/* Subroutine */ int cggesx_(char *jobvsl, char *jobvsr, char *sort, L_fp
selctg, char *sense, integer *n, complex *a, integer *lda, complex *b,
integer *ldb, integer *sdim, complex *alpha, complex *beta, complex *
vsl, integer *ldvsl, complex *vsr, integer *ldvsr, real *rconde, real
*rcondv, complex *work, integer *lwork, real *rwork, integer *iwork,
integer *liwork, logical *bwork, integer *info)
{
/* -- LAPACK driver 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
=======
CGGESX computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, the complex Schur form (S,T),
and, optionally, the left and/or right matrices of Schur vectors (VSL
and VSR). This gives the generalized Schur factorization
(A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
where (VSR)**H is the conjugate-transpose of VSR.
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
triangular matrix S and the upper triangular matrix T; computes
a reciprocal condition number for the average of the selected
eigenvalues (RCONDE); and computes a reciprocal condition number for
the right and left deflating subspaces corresponding to the selected
eigenvalues (RCONDV). The leading columns of VSL and VSR then form
an orthonormal basis for the corresponding left and right eigenspaces
(deflating subspaces).
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 a
reasonable interpretation for beta=0 or for both being zero.
A pair of matrices (S,T) is in generalized complex Schur form if T is
upper triangular with non-negative diagonal and S is upper
triangular.
Arguments
=========
JOBVSL (input) CHARACTER*1
= 'N': do not compute the left Schur vectors;
= 'V': compute the left Schur vectors.
JOBVSR (input) CHARACTER*1
= 'N': do not compute the right Schur vectors;
= 'V': compute the right Schur vectors.
SORT (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELCTG).
SELCTG (input) LOGICAL FUNCTION of two COMPLEX arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned), in this
case INFO is set to N+3 see INFO below).
SENSE (input) CHARACTER
Determines which reciprocal condition numbers are computed.
= 'N' : None are computed;
= 'E' : Computed for average of selected eigenvalues only;
= 'V' : Computed for selected deflating subspaces only;
= 'B' : Computed for both.
If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
SDIM (output) INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.
ALPHA (output) COMPLEX array, dimension (N)
BETA (output) COMPLEX array, dimension (N)
On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the
generalized eigenvalues. ALPHA(j) and BETA(j),j=1,...,N are
the diagonals of the complex Schur form (S,T). BETA(j) will
be non-negative real.
Note: the quotients ALPHA(j)/BETA(j) may easily over- or
underflow, and BETA(j) may even be zero. Thus, the user
should avoid naively computing the ratio alpha/beta.
However, ALPHA will be always less than and usually
comparable with norm(A) in magnitude, and BETA always less
than and usually comparable with norm(B).
VSL (output) COMPLEX array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
VSR (output) COMPLEX array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
RCONDE (output) REAL array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
reciprocal condition numbers for the average of the selected
eigenvalues.
Not referenced if SENSE = 'N' or 'V'.
RCONDV (output) REAL array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition number for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 2*N.
If SENSE = 'E', 'V', or 'B',
LWORK >= MAX(2*N, 2*SDIM*(N-SDIM)).
RWORK (workspace) REAL array, dimension ( 8*N )
Real workspace.
IWORK (workspace/output) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N'.
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array WORK. LIWORK >= N+2.
BWORK (workspace) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. (A,B) are not in Schur
form, but ALPHA(j) and BETA(j) should be correct for
j=INFO+1,...,N.
> N: =N+1: other than QZ iteration failed in CHGEQZ
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE. This could also
be caused due to scaling.
=N+3: reordering failed in CTGSEN.
=====================================================================
Decode the input arguments
Parameter adjustments */
/* Table of constant values */
static complex c_b1 = {0.f,0.f};
static complex c_b2 = {1.f,0.f};
static integer c__1 = 1;
static integer c__0 = 0;
static integer c_n1 = -1;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vsl_dim1, vsl_offset,
vsr_dim1, vsr_offset, i__1, i__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer ijob;
static real anrm, bnrm;
static integer ierr, itau, iwrk, i__;
extern logical lsame_(char *, char *);
static integer ileft, icols;
static logical cursl, ilvsl, ilvsr;
static integer irwrk, irows;
extern /* Subroutine */ int cggbak_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, complex *, integer *,
integer *), cggbal_(char *, integer *, complex *,
integer *, complex *, integer *, integer *, integer *, real *,
real *, real *, integer *), slabad_(real *, real *);
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
static real pl;
extern /* Subroutine */ int cgghrd_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, complex *,
integer *, complex *, integer *, integer *),
clascl_(char *, integer *, integer *, real *, real *, integer *,
integer *, complex *, integer *, integer *);
static real pr;
static logical ilascl, ilbscl;
extern /* Subroutine */ int cgeqrf_(integer *, integer *, complex *,
integer *, complex *, complex *, integer *, integer *), clacpy_(
char *, integer *, integer *, complex *, integer *, complex *,
integer *), claset_(char *, integer *, integer *, complex
*, complex *, complex *, integer *), xerbla_(char *,
integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern doublereal slamch_(char *);
static real bignum;
extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *,
integer *, integer *, complex *, integer *, complex *, integer *,
complex *, complex *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *),
ctgsen_(integer *, logical *, logical *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, complex *,
complex *, integer *, complex *, integer *, integer *, real *,
real *, real *, complex *, integer *, integer *, integer *,
integer *);
static integer ijobvl, iright, ijobvr;
static logical wantsb;
static integer liwmin;
static logical wantse, lastsl;
static real anrmto, bnrmto;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
static integer minwrk, maxwrk;
static logical wantsn;
static real smlnum;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
static logical wantst, wantsv;
static real dif[2];
static integer ihi, ilo;
static real 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 vsl_subscr(a_1,a_2) (a_2)*vsl_dim1 + a_1
#define vsl_ref(a_1,a_2) vsl[vsl_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;
--alpha;
--beta;
vsl_dim1 = *ldvsl;
vsl_offset = 1 + vsl_dim1 * 1;
vsl -= vsl_offset;
vsr_dim1 = *ldvsr;
vsr_offset = 1 + vsr_dim1 * 1;
vsr -= vsr_offset;
--rconde;
--rcondv;
--work;
--rwork;
--iwork;
--bwork;
/* Function Body */
if (lsame_(jobvsl, "N")) {
ijobvl = 1;
ilvsl = FALSE_;
} else if (lsame_(jobvsl, "V")) {
ijobvl = 2;
ilvsl = TRUE_;
} else {
ijobvl = -1;
ilvsl = FALSE_;
}
if (lsame_(jobvsr, "N")) {
ijobvr = 1;
ilvsr = FALSE_;
} else if (lsame_(jobvsr, "V")) {
ijobvr = 2;
ilvsr = TRUE_;
} else {
ijobvr = -1;
ilvsr = FALSE_;
}
wantst = lsame_(sort, "S");
wantsn = lsame_(sense, "N");
wantse = lsame_(sense, "E");
wantsv = lsame_(sense, "V");
wantsb = lsame_(sense, "B");
if (wantsn) {
ijob = 0;
iwork[1] = 1;
} else if (wantse) {
ijob = 1;
} else if (wantsv) {
ijob = 2;
} else if (wantsb) {
ijob = 4;
}
/* Test the input arguments */
*info = 0;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (! wantst && ! lsame_(sort, "N")) {
*info = -3;
} else if (! (wantsn || wantse || wantsv || wantsb) || ! wantst && !
wantsn) {
*info = -5;
} else if (*n < 0) {
*info = -6;
} else if (*lda < max(1,*n)) {
*info = -8;
} else if (*ldb < max(1,*n)) {
*info = -10;
} else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) {
*info = -15;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*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) {
/* Computing MAX */
i__1 = 1, i__2 = *n << 1;
minwrk = max(i__1,i__2);
maxwrk = *n + *n * ilaenv_(&c__1, "CGEQRF", " ", n, &c__1, n, &c__0, (
ftnlen)6, (ftnlen)1);
if (ilvsl) {
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "CUNGQR", " ", n, &
c__1, n, &c_n1, (ftnlen)6, (ftnlen)1);
maxwrk = max(i__1,i__2);
}
work[1].r = (real) maxwrk, work[1].i = 0.f;
}
if (! wantsn) {
liwmin = *n + 2;
} else {
liwmin = 1;
}
iwork[1] = liwmin;
if (*info == 0 && *lwork < minwrk) {
*info = -21;
} else if (*info == 0 && ijob >= 1) {
if (*liwork < liwmin) {
*info = -24;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGGESX", &i__1);
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*sdim = 0;
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1.f / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]);
ilascl = FALSE_;
if (anrm > 0.f && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
clascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]);
ilbscl = FALSE_;
if (bnrm > 0.f && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
clascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute the matrix to make it more nearly triangular
(Real Workspace: need 6*N) */
ileft = 1;
iright = *n + 1;
irwrk = iright + *n;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwrk], &ierr);
/* Reduce B to triangular form (QR decomposition of B)
(Complex Workspace: need N, prefer N*NB) */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = 1;
iwrk = itau + irows;
i__1 = *lwork + 1 - iwrk;
cgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwrk], &
i__1, &ierr);
/* Apply the unitary transformation to matrix A
(Complex Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
cunmqr_("L", "C", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[
itau], &a_ref(ilo, ilo), lda, &work[iwrk], &i__1, &ierr);
/* Initialize VSL
(Complex Workspace: need N, prefer N*NB) */
if (ilvsl) {
claset_("Full", n, n, &c_b1, &c_b2, &vsl[vsl_offset], ldvsl);
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo +
1, ilo), ldvsl);
i__1 = *lwork + 1 - iwrk;
cungqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau]
, &work[iwrk], &i__1, &ierr);
}
/* Initialize VSR */
if (ilvsr) {
claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form
(Workspace: none needed) */
cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &ierr);
*sdim = 0;
/* Perform QZ algorithm, computing Schur vectors if desired
(Complex Workspace: need N)
(Real Workspace: need N) */
iwrk = itau;
i__1 = *lwork + 1 - iwrk;
chgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl, &
vsr[vsr_offset], ldvsr, &work[iwrk], &i__1, &rwork[irwrk], &ierr);
if (ierr != 0) {
if (ierr > 0 && ierr <= *n) {
*info = ierr;
} else if (ierr > *n && ierr <= *n << 1) {
*info = ierr - *n;
} else {
*info = *n + 1;
}
goto L40;
}
/* Sort eigenvalues ALPHA/BETA and compute the reciprocal of
condition number(s) */
if (wantst) {
/* Undo scaling on eigenvalues before SELCTGing */
if (ilascl) {
clascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n,
&ierr);
}
if (ilbscl) {
clascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n,
&ierr);
}
/* Select eigenvalues */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
bwork[i__] = (*selctg)(&alpha[i__], &beta[i__]);
/* L10: */
}
/* Reorder eigenvalues, transform Generalized Schur vectors, and
compute reciprocal condition numbers
(Complex Workspace: If IJOB >= 1, need MAX(1, 2*SDIM*(N-SDIM))
otherwise, need 1 ) */
i__1 = *lwork - iwrk + 1;
ctgsen_(&ijob, &ilvsl, &ilvsr, &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl,
&vsr[vsr_offset], ldvsr, sdim, &pl, &pr, dif, &work[iwrk], &
i__1, &iwork[1], liwork, &ierr);
if (ijob >= 1) {
/* Computing MAX */
i__1 = maxwrk, i__2 = (*sdim << 1) * (*n - *sdim);
maxwrk = max(i__1,i__2);
}
if (ierr == -21) {
/* not enough complex workspace */
*info = -21;
} else {
rconde[1] = pl;
rconde[2] = pl;
rcondv[1] = dif[0];
rcondv[2] = dif[1];
if (ierr == 1) {
*info = *n + 3;
}
}
}
/* Apply permutation to VSL and VSR
(Workspace: none needed) */
if (ilvsl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsl[vsl_offset], ldvsl, &ierr);
}
if (ilvsr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsr[vsr_offset], ldvsr, &ierr);
}
/* Undo scaling */
if (ilascl) {
clascl_("U", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, &
ierr);
clascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n, &
ierr);
}
if (ilbscl) {
clascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
ierr);
clascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
/* L20: */
if (wantst) {
/* Check if reordering is correct */
lastsl = TRUE_;
*sdim = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
cursl = (*selctg)(&alpha[i__], &beta[i__]);
if (cursl) {
++(*sdim);
}
if (cursl && ! lastsl) {
*info = *n + 2;
}
lastsl = cursl;
/* L30: */
}
}
L40:
work[1].r = (real) maxwrk, work[1].i = 0.f;
iwork[1] = liwmin;
return 0;
/* End of CGGESX */
} /* cggesx_ */
/* Subroutine */ int cgges_(char *jobvsl, char *jobvsr, char *sort, L_fp
selctg, integer *n, complex *a, integer *lda, complex *b, integer *
ldb, integer *sdim, complex *alpha, complex *beta, complex *vsl,
integer *ldvsl, complex *vsr, integer *ldvsr, complex *work, integer *
lwork, real *rwork, logical *bwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vsl_dim1, vsl_offset,
vsr_dim1, vsr_offset, i__1, i__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__;
real dif[2];
integer ihi, ilo;
real eps, anrm, bnrm;
integer idum[1], ierr, itau, iwrk;
real pvsl, pvsr;
extern logical lsame_(char *, char *);
integer ileft, icols;
logical cursl, ilvsl, ilvsr;
integer irwrk, irows;
extern /* Subroutine */ int cggbak_(char *, char *, integer *, integer *,
integer *, real *, real *, integer *, complex *, integer *,
integer *), cggbal_(char *, integer *, complex *,
integer *, complex *, integer *, integer *, integer *, real *,
real *, real *, integer *), slabad_(real *, real *);
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
extern /* Subroutine */ int cgghrd_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, complex *,
integer *, complex *, integer *, integer *),
clascl_(char *, integer *, integer *, real *, real *, integer *,
integer *, complex *, integer *, integer *);
logical ilascl, ilbscl;
extern /* Subroutine */ int cgeqrf_(integer *, integer *, complex *,
integer *, complex *, 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 *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
real bignum;
extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *,
integer *, integer *, complex *, integer *, complex *, integer *,
complex *, complex *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *),
ctgsen_(integer *, logical *, logical *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, complex *,
complex *, integer *, complex *, integer *, integer *, real *,
real *, real *, complex *, integer *, integer *, integer *,
integer *);
integer ijobvl, iright, ijobvr;
real anrmto;
integer lwkmin;
logical lastsl;
real bnrmto;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *),
cunmqr_(char *, char *, integer *, integer *, integer *, complex
*, integer *, complex *, complex *, integer *, complex *, integer
*, integer *);
real smlnum;
logical wantst, lquery;
integer lwkopt;
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* .. Function Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGGES computes for a pair of N-by-N complex nonsymmetric matrices */
/* (A,B), the generalized eigenvalues, the generalized complex Schur */
/* form (S, T), and optionally left and/or right Schur vectors (VSL */
/* and VSR). This gives the generalized Schur factorization */
/* (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) */
/* where (VSR)**H is the conjugate-transpose of VSR. */
/* Optionally, it also orders the eigenvalues so that a selected cluster */
/* of eigenvalues appears in the leading diagonal blocks of the upper */
/* triangular matrix S and the upper triangular matrix T. The leading */
/* columns of VSL and VSR then form an unitary basis for the */
/* corresponding left and right eigenspaces (deflating subspaces). */
/* (If only the generalized eigenvalues are needed, use the driver */
/* CGGEV instead, which is faster.) */
/* 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 a */
/* reasonable interpretation for beta=0, and even for both being zero. */
/* A pair of matrices (S,T) is in generalized complex Schur form if S */
/* and T are upper triangular and, in addition, the diagonal elements */
/* of T are non-negative real numbers. */
/* Arguments */
/* ========= */
/* JOBVSL (input) CHARACTER*1 */
/* = 'N': do not compute the left Schur vectors; */
/* = 'V': compute the left Schur vectors. */
/* JOBVSR (input) CHARACTER*1 */
/* = 'N': do not compute the right Schur vectors; */
/* = 'V': compute the right Schur vectors. */
/* SORT (input) CHARACTER*1 */
/* Specifies whether or not to order the eigenvalues on the */
/* diagonal of the generalized Schur form. */
/* = 'N': Eigenvalues are not ordered; */
/* = 'S': Eigenvalues are ordered (see SELCTG). */
/* SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX arguments */
/* SELCTG must be declared EXTERNAL in the calling subroutine. */
/* If SORT = 'N', SELCTG is not referenced. */
/* If SORT = 'S', SELCTG is used to select eigenvalues to sort */
/* to the top left of the Schur form. */
/* An eigenvalue ALPHA(j)/BETA(j) is selected if */
/* SELCTG(ALPHA(j),BETA(j)) is true. */
/* Note that a selected complex eigenvalue may no longer satisfy */
/* SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since */
/* ordering may change the value of complex eigenvalues */
/* (especially if the eigenvalue is ill-conditioned), in this */
/* case INFO is set to N+2 (See INFO below). */
/* N (input) INTEGER */
/* The order of the matrices A, B, VSL, and VSR. N >= 0. */
/* A (input/output) COMPLEX array, dimension (LDA, N) */
/* On entry, the first of the pair of matrices. */
/* On exit, A has been overwritten by its generalized Schur */
/* form S. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) COMPLEX array, dimension (LDB, N) */
/* On entry, the second of the pair of matrices. */
/* On exit, B has been overwritten by its generalized Schur */
/* form T. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* SDIM (output) INTEGER */
/* If SORT = 'N', SDIM = 0. */
/* If SORT = 'S', SDIM = number of eigenvalues (after sorting) */
/* for which SELCTG is true. */
/* ALPHA (output) COMPLEX array, dimension (N) */
/* BETA (output) COMPLEX array, dimension (N) */
/* On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the */
/* generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), */
/* j=1,...,N are the diagonals of the complex Schur form (A,B) */
/* output by CGGES. The BETA(j) will be non-negative real. */
/* Note: the quotients ALPHA(j)/BETA(j) may easily over- or */
/* underflow, and BETA(j) may even be zero. Thus, the user */
/* should avoid naively computing the ratio alpha/beta. */
/* However, ALPHA will be always less than and usually */
/* comparable with norm(A) in magnitude, and BETA always less */
/* than and usually comparable with norm(B). */
/* VSL (output) COMPLEX array, dimension (LDVSL,N) */
/* If JOBVSL = 'V', VSL will contain the left Schur vectors. */
/* Not referenced if JOBVSL = 'N'. */
/* LDVSL (input) INTEGER */
/* The leading dimension of the matrix VSL. LDVSL >= 1, and */
/* if JOBVSL = 'V', LDVSL >= N. */
/* VSR (output) COMPLEX array, dimension (LDVSR,N) */
/* If JOBVSR = 'V', VSR will contain the right Schur vectors. */
/* Not referenced if JOBVSR = 'N'. */
/* LDVSR (input) INTEGER */
/* The leading dimension of the matrix VSR. LDVSR >= 1, and */
/* if JOBVSR = 'V', LDVSR >= N. */
/* WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= max(1,2*N). */
/* For good performance, LWORK must generally be larger. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* RWORK (workspace) REAL array, dimension (8*N) */
/* BWORK (workspace) LOGICAL array, dimension (N) */
/* Not referenced if SORT = 'N'. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* =1,...,N: */
/* The QZ iteration failed. (A,B) are not in Schur */
/* form, but ALPHA(j) and BETA(j) should be correct for */
/* j=INFO+1,...,N. */
/* > N: =N+1: other than QZ iteration failed in CHGEQZ */
/* =N+2: after reordering, roundoff changed values of */
/* some complex eigenvalues so that leading */
/* eigenvalues in the Generalized Schur form no */
/* longer satisfy SELCTG=.TRUE. This could also */
/* be caused due to scaling. */
/* =N+3: reordering falied in CTGSEN. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode the input arguments */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
vsl_dim1 = *ldvsl;
vsl_offset = 1 + vsl_dim1;
vsl -= vsl_offset;
vsr_dim1 = *ldvsr;
vsr_offset = 1 + vsr_dim1;
vsr -= vsr_offset;
--work;
--rwork;
--bwork;
/* Function Body */
if (lsame_(jobvsl, "N")) {
ijobvl = 1;
ilvsl = FALSE_;
} else if (lsame_(jobvsl, "V")) {
ijobvl = 2;
ilvsl = TRUE_;
} else {
ijobvl = -1;
ilvsl = FALSE_;
}
if (lsame_(jobvsr, "N")) {
ijobvr = 1;
ilvsr = FALSE_;
} else if (lsame_(jobvsr, "V")) {
ijobvr = 2;
ilvsr = TRUE_;
} else {
ijobvr = -1;
ilvsr = FALSE_;
}
wantst = lsame_(sort, "S");
/* Test the input arguments */
*info = 0;
lquery = *lwork == -1;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (! wantst && ! lsame_(sort, "N")) {
*info = -3;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) {
*info = -14;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*info = -16;
}
/* 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.) */
if (*info == 0) {
/* Computing MAX */
i__1 = 1, i__2 = *n << 1;
lwkmin = max(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + *n * ilaenv_(&c__1, "CGEQRF", " ", n, &c__1, n,
&c__0);
lwkopt = max(i__1,i__2);
/* Computing MAX */
i__1 = lwkopt, i__2 = *n + *n * ilaenv_(&c__1, "CUNMQR", " ", n, &
c__1, n, &c_n1);
lwkopt = max(i__1,i__2);
if (ilvsl) {
/* Computing MAX */
i__1 = lwkopt, i__2 = *n + *n * ilaenv_(&c__1, "CUNGQR", " ", n, &
c__1, n, &c_n1);
lwkopt = max(i__1,i__2);
}
work[1].r = (real) lwkopt, work[1].i = 0.f;
if (*lwork < lwkmin && ! lquery) {
*info = -18;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGGES ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*sdim = 0;
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1.f / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]);
ilascl = FALSE_;
if (anrm > 0.f && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
clascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]);
ilbscl = FALSE_;
if (bnrm > 0.f && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
clascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute the matrix to make it more nearly triangular */
/* (Real Workspace: need 6*N) */
ileft = 1;
iright = *n + 1;
irwrk = iright + *n;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwrk], &ierr);
/* Reduce B to triangular form (QR decomposition of B) */
/* (Complex Workspace: need N, prefer N*NB) */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = 1;
iwrk = itau + irows;
i__1 = *lwork + 1 - iwrk;
cgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwrk], &i__1, &ierr);
/* Apply the orthogonal transformation to matrix A */
/* (Complex Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
cunmqr_("L", "C", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwrk], &i__1, &
ierr);
/* Initialize VSL */
/* (Complex Workspace: need N, prefer N*NB) */
if (ilvsl) {
claset_("Full", n, n, &c_b1, &c_b2, &vsl[vsl_offset], ldvsl);
if (irows > 1) {
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vsl[
ilo + 1 + ilo * vsl_dim1], ldvsl);
}
i__1 = *lwork + 1 - iwrk;
cungqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, &
work[itau], &work[iwrk], &i__1, &ierr);
}
/* Initialize VSR */
if (ilvsr) {
claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
/* (Workspace: none needed) */
cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &ierr);
*sdim = 0;
/* Perform QZ algorithm, computing Schur vectors if desired */
/* (Complex Workspace: need N) */
/* (Real Workspace: need N) */
iwrk = itau;
i__1 = *lwork + 1 - iwrk;
chgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl, &
vsr[vsr_offset], ldvsr, &work[iwrk], &i__1, &rwork[irwrk], &ierr);
if (ierr != 0) {
if (ierr > 0 && ierr <= *n) {
*info = ierr;
} else if (ierr > *n && ierr <= *n << 1) {
*info = ierr - *n;
} else {
*info = *n + 1;
}
goto L30;
}
/* Sort eigenvalues ALPHA/BETA if desired */
/* (Workspace: none needed) */
if (wantst) {
/* Undo scaling on eigenvalues before selecting */
if (ilascl) {
clascl_("G", &c__0, &c__0, &anrm, &anrmto, n, &c__1, &alpha[1], n,
&ierr);
}
if (ilbscl) {
clascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, &c__1, &beta[1], n,
&ierr);
}
/* Select eigenvalues */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
bwork[i__] = (*selctg)(&alpha[i__], &beta[i__]);
/* L10: */
}
i__1 = *lwork - iwrk + 1;
ctgsen_(&c__0, &ilvsl, &ilvsr, &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vsl[vsl_offset], ldvsl,
&vsr[vsr_offset], ldvsr, sdim, &pvsl, &pvsr, dif, &work[iwrk],
&i__1, idum, &c__1, &ierr);
if (ierr == 1) {
*info = *n + 3;
}
}
/* Apply back-permutation to VSL and VSR */
/* (Workspace: none needed) */
if (ilvsl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsl[vsl_offset], ldvsl, &ierr);
}
if (ilvsr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsr[vsr_offset], ldvsr, &ierr);
}
/* Undo scaling */
if (ilascl) {
clascl_("U", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, &
ierr);
clascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alpha[1], n, &
ierr);
}
if (ilbscl) {
clascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
ierr);
clascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
if (wantst) {
/* Check if reordering is correct */
lastsl = TRUE_;
*sdim = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
cursl = (*selctg)(&alpha[i__], &beta[i__]);
if (cursl) {
++(*sdim);
}
if (cursl && ! lastsl) {
*info = *n + 2;
}
lastsl = cursl;
/* L20: */
}
}
L30:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGGES */
} /* cgges_ */
