *
t1.i + d11.i * t1.r;
z__3.r = d12.r * t2.r - d12.i * t2.i, z__3.i = d12.r *
t2.i + d12.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
i__2 = b_subscr(k + 1, j);
z__2.r = d21.r * t1.r - d21.i * t1.i, z__2.i = d21.r *
t1.i + d21.i * t1.r;
z__3.r = d22.r * t2.r - d22.i * t2.i, z__3.i = d22.r *
t2.i + d22.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
/* L20: */
}
}
/* Multiply by P(K) * inv(U(K)) if K > 1. */
if (k > 1) {
/* Apply the transformations. */
i__1 = k - 1;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(1, k), &c__1, &b_ref(k,
1), ldb, &b_ref(1, 1), ldb);
i__1 = k - 1;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(1, k + 1), &c__1, &
b_ref(k + 1, 1), ldb, &b_ref(1, 1), ldb);
/* Interchange if P(K) != I. */
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
}
k += 2;
}
goto L10;
L30:
/* Compute B := L*B
where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m)) . */
;
} else {
/* Loop backward applying the transformations to B. */
k = *n;
L40:
if (k < 1) {
goto L60;
}
/* Test the pivot index. If greater than zero, a 1 x 1
pivot was used, otherwise a 2 x 2 pivot was used. */
if (ipiv[k] > 0) {
/* 1 x 1 pivot block:
Multiply by the diagonal element if forming L * D. */
if (nounit) {
zscal_(nrhs, &a_ref(k, k), &b_ref(k, 1), ldb);
}
/* Multiply by P(K) * inv(L(K)) if K < N. */
if (k != *n) {
kp = ipiv[k];
/* Apply the transformation. */
i__1 = *n - k;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(k + 1, k), &c__1, &
b_ref(k, 1), ldb, &b_ref(k + 1, 1), ldb);
/* Interchange if a permutation was applied at the
K-th step of the factorization. */
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
}
--k;
} else {
/* 2 x 2 pivot block:
Multiply by the diagonal block if forming L * D. */
if (nounit) {
i__1 = a_subscr(k - 1, k - 1);
d11.r = a[i__1].r, d11.i = a[i__1].i;
i__1 = a_subscr(k, k);
d22.r = a[i__1].r, d22.i = a[i__1].i;
i__1 = a_subscr(k, k - 1);
d21.r = a[i__1].r, d21.i = a[i__1].i;
d12.r = d21.r, d12.i = d21.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = b_subscr(k - 1, j);
t1.r = b[i__2].r, t1.i = b[i__2].i;
i__2 = b_subscr(k, j);
t2.r = b[i__2].r, t2.i = b[i__2].i;
i__2 = b_subscr(k - 1, j);
z__2.r = d11.r * t1.r - d11.i * t1.i, z__2.i = d11.r *
t1.i + d11.i * t1.r;
z__3.r = d12.r * t2.r - d12.i * t2.i, z__3.i = d12.r *
t2.i + d12.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
i__2 = b_subscr(k, j);
z__2.r = d21.r * t1.r - d21.i * t1.i, z__2.i = d21.r *
t1.i + d21.i * t1.r;
z__3.r = d22.r * t2.r - d22.i * t2.i, z__3.i = d22.r *
t2.i + d22.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
/* L50: */
}
}
/* Multiply by P(K) * inv(L(K)) if K < N. */
if (k != *n) {
/* Apply the transformation. */
i__1 = *n - k;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(k + 1, k), &c__1, &
b_ref(k, 1), ldb, &b_ref(k + 1, 1), ldb);
i__1 = *n - k;
zgeru_(&i__1, nrhs, &c_b1, &a_ref(k + 1, k - 1), &c__1, &
b_ref(k - 1, 1), ldb, &b_ref(k + 1, 1), ldb);
/* Interchange if a permutation was applied at the
K-th step of the factorization. */
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
}
k += -2;
}
goto L40;
L60:
;
}
/* ----------------------------------------
Compute B := A' * B (transpose)
---------------------------------------- */
} else if (lsame_(trans, "T")) {
/* Form B := U'*B
where U = P(m)*inv(U(m))* ... *P(1)*inv(U(1))
and U' = inv(U'(1))*P(1)* ... *inv(U'(m))*P(m) */
if (lsame_(uplo, "U")) {
/* Loop backward applying the transformations. */
k = *n;
L70:
if (k < 1) {
goto L90;
}
/* 1 x 1 pivot block. */
if (ipiv[k] > 0) {
if (k > 1) {
/* Interchange if P(K) != I. */
kp = ipiv[k];
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Apply the transformation */
i__1 = k - 1;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b[b_offset], ldb,
&a_ref(1, k), &c__1, &c_b1, &b_ref(k, 1), ldb);
}
if (nounit) {
zscal_(nrhs, &a_ref(k, k), &b_ref(k, 1), ldb);
}
--k;
/* 2 x 2 pivot block. */
} else {
if (k > 2) {
/* Interchange if P(K) != I. */
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k - 1) {
zswap_(nrhs, &b_ref(k - 1, 1), ldb, &b_ref(kp, 1),
ldb);
}
/* Apply the transformations */
i__1 = k - 2;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b[b_offset], ldb,
&a_ref(1, k), &c__1, &c_b1, &b_ref(k, 1), ldb);
i__1 = k - 2;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b[b_offset], ldb,
&a_ref(1, k - 1), &c__1, &c_b1, &b_ref(k - 1, 1),
ldb);
}
/* Multiply by the diagonal block if non-unit. */
if (nounit) {
i__1 = a_subscr(k - 1, k - 1);
d11.r = a[i__1].r, d11.i = a[i__1].i;
i__1 = a_subscr(k, k);
d22.r = a[i__1].r, d22.i = a[i__1].i;
i__1 = a_subscr(k - 1, k);
d12.r = a[i__1].r, d12.i = a[i__1].i;
d21.r = d12.r, d21.i = d12.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = b_subscr(k - 1, j);
t1.r = b[i__2].r, t1.i = b[i__2].i;
i__2 = b_subscr(k, j);
t2.r = b[i__2].r, t2.i = b[i__2].i;
i__2 = b_subscr(k - 1, j);
z__2.r = d11.r * t1.r - d11.i * t1.i, z__2.i = d11.r *
t1.i + d11.i * t1.r;
z__3.r = d12.r * t2.r - d12.i * t2.i, z__3.i = d12.r *
t2.i + d12.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
i__2 = b_subscr(k, j);
z__2.r = d21.r * t1.r - d21.i * t1.i, z__2.i = d21.r *
t1.i + d21.i * t1.r;
z__3.r = d22.r * t2.r - d22.i * t2.i, z__3.i = d22.r *
t2.i + d22.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
/* L80: */
}
}
k += -2;
}
goto L70;
L90:
/* Form B := L'*B
where L = P(1)*inv(L(1))* ... *P(m)*inv(L(m))
and L' = inv(L'(m))*P(m)* ... *inv(L'(1))*P(1) */
;
} else {
/* Loop forward applying the L-transformations. */
k = 1;
L100:
if (k > *n) {
goto L120;
}
/* 1 x 1 pivot block */
if (ipiv[k] > 0) {
if (k < *n) {
/* Interchange if P(K) != I. */
kp = ipiv[k];
if (kp != k) {
zswap_(nrhs, &b_ref(k, 1), ldb, &b_ref(kp, 1), ldb);
}
/* Apply the transformation */
i__1 = *n - k;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b_ref(k + 1, 1),
ldb, &a_ref(k + 1, k), &c__1, &c_b1, &b_ref(k, 1),
ldb);
}
if (nounit) {
zscal_(nrhs, &a_ref(k, k), &b_ref(k, 1), ldb);
}
++k;
/* 2 x 2 pivot block. */
} else {
if (k < *n - 1) {
/* Interchange if P(K) != I. */
kp = (i__1 = ipiv[k], abs(i__1));
if (kp != k + 1) {
zswap_(nrhs, &b_ref(k + 1, 1), ldb, &b_ref(kp, 1),
ldb);
}
/* Apply the transformation */
i__1 = *n - k - 1;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b_ref(k + 2, 1),
ldb, &a_ref(k + 2, k + 1), &c__1, &c_b1, &b_ref(k
+ 1, 1), ldb);
i__1 = *n - k - 1;
zgemv_("Transpose", &i__1, nrhs, &c_b1, &b_ref(k + 2, 1),
ldb, &a_ref(k + 2, k), &c__1, &c_b1, &b_ref(k, 1),
ldb);
}
/* Multiply by the diagonal block if non-unit. */
if (nounit) {
i__1 = a_subscr(k, k);
d11.r = a[i__1].r, d11.i = a[i__1].i;
i__1 = a_subscr(k + 1, k + 1);
d22.r = a[i__1].r, d22.i = a[i__1].i;
i__1 = a_subscr(k + 1, k);
d21.r = a[i__1].r, d21.i = a[i__1].i;
d12.r = d21.r, d12.i = d21.i;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
i__2 = b_subscr(k, j);
t1.r = b[i__2].r, t1.i = b[i__2].i;
i__2 = b_subscr(k + 1, j);
t2.r = b[i__2].r, t2.i = b[i__2].i;
i__2 = b_subscr(k, j);
z__2.r = d11.r * t1.r - d11.i * t1.i, z__2.i = d11.r *
t1.i + d11.i * t1.r;
z__3.r = d12.r * t2.r - d12.i * t2.i, z__3.i = d12.r *
t2.i + d12.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
i__2 = b_subscr(k + 1, j);
z__2.r = d21.r * t1.r - d21.i * t1.i, z__2.i = d21.r *
t1.i + d21.i * t1.r;
z__3.r = d22.r * t2.r - d22.i * t2.i, z__3.i = d22.r *
t2.i + d22.i * t2.r;
z__1.r = z__2.r + z__3.r, z__1.i = z__2.i + z__3.i;
b[i__2].r = z__1.r, b[i__2].i = z__1.i;
/* L110: */
}
}
k += 2;
}
goto L100;
L120:
;
}
}
return 0;
/* End of ZLAVSY */
} /* zlavsy_ */
/* Subroutine */ int zgebrd_(integer *m, integer *n, doublecomplex *a,
integer *lda, doublereal *d__, doublereal *e, doublecomplex *tauq,
doublecomplex *taup, doublecomplex *work, integer *lwork, integer *
info)
{
/* -- 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
Purpose
=======
ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
bidiagonal form B by a unitary transformation: Q**H * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Arguments
=========
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the unitary matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the unitary matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (output) DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ (output) COMPLEX*16 array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
TAUP (output) COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,M,N).
For optimum performance LWORK >= (M+N)*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 had an illegal value.
Further Details
===============
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static doublecomplex c_b1 = {1.,0.};
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, i__5;
doublereal d__1;
doublecomplex z__1;
/* Local variables */
static integer i__, j, nbmin, iinfo, minmn;
extern /* Subroutine */ int zgemm_(char *, char *, integer *, integer *,
integer *, doublecomplex *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *), zgebd2_(integer *, integer *,
doublecomplex *, integer *, doublereal *, doublereal *,
doublecomplex *, doublecomplex *, doublecomplex *, integer *);
static integer nb, nx;
static doublereal ws;
extern /* Subroutine */ int xerbla_(char *, integer *), zlabrd_(
integer *, integer *, integer *, doublecomplex *, integer *,
doublereal *, doublereal *, doublecomplex *, doublecomplex *,
doublecomplex *, integer *, doublecomplex *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ldwrkx, ldwrky, lwkopt;
static logical lquery;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--d__;
--e;
--tauq;
--taup;
--work;
/* Function Body */
*info = 0;
/* Computing MAX */
i__1 = 1, i__2 = ilaenv_(&c__1, "ZGEBRD", " ", m, n, &c_n1, &c_n1, (
ftnlen)6, (ftnlen)1);
nb = max(i__1,i__2);
lwkopt = (*m + *n) * nb;
d__1 = (doublereal) lwkopt;
work[1].r = d__1, work[1].i = 0.;
lquery = *lwork == -1;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
} else /* if(complicated condition) */ {
/* Computing MAX */
i__1 = max(1,*m);
if (*lwork < max(i__1,*n) && ! lquery) {
*info = -10;
}
}
if (*info < 0) {
i__1 = -(*info);
xerbla_("ZGEBRD", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
minmn = min(*m,*n);
if (minmn == 0) {
work[1].r = 1., work[1].i = 0.;
return 0;
}
ws = (doublereal) max(*m,*n);
ldwrkx = *m;
ldwrky = *n;
if (nb > 1 && nb < minmn) {
/* Set the crossover point NX.
Computing MAX */
i__1 = nb, i__2 = ilaenv_(&c__3, "ZGEBRD", " ", m, n, &c_n1, &c_n1, (
ftnlen)6, (ftnlen)1);
nx = max(i__1,i__2);
/* Determine when to switch from blocked to unblocked code. */
if (nx < minmn) {
ws = (doublereal) ((*m + *n) * nb);
if ((doublereal) (*lwork) < ws) {
/* Not enough work space for the optimal NB, consider using
a smaller block size. */
nbmin = ilaenv_(&c__2, "ZGEBRD", " ", m, n, &c_n1, &c_n1, (
ftnlen)6, (ftnlen)1);
if (*lwork >= (*m + *n) * nbmin) {
nb = *lwork / (*m + *n);
} else {
nb = 1;
nx = minmn;
}
}
}
} else {
nx = minmn;
}
i__1 = minmn - nx;
i__2 = nb;
for (i__ = 1; i__2 < 0 ? i__ >= i__1 : i__ <= i__1; i__ += i__2) {
/* Reduce rows and columns i:i+ib-1 to bidiagonal form and return
the matrices X and Y which are needed to update the unreduced
part of the matrix */
i__3 = *m - i__ + 1;
i__4 = *n - i__ + 1;
zlabrd_(&i__3, &i__4, &nb, &a_ref(i__, i__), lda, &d__[i__], &e[i__],
&tauq[i__], &taup[i__], &work[1], &ldwrkx, &work[ldwrkx * nb
+ 1], &ldwrky);
/* Update the trailing submatrix A(i+ib:m,i+ib:n), using
an update of the form A := A - V*Y' - X*U' */
i__3 = *m - i__ - nb + 1;
i__4 = *n - i__ - nb + 1;
z__1.r = -1., z__1.i = 0.;
zgemm_("No transpose", "Conjugate transpose", &i__3, &i__4, &nb, &
z__1, &a_ref(i__ + nb, i__), lda, &work[ldwrkx * nb + nb + 1],
&ldwrky, &c_b1, &a_ref(i__ + nb, i__ + nb), lda);
i__3 = *m - i__ - nb + 1;
i__4 = *n - i__ - nb + 1;
z__1.r = -1., z__1.i = 0.;
zgemm_("No transpose", "No transpose", &i__3, &i__4, &nb, &z__1, &
work[nb + 1], &ldwrkx, &a_ref(i__, i__ + nb), lda, &c_b1, &
a_ref(i__ + nb, i__ + nb), lda);
/* Copy diagonal and off-diagonal elements of B back into A */
if (*m >= *n) {
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
i__4 = a_subscr(j, j);
i__5 = j;
a[i__4].r = d__[i__5], a[i__4].i = 0.;
i__4 = a_subscr(j, j + 1);
i__5 = j;
a[i__4].r = e[i__5], a[i__4].i = 0.;
/* L10: */
}
} else {
i__3 = i__ + nb - 1;
for (j = i__; j <= i__3; ++j) {
i__4 = a_subscr(j, j);
i__5 = j;
a[i__4].r = d__[i__5], a[i__4].i = 0.;
i__4 = a_subscr(j + 1, j);
i__5 = j;
a[i__4].r = e[i__5], a[i__4].i = 0.;
/* L20: */
}
}
/* L30: */
}
/* Use unblocked code to reduce the remainder of the matrix */
i__2 = *m - i__ + 1;
i__1 = *n - i__ + 1;
zgebd2_(&i__2, &i__1, &a_ref(i__, i__), lda, &d__[i__], &e[i__], &tauq[
i__], &taup[i__], &work[1], &iinfo);
work[1].r = ws, work[1].i = 0.;
return 0;
/* End of ZGEBRD */
} /* zgebrd_ */
/* Subroutine */ int cgebd2_(integer *m, integer *n, complex *a, integer *lda,
real *d__, real *e, complex *tauq, complex *taup, complex *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
=======
CGEBD2 reduces a complex general m by n matrix A to upper or lower
real bidiagonal form B by a unitary transformation: Q' * A * P = B.
If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
Arguments
=========
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit,
if m >= n, the diagonal and the first superdiagonal are
overwritten with the upper bidiagonal matrix B; the
elements below the diagonal, with the array TAUQ, represent
the unitary matrix Q as a product of elementary
reflectors, and the elements above the first superdiagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors;
if m < n, the diagonal and the first subdiagonal are
overwritten with the lower bidiagonal matrix B; the
elements below the first subdiagonal, with the array TAUQ,
represent the unitary matrix Q as a product of
elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).
E (output) REAL array, dimension (min(M,N)-1)
The off-diagonal elements of the bidiagonal matrix B:
if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
TAUQ (output) COMPLEX array dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
TAUP (output) COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
WORK (workspace) COMPLEX array, dimension (max(M,N))
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
The matrices Q and P are represented as products of elementary
reflectors:
If m >= n,
Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, and v and u are complex
vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n,
Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v' and G(i) = I - taup * u * u'
where tauq and taup are complex scalars, v and u are complex vectors;
v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
tauq is stored in TAUQ(i) and taup in TAUP(i).
The contents of A on exit are illustrated by the following examples:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
where d and e denote diagonal and off-diagonal elements of B, vi
denotes an element of the vector defining H(i), and ui an element of
the vector defining G(i).
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
complex q__1;
/* Builtin functions */
void r_cnjg(complex *, complex *);
/* Local variables */
static integer i__;
static complex alpha;
extern /* Subroutine */ int clarf_(char *, integer *, integer *, complex *
, integer *, complex *, complex *, integer *, complex *),
clarfg_(integer *, complex *, complex *, integer *, complex *),
clacgv_(integer *, complex *, integer *), xerbla_(char *, integer
*);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--d__;
--e;
--tauq;
--taup;
--work;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info < 0) {
i__1 = -(*info);
xerbla_("CGEBD2", &i__1);
return 0;
}
if (*m >= *n) {
/* Reduce to upper bidiagonal form */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Generate elementary reflector H(i) to annihilate A(i+1:m,i) */
i__2 = a_subscr(i__, i__);
alpha.r = a[i__2].r, alpha.i = a[i__2].i;
/* Computing MIN */
i__2 = i__ + 1;
i__3 = *m - i__ + 1;
clarfg_(&i__3, &alpha, &a_ref(min(i__2,*m), i__), &c__1, &tauq[
i__]);
i__2 = i__;
d__[i__2] = alpha.r;
i__2 = a_subscr(i__, i__);
a[i__2].r = 1.f, a[i__2].i = 0.f;
/* Apply H(i)' to A(i:m,i+1:n) from the left */
i__2 = *m - i__ + 1;
i__3 = *n - i__;
r_cnjg(&q__1, &tauq[i__]);
clarf_("Left", &i__2, &i__3, &a_ref(i__, i__), &c__1, &q__1, &
a_ref(i__, i__ + 1), lda, &work[1]);
i__2 = a_subscr(i__, i__);
i__3 = i__;
a[i__2].r = d__[i__3], a[i__2].i = 0.f;
if (i__ < *n) {
/* Generate elementary reflector G(i) to annihilate
A(i,i+2:n) */
i__2 = *n - i__;
clacgv_(&i__2, &a_ref(i__, i__ + 1), lda);
i__2 = a_subscr(i__, i__ + 1);
alpha.r = a[i__2].r, alpha.i = a[i__2].i;
/* Computing MIN */
i__2 = i__ + 2;
i__3 = *n - i__;
clarfg_(&i__3, &alpha, &a_ref(i__, min(i__2,*n)), lda, &taup[
i__]);
i__2 = i__;
e[i__2] = alpha.r;
i__2 = a_subscr(i__, i__ + 1);
a[i__2].r = 1.f, a[i__2].i = 0.f;
/* Apply G(i) to A(i+1:m,i+1:n) from the right */
i__2 = *m - i__;
i__3 = *n - i__;
clarf_("Right", &i__2, &i__3, &a_ref(i__, i__ + 1), lda, &
taup[i__], &a_ref(i__ + 1, i__ + 1), lda, &work[1]);
i__2 = *n - i__;
clacgv_(&i__2, &a_ref(i__, i__ + 1), lda);
i__2 = a_subscr(i__, i__ + 1);
i__3 = i__;
a[i__2].r = e[i__3], a[i__2].i = 0.f;
} else {
i__2 = i__;
taup[i__2].r = 0.f, taup[i__2].i = 0.f;
}
/* L10: */
}
} else {
/* Reduce to lower bidiagonal form */
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Generate elementary reflector G(i) to annihilate A(i,i+1:n) */
i__2 = *n - i__ + 1;
clacgv_(&i__2, &a_ref(i__, i__), lda);
i__2 = a_subscr(i__, i__);
alpha.r = a[i__2].r, alpha.i = a[i__2].i;
/* Computing MIN */
i__2 = i__ + 1;
i__3 = *n - i__ + 1;
clarfg_(&i__3, &alpha, &a_ref(i__, min(i__2,*n)), lda, &taup[i__])
;
i__2 = i__;
d__[i__2] = alpha.r;
i__2 = a_subscr(i__, i__);
a[i__2].r = 1.f, a[i__2].i = 0.f;
/* Apply G(i) to A(i+1:m,i:n) from the right
Computing MIN */
i__2 = i__ + 1;
i__3 = *m - i__;
i__4 = *n - i__ + 1;
clarf_("Right", &i__3, &i__4, &a_ref(i__, i__), lda, &taup[i__], &
a_ref(min(i__2,*m), i__), lda, &work[1]);
i__2 = *n - i__ + 1;
clacgv_(&i__2, &a_ref(i__, i__), lda);
i__2 = a_subscr(i__, i__);
i__3 = i__;
a[i__2].r = d__[i__3], a[i__2].i = 0.f;
if (i__ < *m) {
/* Generate elementary reflector H(i) to annihilate
A(i+2:m,i) */
i__2 = a_subscr(i__ + 1, i__);
alpha.r = a[i__2].r, alpha.i = a[i__2].i;
/* Computing MIN */
i__2 = i__ + 2;
i__3 = *m - i__;
clarfg_(&i__3, &alpha, &a_ref(min(i__2,*m), i__), &c__1, &
tauq[i__]);
i__2 = i__;
e[i__2] = alpha.r;
i__2 = a_subscr(i__ + 1, i__);
a[i__2].r = 1.f, a[i__2].i = 0.f;
/* Apply H(i)' to A(i+1:m,i+1:n) from the left */
i__2 = *m - i__;
i__3 = *n - i__;
r_cnjg(&q__1, &tauq[i__]);
clarf_("Left", &i__2, &i__3, &a_ref(i__ + 1, i__), &c__1, &
q__1, &a_ref(i__ + 1, i__ + 1), lda, &work[1]);
i__2 = a_subscr(i__ + 1, i__);
i__3 = i__;
a[i__2].r = e[i__3], a[i__2].i = 0.f;
} else {
i__2 = i__;
tauq[i__2].r = 0.f, tauq[i__2].i = 0.f;
}
/* L20: */
}
}
return 0;
/* End of CGEBD2 */
} /* cgebd2_ */
/* Subroutine */ int zlauu2_(char *uplo, integer *n, doublecomplex *a,
integer *lda, 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
September 30, 1994
Purpose
=======
ZLAUU2 computes the product U * U' or L' * L, where the triangular
factor U or L is stored in the upper or lower triangular part of
the array A.
If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
overwriting the factor U in A.
If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
overwriting the factor L in A.
This is the unblocked form of the algorithm, calling Level 2 BLAS.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the triangular factor stored in the array A
is upper or lower triangular:
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the triangular factor U or L. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the triangular factor U or L.
On exit, if UPLO = 'U', the upper triangle of A is
overwritten with the upper triangle of the product U * U';
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L' * L.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static doublecomplex c_b1 = {1.,0.};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublereal d__1;
doublecomplex z__1;
/* Local variables */
static integer i__;
extern logical lsame_(char *, char *);
extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
extern /* Subroutine */ int zgemv_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, integer *, doublecomplex *,
integer *, doublecomplex *, doublecomplex *, integer *);
static logical upper;
extern /* Subroutine */ int xerbla_(char *, integer *), zdscal_(
integer *, doublereal *, doublecomplex *, integer *), zlacgv_(
integer *, doublecomplex *, integer *);
static doublereal aii;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZLAUU2", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
if (upper) {
/* Compute the product U * U'. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = a_subscr(i__, i__);
aii = a[i__2].r;
if (i__ < *n) {
i__2 = a_subscr(i__, i__);
i__3 = *n - i__;
zdotc_(&z__1, &i__3, &a_ref(i__, i__ + 1), lda, &a_ref(i__,
i__ + 1), lda);
d__1 = aii * aii + z__1.r;
a[i__2].r = d__1, a[i__2].i = 0.;
i__2 = *n - i__;
zlacgv_(&i__2, &a_ref(i__, i__ + 1), lda);
i__2 = i__ - 1;
i__3 = *n - i__;
z__1.r = aii, z__1.i = 0.;
zgemv_("No transpose", &i__2, &i__3, &c_b1, &a_ref(1, i__ + 1)
, lda, &a_ref(i__, i__ + 1), lda, &z__1, &a_ref(1,
i__), &c__1);
i__2 = *n - i__;
zlacgv_(&i__2, &a_ref(i__, i__ + 1), lda);
} else {
zdscal_(&i__, &aii, &a_ref(1, i__), &c__1);
}
/* L10: */
}
} else {
/* Compute the product L' * L. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = a_subscr(i__, i__);
aii = a[i__2].r;
if (i__ < *n) {
i__2 = a_subscr(i__, i__);
i__3 = *n - i__;
zdotc_(&z__1, &i__3, &a_ref(i__ + 1, i__), &c__1, &a_ref(i__
+ 1, i__), &c__1);
d__1 = aii * aii + z__1.r;
a[i__2].r = d__1, a[i__2].i = 0.;
i__2 = i__ - 1;
zlacgv_(&i__2, &a_ref(i__, 1), lda);
i__2 = *n - i__;
i__3 = i__ - 1;
z__1.r = aii, z__1.i = 0.;
zgemv_("Conjugate transpose", &i__2, &i__3, &c_b1, &a_ref(i__
+ 1, 1), lda, &a_ref(i__ + 1, i__), &c__1, &z__1, &
a_ref(i__, 1), lda);
i__2 = i__ - 1;
zlacgv_(&i__2, &a_ref(i__, 1), lda);
} else {
zdscal_(&i__, &aii, &a_ref(i__, 1), lda);
}
/* L20: */
}
}
return 0;
/* End of ZLAUU2 */
} /* zlauu2_ */
/* Subroutine */ int zgebal_(char *job, integer *n, doublecomplex *a, integer
*lda, integer *ilo, integer *ihi, doublereal *scale, integer *info)
{
/* -- 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
Purpose
=======
ZGEBAL balances a general complex matrix A. This involves, first,
permuting A by a similarity transformation to isolate eigenvalues
in the first 1 to ILO-1 and last IHI+1 to N elements on the
diagonal; and second, applying a diagonal similarity transformation
to rows and columns ILO to IHI to make the rows and columns as
close in norm as possible. Both steps are optional.
Balancing may reduce the 1-norm of the matrix, and improve the
accuracy of the computed eigenvalues and/or eigenvectors.
Arguments
=========
JOB (input) CHARACTER*1
Specifies the operations to be performed on A:
= 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
for i = 1,...,N;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If JOB = 'N', A is not referenced.
See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
ILO (output) INTEGER
IHI (output) INTEGER
ILO and IHI are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
If JOB = 'N' or 'S', ILO = 1 and IHI = N.
SCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to
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.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
The permutations consist of row and column interchanges which put
the matrix in the form
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ILO and IHI mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
The output matrix is
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
Information about the permutations P and the diagonal matrix D is
returned in the vector SCALE.
This subroutine is based on the EISPACK routine CBAL.
Modified by Tzu-Yi Chen, Computer Science Division, University of
California at Berkeley, USA
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublereal d__1, d__2;
/* Builtin functions */
double d_imag(doublecomplex *), z_abs(doublecomplex *);
/* Local variables */
static integer iexc;
static doublereal c__, f, g;
static integer i__, j, k, l, m;
static doublereal r__, s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int zswap_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *);
static doublereal sfmin1, sfmin2, sfmax1, sfmax2, ca, ra;
extern doublereal dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *), zdscal_(
integer *, doublereal *, doublecomplex *, integer *);
extern integer izamax_(integer *, doublecomplex *, integer *);
static logical noconv;
static integer ica, ira;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--scale;
/* Function Body */
*info = 0;
if (! lsame_(job, "N") && ! lsame_(job, "P") && ! lsame_(job, "S")
&& ! lsame_(job, "B")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*n)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGEBAL", &i__1);
return 0;
}
k = 1;
l = *n;
if (*n == 0) {
goto L210;
}
if (lsame_(job, "N")) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
scale[i__] = 1.;
/* L10: */
}
goto L210;
}
if (lsame_(job, "S")) {
goto L120;
}
/* Permutation to isolate eigenvalues if possible */
goto L50;
/* Row and column exchange. */
L20:
scale[m] = (doublereal) j;
if (j == m) {
goto L30;
}
zswap_(&l, &a_ref(1, j), &c__1, &a_ref(1, m), &c__1);
i__1 = *n - k + 1;
zswap_(&i__1, &a_ref(j, k), lda, &a_ref(m, k), lda);
L30:
switch (iexc) {
case 1: goto L40;
case 2: goto L80;
}
/* Search for rows isolating an eigenvalue and push them down. */
L40:
if (l == 1) {
goto L210;
}
--l;
L50:
for (j = l; j >= 1; --j) {
i__1 = l;
for (i__ = 1; i__ <= i__1; ++i__) {
if (i__ == j) {
goto L60;
}
i__2 = a_subscr(j, i__);
if (a[i__2].r != 0. || d_imag(&a_ref(j, i__)) != 0.) {
goto L70;
}
L60:
;
}
m = l;
iexc = 1;
goto L20;
L70:
;
}
goto L90;
/* Search for columns isolating an eigenvalue and push them left. */
L80:
++k;
L90:
i__1 = l;
for (j = k; j <= i__1; ++j) {
i__2 = l;
for (i__ = k; i__ <= i__2; ++i__) {
if (i__ == j) {
goto L100;
}
i__3 = a_subscr(i__, j);
if (a[i__3].r != 0. || d_imag(&a_ref(i__, j)) != 0.) {
goto L110;
}
L100:
;
}
m = k;
iexc = 2;
goto L20;
L110:
;
}
L120:
i__1 = l;
for (i__ = k; i__ <= i__1; ++i__) {
scale[i__] = 1.;
/* L130: */
}
if (lsame_(job, "P")) {
goto L210;
}
/* Balance the submatrix in rows K to L.
Iterative loop for norm reduction */
sfmin1 = dlamch_("S") / dlamch_("P");
sfmax1 = 1. / sfmin1;
sfmin2 = sfmin1 * 8.;
sfmax2 = 1. / sfmin2;
L140:
noconv = FALSE_;
i__1 = l;
for (i__ = k; i__ <= i__1; ++i__) {
c__ = 0.;
r__ = 0.;
i__2 = l;
for (j = k; j <= i__2; ++j) {
if (j == i__) {
goto L150;
}
i__3 = a_subscr(j, i__);
c__ += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a_ref(j,
i__)), abs(d__2));
i__3 = a_subscr(i__, j);
r__ += (d__1 = a[i__3].r, abs(d__1)) + (d__2 = d_imag(&a_ref(i__,
j)), abs(d__2));
L150:
;
}
ica = izamax_(&l, &a_ref(1, i__), &c__1);
ca = z_abs(&a_ref(ica, i__));
i__2 = *n - k + 1;
ira = izamax_(&i__2, &a_ref(i__, k), lda);
ra = z_abs(&a_ref(i__, ira + k - 1));
/* Guard against zero C or R due to underflow. */
if (c__ == 0. || r__ == 0.) {
goto L200;
}
g = r__ / 8.;
f = 1.;
s = c__ + r__;
L160:
/* Computing MAX */
d__1 = max(f,c__);
/* Computing MIN */
d__2 = min(r__,g);
if (c__ >= g || max(d__1,ca) >= sfmax2 || min(d__2,ra) <= sfmin2) {
goto L170;
}
f *= 8.;
c__ *= 8.;
ca *= 8.;
r__ /= 8.;
g /= 8.;
ra /= 8.;
goto L160;
L170:
g = c__ / 8.;
L180:
/* Computing MIN */
d__1 = min(f,c__), d__1 = min(d__1,g);
if (g < r__ || max(r__,ra) >= sfmax2 || min(d__1,ca) <= sfmin2) {
goto L190;
}
f /= 8.;
c__ /= 8.;
g /= 8.;
ca /= 8.;
r__ *= 8.;
ra *= 8.;
goto L180;
/* Now balance. */
L190:
if (c__ + r__ >= s * .95) {
goto L200;
}
if (f < 1. && scale[i__] < 1.) {
if (f * scale[i__] <= sfmin1) {
goto L200;
}
}
if (f > 1. && scale[i__] > 1.) {
if (scale[i__] >= sfmax1 / f) {
goto L200;
}
}
g = 1. / f;
scale[i__] *= f;
noconv = TRUE_;
i__2 = *n - k + 1;
zdscal_(&i__2, &g, &a_ref(i__, k), lda);
zdscal_(&l, &f, &a_ref(1, i__), &c__1);
L200:
;
}
if (noconv) {
goto L140;
}
L210:
*ilo = k;
*ihi = l;
return 0;
/* End of ZGEBAL */
} /* zgebal_ */
/* Subroutine */ int claqhe_(char *uplo, integer *n, complex *a, integer *lda,
real *s, real *scond, real *amax, char *equed)
{
/* -- 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
=======
CLAQHE equilibrates a Hermitian matrix A using the scaling factors
in the vector S.
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n by n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n by n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if EQUED = 'Y', the equilibrated matrix:
diag(S) * A * diag(S).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(N,1).
S (input) REAL array, dimension (N)
The scale factors for A.
SCOND (input) REAL
Ratio of the smallest S(i) to the largest S(i).
AMAX (input) REAL
Absolute value of largest matrix entry.
EQUED (output) CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters
===================
THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
=====================================================================
Quick return if possible
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4;
real r__1;
complex q__1;
/* Local variables */
static integer i__, j;
static real large;
extern logical lsame_(char *, char *);
static real small, cj;
extern doublereal slamch_(char *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--s;
/* Function Body */
if (*n <= 0) {
*(unsigned char *)equed = 'N';
return 0;
}
/* Initialize LARGE and SMALL. */
small = slamch_("Safe minimum") / slamch_("Precision");
large = 1.f / small;
if (*scond >= .1f && *amax >= small && *amax <= large) {
/* No equilibration */
*(unsigned char *)equed = 'N';
} else {
/* Replace A by diag(S) * A * diag(S). */
if (lsame_(uplo, "U")) {
/* Upper triangle of A is stored. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
cj = s[j];
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
r__1 = cj * s[i__];
i__4 = a_subscr(i__, j);
q__1.r = r__1 * a[i__4].r, q__1.i = r__1 * a[i__4].i;
a[i__3].r = q__1.r, a[i__3].i = q__1.i;
/* L10: */
}
i__2 = a_subscr(j, j);
i__3 = a_subscr(j, j);
r__1 = cj * cj * a[i__3].r;
a[i__2].r = r__1, a[i__2].i = 0.f;
/* L20: */
}
} else {
/* Lower triangle of A is stored. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
cj = s[j];
i__2 = a_subscr(j, j);
i__3 = a_subscr(j, j);
r__1 = cj * cj * a[i__3].r;
a[i__2].r = r__1, a[i__2].i = 0.f;
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
r__1 = cj * s[i__];
i__4 = a_subscr(i__, j);
q__1.r = r__1 * a[i__4].r, q__1.i = r__1 * a[i__4].i;
a[i__3].r = q__1.r, a[i__3].i = q__1.i;
/* L30: */
}
/* L40: */
}
}
*(unsigned char *)equed = 'Y';
}
return 0;
/* End of CLAQHE */
} /* claqhe_ */
/* Subroutine */ int zlahef_(char *uplo, integer *n, integer *nb, integer *kb,
doublecomplex *a, integer *lda, integer *ipiv, doublecomplex *w,
integer *ldw, 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
=======
ZLAHEF computes a partial factorization of a complex Hermitian
matrix A using the Bunch-Kaufman diagonal pivoting method. The
partial factorization has the form:
A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
( 0 U22 ) ( 0 D ) ( U12' U22' )
A = ( L11 0 ) ( D 0 ) ( L11' L21' ) if UPLO = 'L'
( L21 I ) ( 0 A22 ) ( 0 I )
where the order of D is at most NB. The actual order is returned in
the argument KB, and is either NB or NB-1, or N if N <= NB.
Note that U' denotes the conjugate transpose of U.
ZLAHEF is an auxiliary routine called by ZHETRF. It uses blocked code
(calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
A22 (if UPLO = 'L').
Arguments
=========
UPLO (input) CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
NB (input) INTEGER
The maximum number of columns of the matrix A that should be
factored. NB should be at least 2 to allow for 2-by-2 pivot
blocks.
KB (output) INTEGER
The number of columns of A that were actually factored.
KB is either NB-1 or NB, or N if N <= NB.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, A contains details of the partial factorization.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
IPIV (output) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D.
If UPLO = 'U', only the last KB elements of IPIV are set;
if UPLO = 'L', only the first KB elements are set.
If IPIV(k) > 0, then rows and columns k and IPIV(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
W (workspace) COMPLEX*16 array, dimension (LDW,NB)
LDW (input) INTEGER
The leading dimension of the array W. LDW >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
> 0: if INFO = k, D(k,k) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular.
=====================================================================
Parameter adjustments */
/* Table of constant values */
static doublecomplex c_b1 = {1.,0.};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, w_dim1, w_offset, i__1, i__2, i__3, i__4, i__5;
doublereal d__1, d__2, d__3, d__4;
doublecomplex z__1, z__2, z__3, z__4;
/* Builtin functions */
double sqrt(doublereal), d_imag(doublecomplex *);
void d_cnjg(doublecomplex *, doublecomplex *), z_div(doublecomplex *,
doublecomplex *, doublecomplex *);
/* Local variables */
static integer imax, jmax, j, k;
static doublereal t, alpha;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int zgemm_(char *, char *, integer *, integer *,
integer *, doublecomplex *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *);
static integer kstep;
extern /* Subroutine */ int zgemv_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, integer *, doublecomplex *,
integer *, doublecomplex *, doublecomplex *, integer *);
static doublereal r1;
extern /* Subroutine */ int zcopy_(integer *, doublecomplex *, integer *,
doublecomplex *, integer *), zswap_(integer *, doublecomplex *,
integer *, doublecomplex *, integer *);
static doublecomplex d11, d21, d22;
static integer jb, jj, kk, jp, kp;
static doublereal absakk;
static integer kw;
extern /* Subroutine */ int zdscal_(integer *, doublereal *,
doublecomplex *, integer *);
static doublereal colmax;
extern /* Subroutine */ int zlacgv_(integer *, doublecomplex *, integer *)
;
extern integer izamax_(integer *, doublecomplex *, integer *);
static doublereal rowmax;
static integer kkw;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define w_subscr(a_1,a_2) (a_2)*w_dim1 + a_1
#define w_ref(a_1,a_2) w[w_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipiv;
w_dim1 = *ldw;
w_offset = 1 + w_dim1 * 1;
w -= w_offset;
/* Function Body */
*info = 0;
/* Initialize ALPHA for use in choosing pivot block size. */
alpha = (sqrt(17.) + 1.) / 8.;
if (lsame_(uplo, "U")) {
/* Factorize the trailing columns of A using the upper triangle
of A and working backwards, and compute the matrix W = U12*D
for use in updating A11 (note that conjg(W) is actually stored)
K is the main loop index, decreasing from N in steps of 1 or 2
KW is the column of W which corresponds to column K of A */
k = *n;
L10:
kw = *nb + k - *n;
/* Exit from loop */
if (k <= *n - *nb + 1 && *nb < *n || k < 1) {
goto L30;
}
/* Copy column K of A to column KW of W and update it */
i__1 = k - 1;
zcopy_(&i__1, &a_ref(1, k), &c__1, &w_ref(1, kw), &c__1);
i__1 = w_subscr(k, kw);
i__2 = a_subscr(k, k);
d__1 = a[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
if (k < *n) {
i__1 = *n - k;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &k, &i__1, &z__1, &a_ref(1, k + 1), lda, &
w_ref(k, kw + 1), ldw, &c_b1, &w_ref(1, kw), &c__1);
i__1 = w_subscr(k, kw);
i__2 = w_subscr(k, kw);
d__1 = w[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
}
kstep = 1;
/* Determine rows and columns to be interchanged and whether
a 1-by-1 or 2-by-2 pivot block will be used */
i__1 = w_subscr(k, kw);
absakk = (d__1 = w[i__1].r, abs(d__1));
/* IMAX is the row-index of the largest off-diagonal element in
column K, and COLMAX is its absolute value */
if (k > 1) {
i__1 = k - 1;
imax = izamax_(&i__1, &w_ref(1, kw), &c__1);
i__1 = w_subscr(imax, kw);
colmax = (d__1 = w[i__1].r, abs(d__1)) + (d__2 = d_imag(&w_ref(
imax, kw)), abs(d__2));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0.) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
d__1 = a[i__2].r;
a[i__1].r = d__1, a[i__1].i = 0.;
} else {
if (absakk >= alpha * colmax) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else {
/* Copy column IMAX to column KW-1 of W and update it */
i__1 = imax - 1;
zcopy_(&i__1, &a_ref(1, imax), &c__1, &w_ref(1, kw - 1), &
c__1);
i__1 = w_subscr(imax, kw - 1);
i__2 = a_subscr(imax, imax);
d__1 = a[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
i__1 = k - imax;
zcopy_(&i__1, &a_ref(imax, imax + 1), lda, &w_ref(imax + 1,
kw - 1), &c__1);
i__1 = k - imax;
zlacgv_(&i__1, &w_ref(imax + 1, kw - 1), &c__1);
if (k < *n) {
i__1 = *n - k;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &k, &i__1, &z__1, &a_ref(1, k + 1),
lda, &w_ref(imax, kw + 1), ldw, &c_b1, &w_ref(1,
kw - 1), &c__1);
i__1 = w_subscr(imax, kw - 1);
i__2 = w_subscr(imax, kw - 1);
d__1 = w[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
}
/* JMAX is the column-index of the largest off-diagonal
element in row IMAX, and ROWMAX is its absolute value */
i__1 = k - imax;
jmax = imax + izamax_(&i__1, &w_ref(imax + 1, kw - 1), &c__1);
i__1 = w_subscr(jmax, kw - 1);
rowmax = (d__1 = w[i__1].r, abs(d__1)) + (d__2 = d_imag(&
w_ref(jmax, kw - 1)), abs(d__2));
if (imax > 1) {
i__1 = imax - 1;
jmax = izamax_(&i__1, &w_ref(1, kw - 1), &c__1);
/* Computing MAX */
i__1 = w_subscr(jmax, kw - 1);
d__3 = rowmax, d__4 = (d__1 = w[i__1].r, abs(d__1)) + (
d__2 = d_imag(&w_ref(jmax, kw - 1)), abs(d__2));
rowmax = max(d__3,d__4);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else /* if(complicated condition) */ {
i__1 = w_subscr(imax, kw - 1);
if ((d__1 = w[i__1].r, abs(d__1)) >= alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
/* copy column KW-1 of W to column KW */
zcopy_(&k, &w_ref(1, kw - 1), &c__1, &w_ref(1, kw), &
c__1);
} else {
/* interchange rows and columns K-1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
}
kk = k - kstep + 1;
kkw = *nb + kk - *n;
/* Updated column KP is already stored in column KKW of W */
if (kp != kk) {
/* Copy non-updated column KK to column KP */
i__1 = a_subscr(kp, kp);
i__2 = a_subscr(kk, kk);
d__1 = a[i__2].r;
a[i__1].r = d__1, a[i__1].i = 0.;
i__1 = kk - 1 - kp;
zcopy_(&i__1, &a_ref(kp + 1, kk), &c__1, &a_ref(kp, kp + 1),
lda);
i__1 = kk - 1 - kp;
zlacgv_(&i__1, &a_ref(kp, kp + 1), lda);
i__1 = kp - 1;
zcopy_(&i__1, &a_ref(1, kk), &c__1, &a_ref(1, kp), &c__1);
/* Interchange rows KK and KP in last KK columns of A and W */
if (kk < *n) {
i__1 = *n - kk;
zswap_(&i__1, &a_ref(kk, kk + 1), lda, &a_ref(kp, kk + 1),
lda);
}
i__1 = *n - kk + 1;
zswap_(&i__1, &w_ref(kk, kkw), ldw, &w_ref(kp, kkw), ldw);
}
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column KW of W now holds
W(k) = U(k)*D(k)
where U(k) is the k-th column of U
Store U(k) in column k of A */
zcopy_(&k, &w_ref(1, kw), &c__1, &a_ref(1, k), &c__1);
i__1 = a_subscr(k, k);
r1 = 1. / a[i__1].r;
i__1 = k - 1;
zdscal_(&i__1, &r1, &a_ref(1, k), &c__1);
/* Conjugate W(k) */
i__1 = k - 1;
zlacgv_(&i__1, &w_ref(1, kw), &c__1);
} else {
/* 2-by-2 pivot block D(k): columns KW and KW-1 of W now
hold
( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
where U(k) and U(k-1) are the k-th and (k-1)-th columns
of U */
if (k > 2) {
/* Store U(k) and U(k-1) in columns k and k-1 of A */
i__1 = w_subscr(k - 1, kw);
d21.r = w[i__1].r, d21.i = w[i__1].i;
d_cnjg(&z__2, &d21);
z_div(&z__1, &w_ref(k, kw), &z__2);
d11.r = z__1.r, d11.i = z__1.i;
z_div(&z__1, &w_ref(k - 1, kw - 1), &d21);
d22.r = z__1.r, d22.i = z__1.i;
z__1.r = d11.r * d22.r - d11.i * d22.i, z__1.i = d11.r *
d22.i + d11.i * d22.r;
t = 1. / (z__1.r - 1.);
z__2.r = t, z__2.i = 0.;
z_div(&z__1, &z__2, &d21);
d21.r = z__1.r, d21.i = z__1.i;
i__1 = k - 2;
for (j = 1; j <= i__1; ++j) {
i__2 = a_subscr(j, k - 1);
i__3 = w_subscr(j, kw - 1);
z__3.r = d11.r * w[i__3].r - d11.i * w[i__3].i,
z__3.i = d11.r * w[i__3].i + d11.i * w[i__3]
.r;
i__4 = w_subscr(j, kw);
z__2.r = z__3.r - w[i__4].r, z__2.i = z__3.i - w[i__4]
.i;
z__1.r = d21.r * z__2.r - d21.i * z__2.i, z__1.i =
d21.r * z__2.i + d21.i * z__2.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
i__2 = a_subscr(j, k);
d_cnjg(&z__2, &d21);
i__3 = w_subscr(j, kw);
z__4.r = d22.r * w[i__3].r - d22.i * w[i__3].i,
z__4.i = d22.r * w[i__3].i + d22.i * w[i__3]
.r;
i__4 = w_subscr(j, kw - 1);
z__3.r = z__4.r - w[i__4].r, z__3.i = z__4.i - w[i__4]
.i;
z__1.r = z__2.r * z__3.r - z__2.i * z__3.i, z__1.i =
z__2.r * z__3.i + z__2.i * z__3.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
/* L20: */
}
}
/* Copy D(k) to A */
i__1 = a_subscr(k - 1, k - 1);
i__2 = w_subscr(k - 1, kw - 1);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
i__1 = a_subscr(k - 1, k);
i__2 = w_subscr(k - 1, kw);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
i__1 = a_subscr(k, k);
i__2 = w_subscr(k, kw);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
/* Conjugate W(k) and W(k-1) */
i__1 = k - 1;
zlacgv_(&i__1, &w_ref(1, kw), &c__1);
i__1 = k - 2;
zlacgv_(&i__1, &w_ref(1, kw - 1), &c__1);
}
}
/* Store details of the interchanges in IPIV */
if (kstep == 1) {
ipiv[k] = kp;
} else {
ipiv[k] = -kp;
ipiv[k - 1] = -kp;
}
/* Decrease K and return to the start of the main loop */
k -= kstep;
goto L10;
L30:
/* Update the upper triangle of A11 (= A(1:k,1:k)) as
A11 := A11 - U12*D*U12' = A11 - U12*W'
computing blocks of NB columns at a time (note that conjg(W) is
actually stored) */
i__1 = -(*nb);
for (j = (k - 1) / *nb * *nb + 1; i__1 < 0 ? j >= 1 : j <= 1; j +=
i__1) {
/* Computing MIN */
i__2 = *nb, i__3 = k - j + 1;
jb = min(i__2,i__3);
/* Update the upper triangle of the diagonal block */
i__2 = j + jb - 1;
for (jj = j; jj <= i__2; ++jj) {
i__3 = a_subscr(jj, jj);
i__4 = a_subscr(jj, jj);
d__1 = a[i__4].r;
a[i__3].r = d__1, a[i__3].i = 0.;
i__3 = jj - j + 1;
i__4 = *n - k;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &i__3, &i__4, &z__1, &a_ref(j, k + 1),
lda, &w_ref(jj, kw + 1), ldw, &c_b1, &a_ref(j, jj), &
c__1);
i__3 = a_subscr(jj, jj);
i__4 = a_subscr(jj, jj);
d__1 = a[i__4].r;
a[i__3].r = d__1, a[i__3].i = 0.;
/* L40: */
}
/* Update the rectangular superdiagonal block */
i__2 = j - 1;
i__3 = *n - k;
z__1.r = -1., z__1.i = 0.;
zgemm_("No transpose", "Transpose", &i__2, &jb, &i__3, &z__1, &
a_ref(1, k + 1), lda, &w_ref(j, kw + 1), ldw, &c_b1, &
a_ref(1, j), lda);
/* L50: */
}
/* Put U12 in standard form by partially undoing the interchanges
in columns k+1:n */
j = k + 1;
L60:
jj = j;
jp = ipiv[j];
if (jp < 0) {
jp = -jp;
++j;
}
++j;
if (jp != jj && j <= *n) {
i__1 = *n - j + 1;
zswap_(&i__1, &a_ref(jp, j), lda, &a_ref(jj, j), lda);
}
if (j <= *n) {
goto L60;
}
/* Set KB to the number of columns factorized */
*kb = *n - k;
} else {
/* Factorize the leading columns of A using the lower triangle
of A and working forwards, and compute the matrix W = L21*D
for use in updating A22 (note that conjg(W) is actually stored)
K is the main loop index, increasing from 1 in steps of 1 or 2 */
k = 1;
L70:
/* Exit from loop */
if (k >= *nb && *nb < *n || k > *n) {
goto L90;
}
/* Copy column K of A to column K of W and update it */
i__1 = w_subscr(k, k);
i__2 = a_subscr(k, k);
d__1 = a[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
if (k < *n) {
i__1 = *n - k;
zcopy_(&i__1, &a_ref(k + 1, k), &c__1, &w_ref(k + 1, k), &c__1);
}
i__1 = *n - k + 1;
i__2 = k - 1;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &i__1, &i__2, &z__1, &a_ref(k, 1), lda, &w_ref(
k, 1), ldw, &c_b1, &w_ref(k, k), &c__1);
i__1 = w_subscr(k, k);
i__2 = w_subscr(k, k);
d__1 = w[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
kstep = 1;
/* Determine rows and columns to be interchanged and whether
a 1-by-1 or 2-by-2 pivot block will be used */
i__1 = w_subscr(k, k);
absakk = (d__1 = w[i__1].r, abs(d__1));
/* IMAX is the row-index of the largest off-diagonal element in
column K, and COLMAX is its absolute value */
if (k < *n) {
i__1 = *n - k;
imax = k + izamax_(&i__1, &w_ref(k + 1, k), &c__1);
i__1 = w_subscr(imax, k);
colmax = (d__1 = w[i__1].r, abs(d__1)) + (d__2 = d_imag(&w_ref(
imax, k)), abs(d__2));
} else {
colmax = 0.;
}
if (max(absakk,colmax) == 0.) {
/* Column K is zero: set INFO and continue */
if (*info == 0) {
*info = k;
}
kp = k;
i__1 = a_subscr(k, k);
i__2 = a_subscr(k, k);
d__1 = a[i__2].r;
a[i__1].r = d__1, a[i__1].i = 0.;
} else {
if (absakk >= alpha * colmax) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else {
/* Copy column IMAX to column K+1 of W and update it */
i__1 = imax - k;
zcopy_(&i__1, &a_ref(imax, k), lda, &w_ref(k, k + 1), &c__1);
i__1 = imax - k;
zlacgv_(&i__1, &w_ref(k, k + 1), &c__1);
i__1 = w_subscr(imax, k + 1);
i__2 = a_subscr(imax, imax);
d__1 = a[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
if (imax < *n) {
i__1 = *n - imax;
zcopy_(&i__1, &a_ref(imax + 1, imax), &c__1, &w_ref(imax
+ 1, k + 1), &c__1);
}
i__1 = *n - k + 1;
i__2 = k - 1;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &i__1, &i__2, &z__1, &a_ref(k, 1), lda,
&w_ref(imax, 1), ldw, &c_b1, &w_ref(k, k + 1), &c__1);
i__1 = w_subscr(imax, k + 1);
i__2 = w_subscr(imax, k + 1);
d__1 = w[i__2].r;
w[i__1].r = d__1, w[i__1].i = 0.;
/* JMAX is the column-index of the largest off-diagonal
element in row IMAX, and ROWMAX is its absolute value */
i__1 = imax - k;
jmax = k - 1 + izamax_(&i__1, &w_ref(k, k + 1), &c__1);
i__1 = w_subscr(jmax, k + 1);
rowmax = (d__1 = w[i__1].r, abs(d__1)) + (d__2 = d_imag(&
w_ref(jmax, k + 1)), abs(d__2));
if (imax < *n) {
i__1 = *n - imax;
jmax = imax + izamax_(&i__1, &w_ref(imax + 1, k + 1), &
c__1);
/* Computing MAX */
i__1 = w_subscr(jmax, k + 1);
d__3 = rowmax, d__4 = (d__1 = w[i__1].r, abs(d__1)) + (
d__2 = d_imag(&w_ref(jmax, k + 1)), abs(d__2));
rowmax = max(d__3,d__4);
}
if (absakk >= alpha * colmax * (colmax / rowmax)) {
/* no interchange, use 1-by-1 pivot block */
kp = k;
} else /* if(complicated condition) */ {
i__1 = w_subscr(imax, k + 1);
if ((d__1 = w[i__1].r, abs(d__1)) >= alpha * rowmax) {
/* interchange rows and columns K and IMAX, use 1-by-1
pivot block */
kp = imax;
/* copy column K+1 of W to column K */
i__1 = *n - k + 1;
zcopy_(&i__1, &w_ref(k, k + 1), &c__1, &w_ref(k, k), &
c__1);
} else {
/* interchange rows and columns K+1 and IMAX, use 2-by-2
pivot block */
kp = imax;
kstep = 2;
}
}
}
kk = k + kstep - 1;
/* Updated column KP is already stored in column KK of W */
if (kp != kk) {
/* Copy non-updated column KK to column KP */
i__1 = a_subscr(kp, kp);
i__2 = a_subscr(kk, kk);
d__1 = a[i__2].r;
a[i__1].r = d__1, a[i__1].i = 0.;
i__1 = kp - kk - 1;
zcopy_(&i__1, &a_ref(kk + 1, kk), &c__1, &a_ref(kp, kk + 1),
lda);
i__1 = kp - kk - 1;
zlacgv_(&i__1, &a_ref(kp, kk + 1), lda);
if (kp < *n) {
i__1 = *n - kp;
zcopy_(&i__1, &a_ref(kp + 1, kk), &c__1, &a_ref(kp + 1,
kp), &c__1);
}
/* Interchange rows KK and KP in first KK columns of A and W */
i__1 = kk - 1;
zswap_(&i__1, &a_ref(kk, 1), lda, &a_ref(kp, 1), lda);
zswap_(&kk, &w_ref(kk, 1), ldw, &w_ref(kp, 1), ldw);
}
if (kstep == 1) {
/* 1-by-1 pivot block D(k): column k of W now holds
W(k) = L(k)*D(k)
where L(k) is the k-th column of L
Store L(k) in column k of A */
i__1 = *n - k + 1;
zcopy_(&i__1, &w_ref(k, k), &c__1, &a_ref(k, k), &c__1);
if (k < *n) {
i__1 = a_subscr(k, k);
r1 = 1. / a[i__1].r;
i__1 = *n - k;
zdscal_(&i__1, &r1, &a_ref(k + 1, k), &c__1);
/* Conjugate W(k) */
i__1 = *n - k;
zlacgv_(&i__1, &w_ref(k + 1, k), &c__1);
}
} else {
/* 2-by-2 pivot block D(k): columns k and k+1 of W now hold
( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
where L(k) and L(k+1) are the k-th and (k+1)-th columns
of L */
if (k < *n - 1) {
/* Store L(k) and L(k+1) in columns k and k+1 of A */
i__1 = w_subscr(k + 1, k);
d21.r = w[i__1].r, d21.i = w[i__1].i;
z_div(&z__1, &w_ref(k + 1, k + 1), &d21);
d11.r = z__1.r, d11.i = z__1.i;
d_cnjg(&z__2, &d21);
z_div(&z__1, &w_ref(k, k), &z__2);
d22.r = z__1.r, d22.i = z__1.i;
z__1.r = d11.r * d22.r - d11.i * d22.i, z__1.i = d11.r *
d22.i + d11.i * d22.r;
t = 1. / (z__1.r - 1.);
z__2.r = t, z__2.i = 0.;
z_div(&z__1, &z__2, &d21);
d21.r = z__1.r, d21.i = z__1.i;
i__1 = *n;
for (j = k + 2; j <= i__1; ++j) {
i__2 = a_subscr(j, k);
d_cnjg(&z__2, &d21);
i__3 = w_subscr(j, k);
z__4.r = d11.r * w[i__3].r - d11.i * w[i__3].i,
z__4.i = d11.r * w[i__3].i + d11.i * w[i__3]
.r;
i__4 = w_subscr(j, k + 1);
z__3.r = z__4.r - w[i__4].r, z__3.i = z__4.i - w[i__4]
.i;
z__1.r = z__2.r * z__3.r - z__2.i * z__3.i, z__1.i =
z__2.r * z__3.i + z__2.i * z__3.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
i__2 = a_subscr(j, k + 1);
i__3 = w_subscr(j, k + 1);
z__3.r = d22.r * w[i__3].r - d22.i * w[i__3].i,
z__3.i = d22.r * w[i__3].i + d22.i * w[i__3]
.r;
i__4 = w_subscr(j, k);
z__2.r = z__3.r - w[i__4].r, z__2.i = z__3.i - w[i__4]
.i;
z__1.r = d21.r * z__2.r - d21.i * z__2.i, z__1.i =
d21.r * z__2.i + d21.i * z__2.r;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
/* L80: */
}
}
/* Copy D(k) to A */
i__1 = a_subscr(k, k);
i__2 = w_subscr(k, k);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
i__1 = a_subscr(k + 1, k);
i__2 = w_subscr(k + 1, k);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
i__1 = a_subscr(k + 1, k + 1);
i__2 = w_subscr(k + 1, k + 1);
a[i__1].r = w[i__2].r, a[i__1].i = w[i__2].i;
/* Conjugate W(k) and W(k+1) */
i__1 = *n - k;
zlacgv_(&i__1, &w_ref(k + 1, k), &c__1);
i__1 = *n - k - 1;
zlacgv_(&i__1, &w_ref(k + 2, k + 1), &c__1);
}
}
/* Store details of the interchanges in IPIV */
if (kstep == 1) {
ipiv[k] = kp;
} else {
ipiv[k] = -kp;
ipiv[k + 1] = -kp;
}
/* Increase K and return to the start of the main loop */
k += kstep;
goto L70;
L90:
/* Update the lower triangle of A22 (= A(k:n,k:n)) as
A22 := A22 - L21*D*L21' = A22 - L21*W'
computing blocks of NB columns at a time (note that conjg(W) is
actually stored) */
i__1 = *n;
i__2 = *nb;
for (j = k; i__2 < 0 ? j >= i__1 : j <= i__1; j += i__2) {
/* Computing MIN */
i__3 = *nb, i__4 = *n - j + 1;
jb = min(i__3,i__4);
/* Update the lower triangle of the diagonal block */
i__3 = j + jb - 1;
for (jj = j; jj <= i__3; ++jj) {
i__4 = a_subscr(jj, jj);
i__5 = a_subscr(jj, jj);
d__1 = a[i__5].r;
a[i__4].r = d__1, a[i__4].i = 0.;
i__4 = j + jb - jj;
i__5 = k - 1;
z__1.r = -1., z__1.i = 0.;
zgemv_("No transpose", &i__4, &i__5, &z__1, &a_ref(jj, 1),
lda, &w_ref(jj, 1), ldw, &c_b1, &a_ref(jj, jj), &c__1);
i__4 = a_subscr(jj, jj);
i__5 = a_subscr(jj, jj);
d__1 = a[i__5].r;
a[i__4].r = d__1, a[i__4].i = 0.;
/* L100: */
}
/* Update the rectangular subdiagonal block */
if (j + jb <= *n) {
i__3 = *n - j - jb + 1;
i__4 = k - 1;
z__1.r = -1., z__1.i = 0.;
zgemm_("No transpose", "Transpose", &i__3, &jb, &i__4, &z__1,
&a_ref(j + jb, 1), lda, &w_ref(j, 1), ldw, &c_b1, &
a_ref(j + jb, j), lda);
}
/* L110: */
}
/* Put L21 in standard form by partially undoing the interchanges
in columns 1:k-1 */
j = k - 1;
L120:
jj = j;
jp = ipiv[j];
if (jp < 0) {
jp = -jp;
--j;
}
--j;
if (jp != jj && j >= 1) {
zswap_(&j, &a_ref(jp, 1), lda, &a_ref(jj, 1), lda);
}
if (j >= 1) {
goto L120;
}
/* Set KB to the number of columns factorized */
*kb = k - 1;
}
return 0;
/* End of ZLAHEF */
} /* zlahef_ */
/* Subroutine */ int cggsvp_(char *jobu, char *jobv, char *jobq, integer *m,
integer *p, integer *n, complex *a, integer *lda, complex *b, integer
*ldb, real *tola, real *tolb, integer *k, integer *l, complex *u,
integer *ldu, complex *v, integer *ldv, complex *q, integer *ldq,
integer *iwork, real *rwork, complex *tau, complex *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
=======
CGGSVP computes unitary matrices U, V and Q such that
N-K-L K L
U'*A*Q = K ( 0 A12 A13 ) if M-K-L >= 0;
L ( 0 0 A23 )
M-K-L ( 0 0 0 )
N-K-L K L
= K ( 0 A12 A13 ) if M-K-L < 0;
M-K ( 0 0 A23 )
N-K-L K L
V'*B*Q = 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. K+L = the effective
numerical rank of the (M+P)-by-N matrix (A',B')'. Z' denotes the
conjugate transpose of Z.
This decomposition is the preprocessing step for computing the
Generalized Singular Value Decomposition (GSVD), see subroutine
CGGSVD.
Arguments
=========
JOBU (input) CHARACTER*1
= 'U': Unitary matrix U is computed;
= 'N': U is not computed.
JOBV (input) CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1
= 'Q': Unitary matrix Q is computed;
= '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.
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, A contains the triangular (or trapezoidal) matrix
described in the Purpose section.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix described in
the Purpose section.
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 thresholds to determine the effective
numerical rank of matrix B and a subblock of A. Generally,
they are set to
TOLA = MAX(M,N)*norm(A)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.
K (output) INTEGER
L (output) INTEGER
On exit, K and L specify the dimension of the subblocks
described in Purpose section.
K + L = effective numerical rank of (A',B')'.
U (output) COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the unitary matrix 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 (output) COMPLEX array, dimension (LDV,M)
If JOBV = 'V', V contains the unitary matrix 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 (output) COMPLEX array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the unitary matrix 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.
IWORK (workspace) INTEGER array, dimension (N)
RWORK (workspace) REAL array, dimension (2*N)
TAU (workspace) COMPLEX array, dimension (N)
WORK (workspace) COMPLEX array, dimension (max(3*N,M,P))
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
Further Details
===============
The subroutine uses LAPACK subroutine CGEQPF for the QR factorization
with column pivoting to detect the effective numerical rank of the
a matrix. It may be replaced by a better rank determination strategy.
=====================================================================
Test the input parameters
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {0.f,0.f};
static complex c_b2 = {1.f,0.f};
/* 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;
real r__1, r__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static integer i__, j;
extern logical lsame_(char *, char *);
static logical wantq, wantu, wantv;
extern /* Subroutine */ int cgeqr2_(integer *, integer *, complex *,
integer *, complex *, complex *, integer *), cgerq2_(integer *,
integer *, complex *, integer *, complex *, complex *, integer *),
cung2r_(integer *, integer *, integer *, complex *, integer *,
complex *, complex *, integer *), cunm2r_(char *, char *, integer
*, integer *, integer *, complex *, integer *, complex *, complex
*, integer *, complex *, integer *), cunmr2_(char
*, char *, integer *, integer *, integer *, complex *, integer *,
complex *, complex *, integer *, complex *, integer *), cgeqpf_(integer *, integer *, complex *, integer *,
integer *, complex *, complex *, real *, integer *), clacpy_(char
*, integer *, integer *, complex *, integer *, complex *, integer
*), claset_(char *, integer *, integer *, complex *,
complex *, complex *, integer *), xerbla_(char *, integer
*), clapmt_(logical *, integer *, integer *, complex *,
integer *, integer *);
static logical forwrd;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define u_subscr(a_1,a_2) (a_2)*u_dim1 + a_1
#define u_ref(a_1,a_2) u[u_subscr(a_1,a_2)]
#define v_subscr(a_1,a_2) (a_2)*v_dim1 + a_1
#define v_ref(a_1,a_2) v[v_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
u_dim1 = *ldu;
u_offset = 1 + u_dim1 * 1;
u -= u_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1 * 1;
v -= v_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1 * 1;
q -= q_offset;
--iwork;
--rwork;
--tau;
--work;
/* Function Body */
wantu = lsame_(jobu, "U");
wantv = lsame_(jobv, "V");
wantq = lsame_(jobq, "Q");
forwrd = TRUE_;
*info = 0;
if (! (wantu || lsame_(jobu, "N"))) {
*info = -1;
} else if (! (wantv || lsame_(jobv, "N"))) {
*info = -2;
} else if (! (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 = -8;
} else if (*ldb < max(1,*p)) {
*info = -10;
} else if (*ldu < 1 || wantu && *ldu < *m) {
*info = -16;
} else if (*ldv < 1 || wantv && *ldv < *p) {
*info = -18;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -20;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGGSVP", &i__1);
return 0;
}
/* QR with column pivoting of B: B*P = V*( S11 S12 )
( 0 0 ) */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L10: */
}
cgeqpf_(p, n, &b[b_offset], ldb, &iwork[1], &tau[1], &work[1], &rwork[1],
info);
/* Update A := A*P */
clapmt_(&forwrd, m, n, &a[a_offset], lda, &iwork[1]);
/* Determine the effective rank of matrix B. */
*l = 0;
i__1 = min(*p,*n);
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = b_subscr(i__, i__);
if ((r__1 = b[i__2].r, dabs(r__1)) + (r__2 = r_imag(&b_ref(i__, i__)),
dabs(r__2)) > *tolb) {
++(*l);
}
/* L20: */
}
if (wantv) {
/* Copy the details of V, and form V. */
claset_("Full", p, p, &c_b1, &c_b1, &v[v_offset], ldv);
if (*p > 1) {
i__1 = *p - 1;
clacpy_("Lower", &i__1, n, &b_ref(2, 1), ldb, &v_ref(2, 1), ldv);
}
i__1 = min(*p,*n);
cung2r_(p, p, &i__1, &v[v_offset], ldv, &tau[1], &work[1], info);
}
/* Clean up B */
i__1 = *l - 1;
for (j = 1; j <= i__1; ++j) {
i__2 = *l;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
b[i__3].r = 0.f, b[i__3].i = 0.f;
/* L30: */
}
/* L40: */
}
if (*p > *l) {
i__1 = *p - *l;
claset_("Full", &i__1, n, &c_b1, &c_b1, &b_ref(*l + 1, 1), ldb);
}
if (wantq) {
/* Set Q = I and Update Q := Q*P */
claset_("Full", n, n, &c_b1, &c_b2, &q[q_offset], ldq);
clapmt_(&forwrd, n, n, &q[q_offset], ldq, &iwork[1]);
}
if (*p >= *l && *n != *l) {
/* RQ factorization of ( S11 S12 ) = ( 0 S12 )*Z */
cgerq2_(l, n, &b[b_offset], ldb, &tau[1], &work[1], info);
/* Update A := A*Z' */
cunmr2_("Right", "Conjugate transpose", m, n, l, &b[b_offset], ldb, &
tau[1], &a[a_offset], lda, &work[1], info);
if (wantq) {
/* Update Q := Q*Z' */
cunmr2_("Right", "Conjugate transpose", n, n, l, &b[b_offset],
ldb, &tau[1], &q[q_offset], ldq, &work[1], info);
}
/* Clean up B */
i__1 = *n - *l;
claset_("Full", l, &i__1, &c_b1, &c_b1, &b[b_offset], ldb);
i__1 = *n;
for (j = *n - *l + 1; j <= i__1; ++j) {
i__2 = *l;
for (i__ = j - *n + *l + 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
b[i__3].r = 0.f, b[i__3].i = 0.f;
/* L50: */
}
/* L60: */
}
}
/* Let N-L L
A = ( A11 A12 ) M,
then the following does the complete QR decomposition of A11:
A11 = U*( 0 T12 )*P1'
( 0 0 ) */
i__1 = *n - *l;
for (i__ = 1; i__ <= i__1; ++i__) {
iwork[i__] = 0;
/* L70: */
}
i__1 = *n - *l;
cgeqpf_(m, &i__1, &a[a_offset], lda, &iwork[1], &tau[1], &work[1], &rwork[
1], info);
/* Determine the effective rank of A11 */
*k = 0;
/* Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = a_subscr(i__, i__);
if ((r__1 = a[i__2].r, dabs(r__1)) + (r__2 = r_imag(&a_ref(i__, i__)),
dabs(r__2)) > *tola) {
++(*k);
}
/* L80: */
}
/* Update A12 := U'*A12, where A12 = A( 1:M, N-L+1:N )
Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
cunm2r_("Left", "Conjugate transpose", m, l, &i__1, &a[a_offset], lda, &
tau[1], &a_ref(1, *n - *l + 1), lda, &work[1], info);
if (wantu) {
/* Copy the details of U, and form U */
claset_("Full", m, m, &c_b1, &c_b1, &u[u_offset], ldu);
if (*m > 1) {
i__1 = *m - 1;
i__2 = *n - *l;
clacpy_("Lower", &i__1, &i__2, &a_ref(2, 1), lda, &u_ref(2, 1),
ldu);
}
/* Computing MIN */
i__2 = *m, i__3 = *n - *l;
i__1 = min(i__2,i__3);
cung2r_(m, m, &i__1, &u[u_offset], ldu, &tau[1], &work[1], info);
}
if (wantq) {
/* Update Q( 1:N, 1:N-L ) = Q( 1:N, 1:N-L )*P1 */
i__1 = *n - *l;
clapmt_(&forwrd, n, &i__1, &q[q_offset], ldq, &iwork[1]);
}
/* Clean up A: set the strictly lower triangular part of
A(1:K, 1:K) = 0, and A( K+1:M, 1:N-L ) = 0. */
i__1 = *k - 1;
for (j = 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0.f, a[i__3].i = 0.f;
/* L90: */
}
/* L100: */
}
if (*m > *k) {
i__1 = *m - *k;
i__2 = *n - *l;
claset_("Full", &i__1, &i__2, &c_b1, &c_b1, &a_ref(*k + 1, 1), lda);
}
if (*n - *l > *k) {
/* RQ factorization of ( T11 T12 ) = ( 0 T12 )*Z1 */
i__1 = *n - *l;
cgerq2_(k, &i__1, &a[a_offset], lda, &tau[1], &work[1], info);
if (wantq) {
/* Update Q( 1:N,1:N-L ) = Q( 1:N,1:N-L )*Z1' */
i__1 = *n - *l;
cunmr2_("Right", "Conjugate transpose", n, &i__1, k, &a[a_offset],
lda, &tau[1], &q[q_offset], ldq, &work[1], info);
}
/* Clean up A */
i__1 = *n - *l - *k;
claset_("Full", k, &i__1, &c_b1, &c_b1, &a[a_offset], lda);
i__1 = *n - *l;
for (j = *n - *l - *k + 1; j <= i__1; ++j) {
i__2 = *k;
for (i__ = j - *n + *l + *k + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0.f, a[i__3].i = 0.f;
/* L110: */
}
/* L120: */
}
}
if (*m > *k) {
/* QR factorization of A( K+1:M,N-L+1:N ) */
i__1 = *m - *k;
cgeqr2_(&i__1, l, &a_ref(*k + 1, *n - *l + 1), lda, &tau[1], &work[1],
info);
if (wantu) {
/* Update U(:,K+1:M) := U(:,K+1:M)*U1 */
i__1 = *m - *k;
/* Computing MIN */
i__3 = *m - *k;
i__2 = min(i__3,*l);
cunm2r_("Right", "No transpose", m, &i__1, &i__2, &a_ref(*k + 1, *
n - *l + 1), lda, &tau[1], &u_ref(1, *k + 1), ldu, &work[
1], info);
}
/* Clean up */
i__1 = *n;
for (j = *n - *l + 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = j - *n + *k + *l + 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0.f, a[i__3].i = 0.f;
/* L130: */
}
/* L140: */
}
}
return 0;
/* End of CGGSVP */
} /* cggsvp_ */
/* Subroutine */ int ctrsv_(char *uplo, char *trans, char *diag, integer *n,
complex *a, integer *lda, complex *x, integer *incx)
{
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3, i__4, i__5;
complex q__1, q__2, q__3;
/* Builtin functions */
void c_div(complex *, complex *, complex *), r_cnjg(complex *, complex *);
/* Local variables */
static integer info;
static complex temp;
static integer i__, j;
extern logical lsame_(char *, char *);
static integer ix, jx, kx;
extern /* Subroutine */ int xerbla_(char *, integer *);
static logical noconj, nounit;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
/* Purpose
=======
CTRSV solves one of the systems of equations
A*x = b, or A'*x = b, or conjg( A' )*x = b,
where b and x are n element vectors and A is an n by n unit, or
non-unit, upper or lower triangular matrix.
No test for singularity or near-singularity is included in this
routine. Such tests must be performed before calling this routine.
Parameters
==========
UPLO - CHARACTER*1.
On entry, UPLO specifies whether the matrix is an upper or
lower triangular matrix as follows:
UPLO = 'U' or 'u' A is an upper triangular matrix.
UPLO = 'L' or 'l' A is a lower triangular matrix.
Unchanged on exit.
TRANS - CHARACTER*1.
On entry, TRANS specifies the equations to be solved as
follows:
TRANS = 'N' or 'n' A*x = b.
TRANS = 'T' or 't' A'*x = b.
TRANS = 'C' or 'c' conjg( A' )*x = b.
Unchanged on exit.
DIAG - CHARACTER*1.
On entry, DIAG specifies whether or not A is unit
triangular as follows:
DIAG = 'U' or 'u' A is assumed to be unit triangular.
DIAG = 'N' or 'n' A is not assumed to be unit
triangular.
Unchanged on exit.
N - INTEGER.
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
A - COMPLEX array of DIMENSION ( LDA, n ).
Before entry with UPLO = 'U' or 'u', the leading n by n
upper triangular part of the array A must contain the upper
triangular matrix and the strictly lower triangular part of
A is not referenced.
Before entry with UPLO = 'L' or 'l', the leading n by n
lower triangular part of the array A must contain the lower
triangular matrix and the strictly upper triangular part of
A is not referenced.
Note that when DIAG = 'U' or 'u', the diagonal elements of
A are not referenced either, but are assumed to be unity.
Unchanged on exit.
LDA - INTEGER.
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
max( 1, n ).
Unchanged on exit.
X - COMPLEX array of dimension at least
( 1 + ( n - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the n
element right-hand side vector b. On exit, X is overwritten
with the solution vector x.
INCX - INTEGER.
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
Level 2 Blas routine.
-- Written on 22-October-1986.
Jack Dongarra, Argonne National Lab.
Jeremy Du Croz, Nag Central Office.
Sven Hammarling, Nag Central Office.
Richard Hanson, Sandia National Labs.
Test the input parameters.
Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--x;
/* Function Body */
info = 0;
if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) {
info = 1;
} else if (! lsame_(trans, "N") && ! lsame_(trans,
"T") && ! lsame_(trans, "C")) {
info = 2;
} else if (! lsame_(diag, "U") && ! lsame_(diag,
"N")) {
info = 3;
} else if (*n < 0) {
info = 4;
} else if (*lda < max(1,*n)) {
info = 6;
} else if (*incx == 0) {
info = 8;
}
if (info != 0) {
xerbla_("CTRSV ", &info);
return 0;
}
/* Quick return if possible. */
if (*n == 0) {
return 0;
}
noconj = lsame_(trans, "T");
nounit = lsame_(diag, "N");
/* Set up the start point in X if the increment is not unity. This
will be ( N - 1 )*INCX too small for descending loops. */
if (*incx <= 0) {
kx = 1 - (*n - 1) * *incx;
} else if (*incx != 1) {
kx = 1;
}
/* Start the operations. In this version the elements of A are
accessed sequentially with one pass through A. */
if (lsame_(trans, "N")) {
/* Form x := inv( A )*x. */
if (lsame_(uplo, "U")) {
if (*incx == 1) {
for (j = *n; j >= 1; --j) {
i__1 = j;
if (x[i__1].r != 0.f || x[i__1].i != 0.f) {
if (nounit) {
i__1 = j;
c_div(&q__1, &x[j], &a_ref(j, j));
x[i__1].r = q__1.r, x[i__1].i = q__1.i;
}
i__1 = j;
temp.r = x[i__1].r, temp.i = x[i__1].i;
for (i__ = j - 1; i__ >= 1; --i__) {
i__1 = i__;
i__2 = i__;
i__3 = a_subscr(i__, j);
q__2.r = temp.r * a[i__3].r - temp.i * a[i__3].i,
q__2.i = temp.r * a[i__3].i + temp.i * a[
i__3].r;
q__1.r = x[i__2].r - q__2.r, q__1.i = x[i__2].i -
q__2.i;
x[i__1].r = q__1.r, x[i__1].i = q__1.i;
/* L10: */
}
}
/* L20: */
}
} else {
jx = kx + (*n - 1) * *incx;
for (j = *n; j >= 1; --j) {
i__1 = jx;
if (x[i__1].r != 0.f || x[i__1].i != 0.f) {
if (nounit) {
i__1 = jx;
c_div(&q__1, &x[jx], &a_ref(j, j));
x[i__1].r = q__1.r, x[i__1].i = q__1.i;
}
i__1 = jx;
temp.r = x[i__1].r, temp.i = x[i__1].i;
ix = jx;
for (i__ = j - 1; i__ >= 1; --i__) {
ix -= *incx;
i__1 = ix;
i__2 = ix;
i__3 = a_subscr(i__, j);
q__2.r = temp.r * a[i__3].r - temp.i * a[i__3].i,
q__2.i = temp.r * a[i__3].i + temp.i * a[
i__3].r;
q__1.r = x[i__2].r - q__2.r, q__1.i = x[i__2].i -
q__2.i;
x[i__1].r = q__1.r, x[i__1].i = q__1.i;
/* L30: */
}
}
jx -= *incx;
/* L40: */
}
}
} else {
if (*incx == 1) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j;
if (x[i__2].r != 0.f || x[i__2].i != 0.f) {
if (nounit) {
i__2 = j;
c_div(&q__1, &x[j], &a_ref(j, j));
x[i__2].r = q__1.r, x[i__2].i = q__1.i;
}
i__2 = j;
temp.r = x[i__2].r, temp.i = x[i__2].i;
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = a_subscr(i__, j);
q__2.r = temp.r * a[i__5].r - temp.i * a[i__5].i,
q__2.i = temp.r * a[i__5].i + temp.i * a[
i__5].r;
q__1.r = x[i__4].r - q__2.r, q__1.i = x[i__4].i -
q__2.i;
x[i__3].r = q__1.r, x[i__3].i = q__1.i;
/* L50: */
}
}
/* L60: */
}
} else {
jx = kx;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = jx;
if (x[i__2].r != 0.f || x[i__2].i != 0.f) {
if (nounit) {
i__2 = jx;
c_div(&q__1, &x[jx], &a_ref(j, j));
x[i__2].r = q__1.r, x[i__2].i = q__1.i;
}
i__2 = jx;
temp.r = x[i__2].r, temp.i = x[i__2].i;
ix = jx;
i__2 = *n;
for (i__ = j + 1; i__ <= i__2; ++i__) {
ix += *incx;
i__3 = ix;
i__4 = ix;
i__5 = a_subscr(i__, j);
q__2.r = temp.r * a[i__5].r - temp.i * a[i__5].i,
q__2.i = temp.r * a[i__5].i + temp.i * a[
i__5].r;
q__1.r = x[i__4].r - q__2.r, q__1.i = x[i__4].i -
q__2.i;
x[i__3].r = q__1.r, x[i__3].i = q__1.i;
/* L70: */
}
}
jx += *incx;
/* L80: */
}
}
}
} else {
/* Form x := inv( A' )*x or x := inv( conjg( A' ) )*x. */
if (lsame_(uplo, "U")) {
if (*incx == 1) {
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = j;
temp.r = x[i__2].r, temp.i = x[i__2].i;
if (noconj) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = i__;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[
i__4].i, q__2.i = a[i__3].r * x[i__4].i +
a[i__3].i * x[i__4].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
/* L90: */
}
if (nounit) {
c_div(&q__1, &temp, &a_ref(j, j));
temp.r = q__1.r, temp.i = q__1.i;
}
} else {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
r_cnjg(&q__3, &a_ref(i__, j));
i__3 = i__;
q__2.r = q__3.r * x[i__3].r - q__3.i * x[i__3].i,
q__2.i = q__3.r * x[i__3].i + q__3.i * x[
i__3].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
/* L100: */
}
if (nounit) {
r_cnjg(&q__2, &a_ref(j, j));
c_div(&q__1, &temp, &q__2);
temp.r = q__1.r, temp.i = q__1.i;
}
}
i__2 = j;
x[i__2].r = temp.r, x[i__2].i = temp.i;
/* L110: */
}
} else {
jx = kx;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
ix = kx;
i__2 = jx;
temp.r = x[i__2].r, temp.i = x[i__2].i;
if (noconj) {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = a_subscr(i__, j);
i__4 = ix;
q__2.r = a[i__3].r * x[i__4].r - a[i__3].i * x[
i__4].i, q__2.i = a[i__3].r * x[i__4].i +
a[i__3].i * x[i__4].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
ix += *incx;
/* L120: */
}
if (nounit) {
c_div(&q__1, &temp, &a_ref(j, j));
temp.r = q__1.r, temp.i = q__1.i;
}
} else {
i__2 = j - 1;
for (i__ = 1; i__ <= i__2; ++i__) {
r_cnjg(&q__3, &a_ref(i__, j));
i__3 = ix;
q__2.r = q__3.r * x[i__3].r - q__3.i * x[i__3].i,
q__2.i = q__3.r * x[i__3].i + q__3.i * x[
i__3].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
ix += *incx;
/* L130: */
}
if (nounit) {
r_cnjg(&q__2, &a_ref(j, j));
c_div(&q__1, &temp, &q__2);
temp.r = q__1.r, temp.i = q__1.i;
}
}
i__2 = jx;
x[i__2].r = temp.r, x[i__2].i = temp.i;
jx += *incx;
/* L140: */
}
}
} else {
if (*incx == 1) {
for (j = *n; j >= 1; --j) {
i__1 = j;
temp.r = x[i__1].r, temp.i = x[i__1].i;
if (noconj) {
i__1 = j + 1;
for (i__ = *n; i__ >= i__1; --i__) {
i__2 = a_subscr(i__, j);
i__3 = i__;
q__2.r = a[i__2].r * x[i__3].r - a[i__2].i * x[
i__3].i, q__2.i = a[i__2].r * x[i__3].i +
a[i__2].i * x[i__3].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
/* L150: */
}
if (nounit) {
c_div(&q__1, &temp, &a_ref(j, j));
temp.r = q__1.r, temp.i = q__1.i;
}
} else {
i__1 = j + 1;
for (i__ = *n; i__ >= i__1; --i__) {
r_cnjg(&q__3, &a_ref(i__, j));
i__2 = i__;
q__2.r = q__3.r * x[i__2].r - q__3.i * x[i__2].i,
q__2.i = q__3.r * x[i__2].i + q__3.i * x[
i__2].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
/* L160: */
}
if (nounit) {
r_cnjg(&q__2, &a_ref(j, j));
c_div(&q__1, &temp, &q__2);
temp.r = q__1.r, temp.i = q__1.i;
}
}
i__1 = j;
x[i__1].r = temp.r, x[i__1].i = temp.i;
/* L170: */
}
} else {
kx += (*n - 1) * *incx;
jx = kx;
for (j = *n; j >= 1; --j) {
ix = kx;
i__1 = jx;
temp.r = x[i__1].r, temp.i = x[i__1].i;
if (noconj) {
i__1 = j + 1;
for (i__ = *n; i__ >= i__1; --i__) {
i__2 = a_subscr(i__, j);
i__3 = ix;
q__2.r = a[i__2].r * x[i__3].r - a[i__2].i * x[
i__3].i, q__2.i = a[i__2].r * x[i__3].i +
a[i__2].i * x[i__3].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
ix -= *incx;
/* L180: */
}
if (nounit) {
c_div(&q__1, &temp, &a_ref(j, j));
temp.r = q__1.r, temp.i = q__1.i;
}
} else {
i__1 = j + 1;
for (i__ = *n; i__ >= i__1; --i__) {
r_cnjg(&q__3, &a_ref(i__, j));
i__2 = ix;
q__2.r = q__3.r * x[i__2].r - q__3.i * x[i__2].i,
q__2.i = q__3.r * x[i__2].i + q__3.i * x[
i__2].r;
q__1.r = temp.r - q__2.r, q__1.i = temp.i -
q__2.i;
temp.r = q__1.r, temp.i = q__1.i;
ix -= *incx;
/* L190: */
}
if (nounit) {
r_cnjg(&q__2, &a_ref(j, j));
c_div(&q__1, &temp, &q__2);
temp.r = q__1.r, temp.i = q__1.i;
}
}
i__1 = jx;
x[i__1].r = temp.r, x[i__1].i = temp.i;
jx -= *incx;
/* L200: */
}
}
}
}
return 0;
/* End of CTRSV . */
} /* ctrsv_ */
/* Subroutine */ int zgeequ_(integer *m, integer *n, doublecomplex *a,
integer *lda, doublereal *r__, doublereal *c__, doublereal *rowcnd,
doublereal *colcnd, doublereal *amax, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
March 31, 1993
Purpose
=======
ZGEEQU computes row and column scalings intended to equilibrate an
M-by-N matrix A and reduce its condition number. R returns the row
scale factors and C the column scale factors, chosen to try to make
the largest element in each row and column of the matrix B with
elements B(i,j)=R(i)*A(i,j)*C(j) have absolute value 1.
R(i) and C(j) are restricted to be between SMLNUM = smallest safe
number and BIGNUM = largest safe number. Use of these scaling
factors is not guaranteed to reduce the condition number of A but
works well in practice.
Arguments
=========
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.
A (input) COMPLEX*16 array, dimension (LDA,N)
The M-by-N matrix whose equilibration factors are
to be computed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
R (output) DOUBLE PRECISION array, dimension (M)
If INFO = 0 or INFO > M, R contains the row scale factors
for A.
C (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, C contains the column scale factors for A.
ROWCND (output) DOUBLE PRECISION
If INFO = 0 or INFO > M, ROWCND contains the ratio of the
smallest R(i) to the largest R(i). If ROWCND >= 0.1 and
AMAX is neither too large nor too small, it is not worth
scaling by R.
COLCND (output) DOUBLE PRECISION
If INFO = 0, COLCND contains the ratio of the smallest
C(i) to the largest C(i). If COLCND >= 0.1, it is not
worth scaling by C.
AMAX (output) DOUBLE PRECISION
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= M: the i-th row of A is exactly zero
> M: the (i-M)-th column of A is exactly zero
=====================================================================
Test the input parameters.
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublereal d__1, d__2, d__3, d__4;
/* Builtin functions */
double d_imag(doublecomplex *);
/* Local variables */
static integer i__, j;
static doublereal rcmin, rcmax;
extern doublereal dlamch_(char *);
extern /* Subroutine */ int xerbla_(char *, integer *);
static doublereal bignum, smlnum;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--r__;
--c__;
/* Function Body */
*info = 0;
if (*m < 0) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*lda < max(1,*m)) {
*info = -4;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZGEEQU", &i__1);
return 0;
}
/* Quick return if possible */
if (*m == 0 || *n == 0) {
*rowcnd = 1.;
*colcnd = 1.;
*amax = 0.;
return 0;
}
/* Get machine constants. */
smlnum = dlamch_("S");
bignum = 1. / smlnum;
/* Compute row scale factors. */
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
r__[i__] = 0.;
/* L10: */
}
/* Find the maximum element in each row. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = a_subscr(i__, j);
d__3 = r__[i__], d__4 = (d__1 = a[i__3].r, abs(d__1)) + (d__2 =
d_imag(&a_ref(i__, j)), abs(d__2));
r__[i__] = max(d__3,d__4);
/* L20: */
}
/* L30: */
}
/* Find the maximum and minimum scale factors. */
rcmin = bignum;
rcmax = 0.;
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
d__1 = rcmax, d__2 = r__[i__];
rcmax = max(d__1,d__2);
/* Computing MIN */
d__1 = rcmin, d__2 = r__[i__];
rcmin = min(d__1,d__2);
/* L40: */
}
*amax = rcmax;
if (rcmin == 0.) {
/* Find the first zero scale factor and return an error code. */
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
if (r__[i__] == 0.) {
*info = i__;
return 0;
}
/* L50: */
}
} else {
/* Invert the scale factors. */
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MIN
Computing MAX */
d__2 = r__[i__];
d__1 = max(d__2,smlnum);
r__[i__] = 1. / min(d__1,bignum);
/* L60: */
}
/* Compute ROWCND = min(R(I)) / max(R(I)) */
*rowcnd = max(rcmin,smlnum) / min(rcmax,bignum);
}
/* Compute column scale factors */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
c__[j] = 0.;
/* L70: */
}
/* Find the maximum element in each column,
assuming the row scaling computed above. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = a_subscr(i__, j);
d__3 = c__[j], d__4 = ((d__1 = a[i__3].r, abs(d__1)) + (d__2 =
d_imag(&a_ref(i__, j)), abs(d__2))) * r__[i__];
c__[j] = max(d__3,d__4);
/* L80: */
}
/* L90: */
}
/* Find the maximum and minimum scale factors. */
rcmin = bignum;
rcmax = 0.;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
d__1 = rcmin, d__2 = c__[j];
rcmin = min(d__1,d__2);
/* Computing MAX */
d__1 = rcmax, d__2 = c__[j];
rcmax = max(d__1,d__2);
/* L100: */
}
if (rcmin == 0.) {
/* Find the first zero scale factor and return an error code. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (c__[j] == 0.) {
*info = *m + j;
return 0;
}
/* L110: */
}
} else {
/* Invert the scale factors. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MIN
Computing MAX */
d__2 = c__[j];
d__1 = max(d__2,smlnum);
c__[j] = 1. / min(d__1,bignum);
/* L120: */
}
/* Compute COLCND = min(C(J)) / max(C(J)) */
*colcnd = max(rcmin,smlnum) / min(rcmax,bignum);
}
return 0;
/* End of ZGEEQU */
} /* zgeequ_ */
/* Subroutine */ int cget07_(char *trans, integer *n, integer *nrhs, complex *
a, integer *lda, complex *b, integer *ldb, complex *x, integer *ldx,
complex *xact, integer *ldxact, real *ferr, real *berr, real *reslts)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset, xact_dim1,
xact_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1, q__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static real diff, axbi;
static integer imax;
static real unfl, ovfl;
static integer i__, j, k;
extern logical lsame_(char *, char *);
static real xnorm;
extern integer icamax_(integer *, complex *, integer *);
extern doublereal slamch_(char *);
static real errbnd;
static logical notran;
static real eps, tmp;
#define xact_subscr(a_1,a_2) (a_2)*xact_dim1 + a_1
#define xact_ref(a_1,a_2) xact[xact_subscr(a_1,a_2)]
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define x_subscr(a_1,a_2) (a_2)*x_dim1 + a_1
#define x_ref(a_1,a_2) x[x_subscr(a_1,a_2)]
/* -- LAPACK test 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
=======
CGET07 tests the error bounds from iterative refinement for the
computed solution to a system of equations op(A)*X = B, where A is a
general n by n matrix and op(A) = A or A**T, depending on TRANS.
RESLTS(1) = test of the error bound
= norm(X - XACT) / ( norm(X) * FERR )
A large value is returned if this ratio is not less than one.
RESLTS(2) = residual from the iterative refinement routine
= the maximum of BERR / ( (n+1)*EPS + (*) ), where
(*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
Arguments
=========
TRANS (input) CHARACTER*1
Specifies the form of the system of equations.
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose = Transpose)
N (input) INTEGER
The number of rows of the matrices X and XACT. N >= 0.
NRHS (input) INTEGER
The number of columns of the matrices X and XACT. NRHS >= 0.
A (input) COMPLEX array, dimension (LDA,N)
The original n by n matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) COMPLEX array, dimension (LDB,NRHS)
The right hand side vectors for the system of linear
equations.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (input) COMPLEX array, dimension (LDX,NRHS)
The computed solution vectors. Each vector is stored as a
column of the matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
XACT (input) COMPLEX array, dimension (LDX,NRHS)
The exact solution vectors. Each vector is stored as a
column of the matrix XACT.
LDXACT (input) INTEGER
The leading dimension of the array XACT. LDXACT >= max(1,N).
FERR (input) REAL array, dimension (NRHS)
The estimated forward error bounds for each solution vector
X. If XTRUE is the true solution, FERR bounds the magnitude
of the largest entry in (X - XTRUE) divided by the magnitude
of the largest entry in X.
BERR (input) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector (i.e., the smallest relative change in any entry of A
or B that makes X an exact solution).
RESLTS (output) REAL array, dimension (2)
The maximum over the NRHS solution vectors of the ratios:
RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
=====================================================================
Quick exit if N = 0 or NRHS = 0.
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;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
xact_dim1 = *ldxact;
xact_offset = 1 + xact_dim1 * 1;
xact -= xact_offset;
--ferr;
--berr;
--reslts;
/* Function Body */
if (*n <= 0 || *nrhs <= 0) {
reslts[1] = 0.f;
reslts[2] = 0.f;
return 0;
}
eps = slamch_("Epsilon");
unfl = slamch_("Safe minimum");
ovfl = 1.f / unfl;
notran = lsame_(trans, "N");
/* Test 1: Compute the maximum of
norm(X - XACT) / ( norm(X) * FERR )
over all the vectors X and XACT using the infinity-norm. */
errbnd = 0.f;
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
imax = icamax_(n, &x_ref(1, j), &c__1);
/* Computing MAX */
i__2 = x_subscr(imax, j);
r__3 = (r__1 = x[i__2].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(imax, j)
), dabs(r__2));
xnorm = dmax(r__3,unfl);
diff = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = x_subscr(i__, j);
i__4 = xact_subscr(i__, j);
q__2.r = x[i__3].r - xact[i__4].r, q__2.i = x[i__3].i - xact[i__4]
.i;
q__1.r = q__2.r, q__1.i = q__2.i;
/* Computing MAX */
r__3 = diff, r__4 = (r__1 = q__1.r, dabs(r__1)) + (r__2 = r_imag(&
q__1), dabs(r__2));
diff = dmax(r__3,r__4);
/* L10: */
}
if (xnorm > 1.f) {
goto L20;
} else if (diff <= ovfl * xnorm) {
goto L20;
} else {
errbnd = 1.f / eps;
goto L30;
}
L20:
if (diff / xnorm <= ferr[j]) {
/* Computing MAX */
r__1 = errbnd, r__2 = diff / xnorm / ferr[j];
errbnd = dmax(r__1,r__2);
} else {
errbnd = 1.f / eps;
}
L30:
;
}
reslts[1] = errbnd;
/* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
(*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) */
i__1 = *nrhs;
for (k = 1; k <= i__1; ++k) {
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, k);
tmp = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&b_ref(i__,
k)), dabs(r__2));
if (notran) {
i__3 = *n;
for (j = 1; j <= i__3; ++j) {
i__4 = a_subscr(i__, j);
i__5 = x_subscr(j, k);
tmp += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
a_ref(i__, j)), dabs(r__2))) * ((r__3 = x[i__5].r,
dabs(r__3)) + (r__4 = r_imag(&x_ref(j, k)), dabs(
r__4)));
/* L40: */
}
} else {
i__3 = *n;
for (j = 1; j <= i__3; ++j) {
i__4 = a_subscr(j, i__);
i__5 = x_subscr(j, k);
tmp += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
a_ref(j, i__)), dabs(r__2))) * ((r__3 = x[i__5].r,
dabs(r__3)) + (r__4 = r_imag(&x_ref(j, k)), dabs(
r__4)));
/* L50: */
}
}
if (i__ == 1) {
axbi = tmp;
} else {
axbi = dmin(axbi,tmp);
}
/* L60: */
}
/* Computing MAX */
r__1 = axbi, r__2 = (*n + 1) * unfl;
tmp = berr[k] / ((*n + 1) * eps + (*n + 1) * unfl / dmax(r__1,r__2));
if (k == 1) {
reslts[2] = tmp;
} else {
reslts[2] = dmax(reslts[2],tmp);
}
/* L70: */
}
return 0;
/* End of CGET07 */
} /* cget07_ */
/* Subroutine */ int cherfs_(char *uplo, integer *n, integer *nrhs, complex *
a, integer *lda, complex *af, integer *ldaf, integer *ipiv, complex *
b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr,
complex *work, real *rwork, 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
=======
CHERFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian indefinite, and
provides error bounds and backward error estimates for the solution.
Arguments
=========
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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A (input) COMPLEX array, dimension (LDA,N)
The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input) COMPLEX array, dimension (LDAF,N)
The factored form of the matrix A. AF contains the block
diagonal matrix D and the multipliers used to obtain the
factor U or L from the factorization A = U*D*U**H or
A = L*D*L**H as computed by CHETRF.
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF.
B (input) COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (input/output) COMPLEX array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by CHETRS.
On exit, the improved solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Internal Parameters
===================
ITMAX is the maximum number of steps of iterative refinement.
=====================================================================
Test the input parameters.
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {1.f,0.f};
static integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1,
x_offset, i__1, i__2, i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static integer kase;
static real safe1, safe2;
static integer i__, j, k;
static real s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int chemv_(char *, integer *, complex *, complex *
, integer *, complex *, integer *, complex *, complex *, integer *
), ccopy_(integer *, complex *, integer *, complex *,
integer *), caxpy_(integer *, complex *, complex *, integer *,
complex *, integer *);
static integer count;
static logical upper;
extern /* Subroutine */ int clacon_(integer *, complex *, complex *, real
*, integer *);
static real xk;
extern doublereal slamch_(char *);
static integer nz;
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *), chetrs_(
char *, integer *, integer *, complex *, integer *, integer *,
complex *, integer *, integer *);
static real lstres, eps;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define x_subscr(a_1,a_2) (a_2)*x_dim1 + a_1
#define x_ref(a_1,a_2) x[x_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
af_dim1 = *ldaf;
af_offset = 1 + af_dim1 * 1;
af -= af_offset;
--ipiv;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (*n < 0) {
*info = -2;
} else if (*nrhs < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldaf < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldx < max(1,*n)) {
*info = -12;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CHERFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
count = 1;
lstres = 3.f;
L20:
/* Loop until stopping criterion is satisfied.
Compute residual R = B - A * X */
ccopy_(n, &b_ref(1, j), &c__1, &work[1], &c__1);
q__1.r = -1.f, q__1.i = 0.f;
chemv_(uplo, n, &q__1, &a[a_offset], lda, &x_ref(1, j), &c__1, &c_b1,
&work[1], &c__1);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. If the i-th component of the denominator is less
than SAFE2, then SAFE1 is added to the i-th components of the
numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
b_ref(i__, j)), dabs(r__2));
/* L30: */
}
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(k,
j)), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * xk;
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
a_ref(i__, k)), dabs(r__2))) * ((r__3 = x[i__5].r,
dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)),
dabs(r__4)));
/* L40: */
}
i__3 = a_subscr(k, k);
rwork[k] = rwork[k] + (r__1 = a[i__3].r, dabs(r__1)) * xk + s;
/* L50: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(k,
j)), dabs(r__2));
i__3 = a_subscr(k, k);
rwork[k] += (r__1 = a[i__3].r, dabs(r__1)) * xk;
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * xk;
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 = r_imag(&
a_ref(i__, k)), dabs(r__2))) * ((r__3 = x[i__5].r,
dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)),
dabs(r__4)));
/* L60: */
}
rwork[k] += s;
/* L70: */
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L80: */
}
berr[j] = s;
/* Test stopping criterion. Continue iterating if
1) The residual BERR(J) is larger than machine epsilon, and
2) BERR(J) decreased by at least a factor of 2 during the
last iteration, and
3) At most ITMAX iterations tried. */
if (berr[j] > eps && berr[j] * 2.f <= lstres && count <= 5) {
/* Update solution and try again. */
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[1],
n, info);
caxpy_(n, &c_b1, &work[1], &c__1, &x_ref(1, j), &c__1);
lstres = berr[j];
++count;
goto L20;
}
/* Bound error from formula
norm(X - XTRUE) / norm(X) .le. FERR =
norm( abs(inv(A))*
( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(A) is the inverse of A
abs(Z) is the componentwise absolute value of the matrix or
vector Z
NZ is the maximum number of nonzeros in any row of A, plus 1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(A)*abs(X) + abs(B) is less than SAFE2.
Use CLACON to estimate the infinity-norm of the matrix
inv(A) * diag(W),
where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L90: */
}
kase = 0;
L100:
clacon_(n, &work[*n + 1], &work[1], &ferr[j], &kase);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(A'). */
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L110: */
}
} else if (kase == 2) {
/* Multiply by inv(A)*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L120: */
}
chetrs_(uplo, n, &c__1, &af[af_offset], ldaf, &ipiv[1], &work[
1], n, info);
}
goto L100;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = x_subscr(i__, j);
r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&x_ref(i__, j)), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L130: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L140: */
}
return 0;
/* End of CHERFS */
} /* cherfs_ */
/* Subroutine */ int zungr2_(integer *m, integer *n, integer *k,
doublecomplex *a, integer *lda, doublecomplex *tau, doublecomplex *
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
=======
ZUNGR2 generates an m by n complex 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 ZGERQF.
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) COMPLEX*16 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 ZGERQF 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) COMPLEX*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by ZGERQF.
WORK (workspace) COMPLEX*16 array, dimension (M)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument has an illegal value
=====================================================================
Test the input arguments
Parameter adjustments */
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublecomplex z__1, z__2;
/* Builtin functions */
void d_cnjg(doublecomplex *, doublecomplex *);
/* Local variables */
static integer i__, j, l;
extern /* Subroutine */ int zscal_(integer *, doublecomplex *,
doublecomplex *, integer *), zlarf_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *, doublecomplex *);
static integer ii;
extern /* Subroutine */ int xerbla_(char *, integer *), zlacgv_(
integer *, doublecomplex *, integer *);
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*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;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZUNGR2", &i__1);
return 0;
}
/* Quick return if possible */
if (*m <= 0) {
return 0;
}
if (*k < *m) {
/* Initialise rows 1:m-k to rows of the unit matrix */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *m - *k;
for (l = 1; l <= i__2; ++l) {
i__3 = a_subscr(l, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* L10: */
}
if (j > *n - *m && j <= *n - *k) {
i__2 = a_subscr(*m - *n + j, j);
a[i__2].r = 1., a[i__2].i = 0.;
}
/* L20: */
}
}
i__1 = *k;
for (i__ = 1; i__ <= i__1; ++i__) {
ii = *m - *k + i__;
/* Apply H(i)' to A(1:m-k+i,1:n-k+i) from the right */
i__2 = *n - *m + ii - 1;
zlacgv_(&i__2, &a_ref(ii, 1), lda);
i__2 = a_subscr(ii, *n - *m + ii);
a[i__2].r = 1., a[i__2].i = 0.;
i__2 = ii - 1;
i__3 = *n - *m + ii;
d_cnjg(&z__1, &tau[i__]);
zlarf_("Right", &i__2, &i__3, &a_ref(ii, 1), lda, &z__1, &a[a_offset],
lda, &work[1]);
i__2 = *n - *m + ii - 1;
i__3 = i__;
z__1.r = -tau[i__3].r, z__1.i = -tau[i__3].i;
zscal_(&i__2, &z__1, &a_ref(ii, 1), lda);
i__2 = *n - *m + ii - 1;
zlacgv_(&i__2, &a_ref(ii, 1), lda);
i__2 = a_subscr(ii, *n - *m + ii);
d_cnjg(&z__2, &tau[i__]);
z__1.r = 1. - z__2.r, z__1.i = 0. - z__2.i;
a[i__2].r = z__1.r, a[i__2].i = z__1.i;
/* Set A(m-k+i,n-k+i+1:n) to zero */
i__2 = *n;
for (l = *n - *m + ii + 1; l <= i__2; ++l) {
i__3 = a_subscr(ii, l);
a[i__3].r = 0., a[i__3].i = 0.;
/* L30: */
}
/* L40: */
}
return 0;
/* End of ZUNGR2 */
} /* zungr2_ */
/* Subroutine */ int zungrq_(integer *m, integer *n, integer *k,
doublecomplex *a, integer *lda, doublecomplex *tau, doublecomplex *
work, integer *lwork, integer *info)
{
/* -- 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
Purpose
=======
ZUNGRQ generates an M-by-N complex 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 ZGERQF.
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) COMPLEX*16 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 ZGERQF 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) COMPLEX*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i), as returned by ZGERQF.
WORK (workspace/output) COMPLEX*16 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.
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
=====================================================================
Test the input arguments
Parameter adjustments */
/* 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, i__5;
/* Local variables */
static integer i__, j, l, nbmin, iinfo, ib, nb, ii, kk;
extern /* Subroutine */ int zungr2_(integer *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *);
static integer nx;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern /* Subroutine */ int zlarfb_(char *, char *, char *, char *,
integer *, integer *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *);
static integer ldwork;
extern /* Subroutine */ int zlarft_(char *, char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, doublecomplex *,
integer *);
static integer lwkopt;
static logical lquery;
static integer iws;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--tau;
--work;
/* Function Body */
*info = 0;
nb = ilaenv_(&c__1, "ZUNGRQ", " ", m, n, k, &c_n1, (ftnlen)6, (ftnlen)1);
lwkopt = max(1,*m) * nb;
work[1].r = (doublereal) lwkopt, work[1].i = 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;
} else if (*lwork < max(1,*m) && ! lquery) {
*info = -8;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZUNGRQ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*m <= 0) {
work[1].r = 1., work[1].i = 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, "ZUNGRQ", " ", m, n, k, &c_n1, (
ftnlen)6, (ftnlen)1);
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, "ZUNGRQ", " ", m, n, k, &c_n1,
(ftnlen)6, (ftnlen)1);
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__) {
i__3 = a_subscr(i__, j);
a[i__3].r = 0., a[i__3].i = 0.;
/* 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;
zungr2_(&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;
zlarft_("Backward", "Rowwise", &i__3, &ib, &a_ref(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;
zlarfb_("Right", "Conjugate transpose", "Backward", "Rowwise",
&i__3, &i__4, &ib, &a_ref(ii, 1), 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;
zungr2_(&ib, &i__3, &ib, &a_ref(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 <= i__3; ++l) {
i__4 = ii + ib - 1;
for (j = ii; j <= i__4; ++j) {
i__5 = a_subscr(j, l);
a[i__5].r = 0., a[i__5].i = 0.;
/* L30: */
}
/* L40: */
}
/* L50: */
}
}
work[1].r = (doublereal) iws, work[1].i = 0.;
return 0;
/* End of ZUNGRQ */
} /* zungrq_ */
/* Subroutine */ int ctrrfs_(char *uplo, char *trans, char *diag, integer *n,
integer *nrhs, complex *a, integer *lda, complex *b, integer *ldb,
complex *x, integer *ldx, real *ferr, real *berr, complex *work, real
*rwork, 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
=======
CTRRFS provides error bounds and backward error estimates for the
solution to a system of linear equations with a triangular
coefficient matrix.
The solution matrix X must be computed by CTRTRS or some other
means before entering this routine. CTRRFS does not do iterative
refinement because doing so cannot improve the backward error.
Arguments
=========
UPLO (input) CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Conjugate transpose)
DIAG (input) CHARACTER*1
= 'N': A is non-unit triangular;
= 'U': A is unit triangular.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A (input) COMPLEX array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) COMPLEX array, dimension (LDB,NRHS)
The right hand side matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (input) COMPLEX array, dimension (LDX,NRHS)
The solution matrix X.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for RCOND, and is almost always a slight
overestimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
WORK (workspace) COMPLEX array, dimension (2*N)
RWORK (workspace) REAL array, dimension (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 integer c__1 = 1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset, i__1, i__2,
i__3, i__4, i__5;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static integer kase;
static real safe1, safe2;
static integer i__, j, k;
static real s;
extern logical lsame_(char *, char *);
extern /* Subroutine */ int ccopy_(integer *, complex *, integer *,
complex *, integer *), caxpy_(integer *, complex *, complex *,
integer *, complex *, integer *);
static logical upper;
extern /* Subroutine */ int ctrmv_(char *, char *, char *, integer *,
complex *, integer *, complex *, integer *), ctrsv_(char *, char *, char *, integer *, complex *,
integer *, complex *, integer *), clacon_(
integer *, complex *, complex *, real *, integer *);
static real xk;
extern doublereal slamch_(char *);
static integer nz;
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
static logical notran;
static char transn[1], transt[1];
static logical nounit;
static real lstres, eps;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define x_subscr(a_1,a_2) (a_2)*x_dim1 + a_1
#define x_ref(a_1,a_2) x[x_subscr(a_1,a_2)]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1 * 1;
x -= x_offset;
--ferr;
--berr;
--work;
--rwork;
/* Function Body */
*info = 0;
upper = lsame_(uplo, "U");
notran = lsame_(trans, "N");
nounit = lsame_(diag, "N");
if (! upper && ! lsame_(uplo, "L")) {
*info = -1;
} else if (! notran && ! lsame_(trans, "T") && !
lsame_(trans, "C")) {
*info = -2;
} else if (! nounit && ! lsame_(diag, "U")) {
*info = -3;
} else if (*n < 0) {
*info = -4;
} else if (*nrhs < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldx < max(1,*n)) {
*info = -11;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTRRFS", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0 || *nrhs == 0) {
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
ferr[j] = 0.f;
berr[j] = 0.f;
/* L10: */
}
return 0;
}
if (notran) {
*(unsigned char *)transn = 'N';
*(unsigned char *)transt = 'C';
} else {
*(unsigned char *)transn = 'C';
*(unsigned char *)transt = 'N';
}
/* NZ = maximum number of nonzero elements in each row of A, plus 1 */
nz = *n + 1;
eps = slamch_("Epsilon");
safmin = slamch_("Safe minimum");
safe1 = nz * safmin;
safe2 = safe1 / eps;
/* Do for each right hand side */
i__1 = *nrhs;
for (j = 1; j <= i__1; ++j) {
/* Compute residual R = B - op(A) * X,
where op(A) = A, A**T, or A**H, depending on TRANS. */
ccopy_(n, &x_ref(1, j), &c__1, &work[1], &c__1);
ctrmv_(uplo, trans, diag, n, &a[a_offset], lda, &work[1], &c__1);
q__1.r = -1.f, q__1.i = 0.f;
caxpy_(n, &q__1, &b_ref(1, j), &c__1, &work[1], &c__1);
/* Compute componentwise relative backward error from formula
max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
where abs(Z) is the componentwise absolute value of the matrix
or vector Z. If the i-th component of the denominator is less
than SAFE2, then SAFE1 is added to the i-th components of the
numerator and denominator before dividing. */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = b_subscr(i__, j);
rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
b_ref(i__, j)), dabs(r__2));
/* L20: */
}
if (notran) {
/* Compute abs(A)*abs(X) + abs(B). */
if (upper) {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&a_ref(i__, k)), dabs(r__2))
) * xk;
/* L30: */
}
/* L40: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&a_ref(i__, k)), dabs(r__2))
) * xk;
/* L50: */
}
rwork[k] += xk;
/* L60: */
}
}
} else {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&a_ref(i__, k)), dabs(r__2))
) * xk;
/* L70: */
}
/* L80: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
rwork[i__] += ((r__1 = a[i__4].r, dabs(r__1)) + (
r__2 = r_imag(&a_ref(i__, k)), dabs(r__2))
) * xk;
/* L90: */
}
rwork[k] += xk;
/* L100: */
}
}
}
} else {
/* Compute abs(A**H)*abs(X) + abs(B). */
if (upper) {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = k;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * ((
r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x_ref(i__, j)), dabs(r__4)));
/* L110: */
}
rwork[k] += s;
/* L120: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = k - 1;
for (i__ = 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * ((
r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x_ref(i__, j)), dabs(r__4)));
/* L130: */
}
rwork[k] += s;
/* L140: */
}
}
} else {
if (nounit) {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
s = 0.f;
i__3 = *n;
for (i__ = k; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * ((
r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x_ref(i__, j)), dabs(r__4)));
/* L150: */
}
rwork[k] += s;
/* L160: */
}
} else {
i__2 = *n;
for (k = 1; k <= i__2; ++k) {
i__3 = x_subscr(k, j);
s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&
x_ref(k, j)), dabs(r__2));
i__3 = *n;
for (i__ = k + 1; i__ <= i__3; ++i__) {
i__4 = a_subscr(i__, k);
i__5 = x_subscr(i__, j);
s += ((r__1 = a[i__4].r, dabs(r__1)) + (r__2 =
r_imag(&a_ref(i__, k)), dabs(r__2))) * ((
r__3 = x[i__5].r, dabs(r__3)) + (r__4 =
r_imag(&x_ref(i__, j)), dabs(r__4)));
/* L170: */
}
rwork[k] += s;
/* L180: */
}
}
}
}
s = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2))) / rwork[i__];
s = dmax(r__3,r__4);
} else {
/* Computing MAX */
i__3 = i__;
r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__]
+ safe1);
s = dmax(r__3,r__4);
}
/* L190: */
}
berr[j] = s;
/* Bound error from formula
norm(X - XTRUE) / norm(X) .le. FERR =
norm( abs(inv(op(A)))*
( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
where
norm(Z) is the magnitude of the largest component of Z
inv(op(A)) is the inverse of op(A)
abs(Z) is the componentwise absolute value of the matrix or
vector Z
NZ is the maximum number of nonzeros in any row of A, plus 1
EPS is machine epsilon
The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
is incremented by SAFE1 if the i-th component of
abs(op(A))*abs(X) + abs(B) is less than SAFE2.
Use CLACON to estimate the infinity-norm of the matrix
inv(op(A)) * diag(W),
where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
if (rwork[i__] > safe2) {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__];
} else {
i__3 = i__;
rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[
i__] + safe1;
}
/* L200: */
}
kase = 0;
L210:
clacon_(n, &work[*n + 1], &work[1], &ferr[j], &kase);
if (kase != 0) {
if (kase == 1) {
/* Multiply by diag(W)*inv(op(A)**H). */
ctrsv_(uplo, transt, diag, n, &a[a_offset], lda, &work[1], &
c__1);
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L220: */
}
} else {
/* Multiply by inv(op(A))*diag(W). */
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
i__3 = i__;
i__4 = i__;
i__5 = i__;
q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4]
* work[i__5].i;
work[i__3].r = q__1.r, work[i__3].i = q__1.i;
/* L230: */
}
ctrsv_(uplo, transn, diag, n, &a[a_offset], lda, &work[1], &
c__1);
}
goto L210;
}
/* Normalize error. */
lstres = 0.f;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
i__3 = x_subscr(i__, j);
r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&x_ref(i__, j)), dabs(r__2));
lstres = dmax(r__3,r__4);
/* L240: */
}
if (lstres != 0.f) {
ferr[j] /= lstres;
}
/* L250: */
}
return 0;
/* End of CTRRFS */
} /* ctrrfs_ */