/* 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 cchkgl_(integer *nin, integer *nout)
{
/* Format strings */
static char fmt_9999[] = "(\002 .. test output of CGGBAL .. \002)";
static char fmt_9998[] = "(\002 ratio of largest test error "
" = \002,e12.3)";
static char fmt_9997[] = "(\002 example number where info is not zero "
" = \002,i4)";
static char fmt_9996[] = "(\002 example number where ILO or IHI is wrong"
" = \002,i4)";
static char fmt_9995[] = "(\002 example number having largest error "
" = \002,i4)";
static char fmt_9994[] = "(\002 number of examples where info is not 0 "
" = \002,i4)";
static char fmt_9993[] = "(\002 total number of examples tested "
" = \002,i4)";
/* System generated locals */
integer i__1, i__2, i__3, i__4;
real r__1, r__2, r__3;
complex q__1;
/* Builtin functions */
integer s_rsle(cilist *), do_lio(integer *, integer *, char *, ftnlen),
e_rsle(void);
double c_abs(complex *);
integer s_wsfe(cilist *), e_wsfe(void), do_fio(integer *, char *, ftnlen);
/* Local variables */
complex a[400] /* was [20][20] */, b[400] /* was [20][20] */;
integer i__, j, n;
complex ain[400] /* was [20][20] */, bin[400] /* was [20][20] */;
integer ihi, ilo;
real eps;
integer knt, info, lmax[3];
real rmax, vmax, work[120];
integer ihiin, ninfo, iloin;
real anorm, bnorm;
extern /* Subroutine */ int cggbal_(char *, integer *, complex *, integer
*, complex *, integer *, integer *, integer *, real *, real *,
real *, integer *);
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
real lscale[20];
extern doublereal slamch_(char *);
real rscale[20], lsclin[20], rsclin[20];
/* Fortran I/O blocks */
static cilist io___6 = { 0, 0, 0, 0, 0 };
static cilist io___9 = { 0, 0, 0, 0, 0 };
static cilist io___12 = { 0, 0, 0, 0, 0 };
static cilist io___14 = { 0, 0, 0, 0, 0 };
static cilist io___17 = { 0, 0, 0, 0, 0 };
static cilist io___19 = { 0, 0, 0, 0, 0 };
static cilist io___21 = { 0, 0, 0, 0, 0 };
static cilist io___23 = { 0, 0, 0, 0, 0 };
static cilist io___34 = { 0, 0, 0, fmt_9999, 0 };
static cilist io___35 = { 0, 0, 0, fmt_9998, 0 };
static cilist io___36 = { 0, 0, 0, fmt_9997, 0 };
static cilist io___37 = { 0, 0, 0, fmt_9996, 0 };
static cilist io___38 = { 0, 0, 0, fmt_9995, 0 };
static cilist io___39 = { 0, 0, 0, fmt_9994, 0 };
static cilist io___40 = { 0, 0, 0, fmt_9993, 0 };
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CCHKGL tests CGGBAL, a routine for balancing a matrix pair (A, B). */
/* Arguments */
/* ========= */
/* NIN (input) INTEGER */
/* The logical unit number for input. NIN > 0. */
/* NOUT (input) INTEGER */
/* The logical unit number for output. NOUT > 0. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
lmax[0] = 0;
lmax[1] = 0;
lmax[2] = 0;
ninfo = 0;
knt = 0;
rmax = 0.f;
eps = slamch_("Precision");
L10:
io___6.ciunit = *nin;
s_rsle(&io___6);
do_lio(&c__3, &c__1, (char *)&n, (ftnlen)sizeof(integer));
e_rsle();
if (n == 0) {
goto L90;
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___9.ciunit = *nin;
s_rsle(&io___9);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__6, &c__1, (char *)&a[i__ + j * 20 - 21], (ftnlen)
sizeof(complex));
}
e_rsle();
/* L20: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___12.ciunit = *nin;
s_rsle(&io___12);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__6, &c__1, (char *)&b[i__ + j * 20 - 21], (ftnlen)
sizeof(complex));
}
e_rsle();
/* L30: */
}
io___14.ciunit = *nin;
s_rsle(&io___14);
do_lio(&c__3, &c__1, (char *)&iloin, (ftnlen)sizeof(integer));
do_lio(&c__3, &c__1, (char *)&ihiin, (ftnlen)sizeof(integer));
e_rsle();
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___17.ciunit = *nin;
s_rsle(&io___17);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__6, &c__1, (char *)&ain[i__ + j * 20 - 21], (ftnlen)
sizeof(complex));
}
e_rsle();
/* L40: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___19.ciunit = *nin;
s_rsle(&io___19);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__6, &c__1, (char *)&bin[i__ + j * 20 - 21], (ftnlen)
sizeof(complex));
}
e_rsle();
/* L50: */
}
io___21.ciunit = *nin;
s_rsle(&io___21);
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
do_lio(&c__4, &c__1, (char *)&lsclin[i__ - 1], (ftnlen)sizeof(real));
}
e_rsle();
io___23.ciunit = *nin;
s_rsle(&io___23);
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
do_lio(&c__4, &c__1, (char *)&rsclin[i__ - 1], (ftnlen)sizeof(real));
}
e_rsle();
anorm = clange_("M", &n, &n, a, &c__20, work);
bnorm = clange_("M", &n, &n, b, &c__20, work);
++knt;
cggbal_("B", &n, a, &c__20, b, &c__20, &ilo, &ihi, lscale, rscale, work, &
info);
if (info != 0) {
++ninfo;
lmax[0] = knt;
}
if (ilo != iloin || ihi != ihiin) {
++ninfo;
lmax[1] = knt;
}
vmax = 0.f;
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = n;
for (j = 1; j <= i__2; ++j) {
/* Computing MAX */
i__3 = i__ + j * 20 - 21;
i__4 = i__ + j * 20 - 21;
q__1.r = a[i__3].r - ain[i__4].r, q__1.i = a[i__3].i - ain[i__4]
.i;
r__1 = vmax, r__2 = c_abs(&q__1);
vmax = dmax(r__1,r__2);
/* Computing MAX */
i__3 = i__ + j * 20 - 21;
i__4 = i__ + j * 20 - 21;
q__1.r = b[i__3].r - bin[i__4].r, q__1.i = b[i__3].i - bin[i__4]
.i;
r__1 = vmax, r__2 = c_abs(&q__1);
vmax = dmax(r__1,r__2);
/* L60: */
}
/* L70: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
r__2 = vmax, r__3 = (r__1 = lscale[i__ - 1] - lsclin[i__ - 1], dabs(
r__1));
vmax = dmax(r__2,r__3);
/* Computing MAX */
r__2 = vmax, r__3 = (r__1 = rscale[i__ - 1] - rsclin[i__ - 1], dabs(
r__1));
vmax = dmax(r__2,r__3);
/* L80: */
}
vmax /= eps * dmax(anorm,bnorm);
if (vmax > rmax) {
lmax[2] = knt;
rmax = vmax;
}
goto L10;
L90:
io___34.ciunit = *nout;
s_wsfe(&io___34);
e_wsfe();
io___35.ciunit = *nout;
s_wsfe(&io___35);
do_fio(&c__1, (char *)&rmax, (ftnlen)sizeof(real));
e_wsfe();
io___36.ciunit = *nout;
s_wsfe(&io___36);
do_fio(&c__1, (char *)&lmax[0], (ftnlen)sizeof(integer));
e_wsfe();
io___37.ciunit = *nout;
s_wsfe(&io___37);
do_fio(&c__1, (char *)&lmax[1], (ftnlen)sizeof(integer));
e_wsfe();
io___38.ciunit = *nout;
s_wsfe(&io___38);
do_fio(&c__1, (char *)&lmax[2], (ftnlen)sizeof(integer));
e_wsfe();
io___39.ciunit = *nout;
s_wsfe(&io___39);
do_fio(&c__1, (char *)&ninfo, (ftnlen)sizeof(integer));
e_wsfe();
io___40.ciunit = *nout;
s_wsfe(&io___40);
do_fio(&c__1, (char *)&knt, (ftnlen)sizeof(integer));
e_wsfe();
return 0;
/* End of CCHKGL */
} /* cchkgl_ */
/* Subroutine */ int cgegs_(char *jobvsl, char *jobvsr, integer *n, complex *
a, integer *lda, complex *b, integer *ldb, complex *alpha, complex *
beta, complex *vsl, integer *ldvsl, complex *vsr, integer *ldvsr,
complex *work, integer *lwork, real *rwork, 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, i__3;
/* Local variables */
integer nb, nb1, nb2, nb3, ihi, ilo;
real eps, anrm, bnrm;
integer itau, lopt;
extern logical lsame_(char *, char *);
integer ileft, iinfo, icols;
logical ilvsl;
integer iwork;
logical ilvsr;
integer 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 *);
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 *);
real safmin;
extern /* Subroutine */ int 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 *);
integer ijobvl, iright, ijobvr;
real anrmto;
integer lwkmin;
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;
integer irwork, lwkopt;
logical lquery;
/* -- LAPACK driver routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine CGGES. */
/* CGEGS computes the eigenvalues, Schur form, and, optionally, the */
/* left and or/right Schur vectors of a complex matrix pair (A,B). */
/* Given two square matrices A and B, the generalized Schur */
/* factorization has the form */
/* A = Q*S*Z**H, B = Q*T*Z**H */
/* where Q and Z are unitary matrices and S and T are upper triangular. */
/* The columns of Q are the left Schur vectors */
/* and the columns of Z are the right Schur vectors. */
/* If only the eigenvalues of (A,B) are needed, the driver routine */
/* CGEGV should be used instead. See CGEGV for a description of the */
/* eigenvalues of the generalized nonsymmetric eigenvalue problem */
/* (GNEP). */
/* Arguments */
/* ========= */
/* JOBVSL (input) CHARACTER*1 */
/* = 'N': do not compute the left Schur vectors; */
/* = 'V': compute the left Schur vectors (returned in VSL). */
/* JOBVSR (input) CHARACTER*1 */
/* = 'N': do not compute the right Schur vectors; */
/* = 'V': compute the right Schur vectors (returned in VSR). */
/* 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 matrix A. */
/* On exit, the upper triangular matrix S from the generalized */
/* Schur factorization. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) COMPLEX array, dimension (LDB, N) */
/* On entry, the matrix B. */
/* On exit, the upper triangular matrix T from the generalized */
/* Schur factorization. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* ALPHA (output) COMPLEX array, dimension (N) */
/* The complex scalars alpha that define the eigenvalues of */
/* GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur */
/* form of A. */
/* BETA (output) COMPLEX array, dimension (N) */
/* The non-negative real scalars beta that define the */
/* eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element */
/* of the triangular factor T. */
/* Together, the quantities alpha = ALPHA(j) and beta = BETA(j) */
/* represent the j-th eigenvalue of the matrix pair (A,B), in */
/* one of the forms lambda = alpha/beta or mu = beta/alpha. */
/* Since either lambda or mu may overflow, they should not, */
/* in general, be computed. */
/* VSL (output) COMPLEX array, dimension (LDVSL,N) */
/* If JOBVSL = 'V', the matrix of left Schur vectors Q. */
/* 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', the matrix of right Schur vectors Z. */
/* 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. */
/* To compute the optimal value of LWORK, call ILAENV to get */
/* blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute: */
/* NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR; */
/* the optimal LWORK is N*(NB+1). */
/* 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 (3*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: errors that usually indicate LAPACK problems: */
/* =N+1: error return from CGGBAL */
/* =N+2: error return from CGEQRF */
/* =N+3: error return from CUNMQR */
/* =N+4: error return from CUNGQR */
/* =N+5: error return from CGGHRD */
/* =N+6: error return from CHGEQZ (other than failed */
/* iteration) */
/* =N+7: error return from CGGBAK (computing VSL) */
/* =N+8: error return from CGGBAK (computing VSR) */
/* =N+9: error return from CLASCL (various places) */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. 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;
/* 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_;
}
/* Test the input arguments */
/* Computing MAX */
i__1 = *n << 1;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1].r = (real) lwkopt, work[1].i = 0.f;
lquery = *lwork == -1;
*info = 0;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -7;
} else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) {
*info = -11;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*info = -13;
} else if (*lwork < lwkmin && ! lquery) {
*info = -15;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "CGEQRF", " ", n, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "CUNMQR", " ", n, n, n, &c_n1);
nb3 = ilaenv_(&c__1, "CUNGQR", " ", n, n, n, &c_n1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
lopt = *n * (nb + 1);
work[1].r = (real) lopt, work[1].i = 0.f;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEGS ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("E") * slamch_("B");
safmin = slamch_("S");
smlnum = *n * safmin / 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_n1, &c_n1, &anrm, &anrmto, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
/* 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_n1, &c_n1, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular */
ileft = 1;
iright = *n + 1;
irwork = iright + *n;
iwork = 1;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L10;
}
/* Reduce B to triangular form, and initialize VSL and/or VSR */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
cgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L10;
}
i__1 = *lwork + 1 - iwork;
cunmqr_("L", "C", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwork], &i__1, &
iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L10;
}
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[ilo + 1 + ilo * b_dim1], ldb, &vsl[ilo
+ 1 + ilo * vsl_dim1], ldvsl);
i__1 = *lwork + 1 - iwork;
cungqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, &
work[itau], &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L10;
}
}
if (ilvsr) {
claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 5;
goto L10;
}
/* Perform QZ algorithm, computing Schur vectors if desired */
iwork = itau;
i__1 = *lwork + 1 - iwork;
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[iwork], &i__1, &rwork[irwork], &
iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
if (iinfo > 0 && iinfo <= *n) {
*info = iinfo;
} else if (iinfo > *n && iinfo <= *n << 1) {
*info = iinfo - *n;
} else {
*info = *n + 6;
}
goto L10;
}
/* Apply permutation to VSL and VSR */
if (ilvsl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsl[vsl_offset], ldvsl, &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L10;
}
}
if (ilvsr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsr[vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L10;
}
}
/* Undo scaling */
if (ilascl) {
clascl_("U", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
clascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alpha[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
if (ilbscl) {
clascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
clascl_("G", &c_n1, &c_n1, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
L10:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGEGS */
} /* cgegs_ */
/* 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_ */
/* Subroutine */ int cggevx_(char *balanc, char *jobvl, char *jobvr, char *
sense, integer *n, complex *a, integer *lda, complex *b, integer *ldb,
complex *alpha, complex *beta, complex *vl, integer *ldvl, complex *
vr, integer *ldvr, integer *ilo, integer *ihi, real *lscale, real *
rscale, real *abnrm, real *bbnrm, real *rconde, real *rcondv, complex
*work, integer *lwork, real *rwork, integer *iwork, logical *bwork,
integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2, i__3, i__4;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double sqrt(doublereal), r_imag(complex *);
/* Local variables */
integer i__, j, m, jc, in, jr;
real eps;
logical ilv;
real anrm, bnrm;
integer ierr, itau;
real temp;
logical ilvl, ilvr;
integer iwrk, iwrk1;
extern logical lsame_(char *, char *);
integer icols;
logical noscl;
integer 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 *), clacpy_(
char *, integer *, integer *, complex *, integer *, complex *,
integer *), claset_(char *, integer *, integer *, complex
*, complex *, complex *, integer *), ctgevc_(char *, char
*, logical *, integer *, complex *, integer *, complex *, integer
*, complex *, integer *, complex *, integer *, integer *, integer
*, complex *, real *, integer *);
logical ldumma[1];
char chtemp[1];
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 *),
ctgsna_(char *, char *, logical *, integer *, complex *, integer *
, complex *, integer *, complex *, integer *, complex *, integer *
, real *, real *, integer *, integer *, complex *, integer *,
integer *, integer *);
integer ijobvl;
extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, real *, integer *, integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
extern doublereal slamch_(char *);
integer ijobvr;
logical wantsb;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
real anrmto;
logical wantse;
real bnrmto;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
integer minwrk, maxwrk;
logical wantsn;
real smlnum;
logical lquery, wantsv;
/* -- LAPACK driver routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices */
/* (A,B) the generalized eigenvalues, and optionally, the left and/or */
/* right generalized eigenvectors. */
/* Optionally, it also computes a balancing transformation to improve */
/* the conditioning of the eigenvalues and eigenvectors (ILO, IHI, */
/* LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for */
/* the eigenvalues (RCONDE), and reciprocal condition numbers for the */
/* right eigenvectors (RCONDV). */
/* A generalized eigenvalue for a pair of matrices (A,B) is a scalar */
/* lambda or a ratio alpha/beta = lambda, such that A - lambda*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. */
/* The right eigenvector v(j) corresponding to the eigenvalue lambda(j) */
/* of (A,B) satisfies */
/* A * v(j) = lambda(j) * B * v(j) . */
/* The left eigenvector u(j) corresponding to the eigenvalue lambda(j) */
/* of (A,B) satisfies */
/* u(j)**H * A = lambda(j) * u(j)**H * B. */
/* where u(j)**H is the conjugate-transpose of u(j). */
/* Arguments */
/* ========= */
/* BALANC (input) CHARACTER*1 */
/* Specifies the balance option to be performed: */
/* = 'N': do not diagonally scale or permute; */
/* = 'P': permute only; */
/* = 'S': scale only; */
/* = 'B': both permute and scale. */
/* Computed reciprocal condition numbers will be for the */
/* matrices after permuting and/or balancing. Permuting does */
/* not change condition numbers (in exact arithmetic), but */
/* balancing does. */
/* JOBVL (input) CHARACTER*1 */
/* = 'N': do not compute the left generalized eigenvectors; */
/* = 'V': compute the left generalized eigenvectors. */
/* JOBVR (input) CHARACTER*1 */
/* = 'N': do not compute the right generalized eigenvectors; */
/* = 'V': compute the right generalized eigenvectors. */
/* SENSE (input) CHARACTER*1 */
/* Determines which reciprocal condition numbers are computed. */
/* = 'N': none are computed; */
/* = 'E': computed for eigenvalues only; */
/* = 'V': computed for eigenvectors only; */
/* = 'B': computed for eigenvalues and eigenvectors. */
/* N (input) INTEGER */
/* The order of the matrices A, B, VL, and VR. N >= 0. */
/* A (input/output) COMPLEX array, dimension (LDA, N) */
/* On entry, the matrix A in the pair (A,B). */
/* On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' */
/* or both, then A contains the first part of the complex Schur */
/* form of the "balanced" versions of the input A and B. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) COMPLEX array, dimension (LDB, N) */
/* On entry, the matrix B in the pair (A,B). */
/* On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' */
/* or both, then B contains the second part of the complex */
/* Schur form of the "balanced" versions of the input A and B. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* 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. */
/* Note: the quotient 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). */
/* VL (output) COMPLEX array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left generalized eigenvectors u(j) are */
/* stored one after another in the columns of VL, in the same */
/* order as their eigenvalues. */
/* Each eigenvector will be scaled so the largest component */
/* will have abs(real part) + abs(imag. part) = 1. */
/* Not referenced if JOBVL = 'N'. */
/* LDVL (input) INTEGER */
/* The leading dimension of the matrix VL. LDVL >= 1, and */
/* if JOBVL = 'V', LDVL >= N. */
/* VR (output) COMPLEX array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right generalized eigenvectors v(j) are */
/* stored one after another in the columns of VR, in the same */
/* order as their eigenvalues. */
/* Each eigenvector will be scaled so the largest component */
/* will have abs(real part) + abs(imag. part) = 1. */
/* Not referenced if JOBVR = 'N'. */
/* LDVR (input) INTEGER */
/* The leading dimension of the matrix VR. LDVR >= 1, and */
/* if JOBVR = 'V', LDVR >= N. */
/* ILO (output) INTEGER */
/* IHI (output) INTEGER */
/* ILO and IHI are integer values 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 BALANC = 'N' or 'S', ILO = 1 and IHI = N. */
/* LSCALE (output) REAL array, dimension (N) */
/* Details of the permutations and scaling factors applied */
/* to the left side of A and B. If PL(j) is the index of the */
/* row interchanged with row j, and DL(j) is the scaling */
/* factor applied to row j, then */
/* LSCALE(j) = PL(j) for j = 1,...,ILO-1 */
/* = DL(j) for j = ILO,...,IHI */
/* = PL(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) REAL array, dimension (N) */
/* Details of the permutations and scaling factors applied */
/* to the right side of A and B. If PR(j) is the index of the */
/* column interchanged with column j, and DR(j) is the scaling */
/* factor applied to column j, then */
/* RSCALE(j) = PR(j) for j = 1,...,ILO-1 */
/* = DR(j) for j = ILO,...,IHI */
/* = PR(j) for j = IHI+1,...,N */
/* The order in which the interchanges are made is N to IHI+1, */
/* then 1 to ILO-1. */
/* ABNRM (output) REAL */
/* The one-norm of the balanced matrix A. */
/* BBNRM (output) REAL */
/* The one-norm of the balanced matrix B. */
/* RCONDE (output) REAL array, dimension (N) */
/* If SENSE = 'E' or 'B', the reciprocal condition numbers of */
/* the eigenvalues, stored in consecutive elements of the array. */
/* If SENSE = 'N' or 'V', RCONDE is not referenced. */
/* RCONDV (output) REAL array, dimension (N) */
/* If SENSE = 'V' or 'B', the estimated reciprocal condition */
/* numbers of the eigenvectors, stored in consecutive elements */
/* of the array. If the eigenvalues cannot be reordered to */
/* compute RCONDV(j), RCONDV(j) is set to 0; this can only occur */
/* when the true value would be very small anyway. */
/* If SENSE = 'N' or 'E', RCONDV is not referenced. */
/* 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). */
/* If SENSE = 'E', LWORK >= max(1,4*N). */
/* If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*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) REAL array, dimension (lrwork) */
/* lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', */
/* and at least max(1,2*N) otherwise. */
/* Real workspace. */
/* IWORK (workspace) INTEGER array, dimension (N+2) */
/* If SENSE = 'E', IWORK is not referenced. */
/* BWORK (workspace) LOGICAL array, dimension (N) */
/* If SENSE = 'N', BWORK is not referenced. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* = 1,...,N: */
/* The QZ iteration failed. No eigenvectors have been */
/* calculated, 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: error return from CTGEVC. */
/* Further Details */
/* =============== */
/* Balancing a matrix pair (A,B) includes, first, permuting rows and */
/* columns to isolate eigenvalues, second, applying diagonal similarity */
/* transformation to the rows and columns to make the rows and columns */
/* as close in norm as possible. The computed reciprocal condition */
/* numbers correspond to the balanced matrix. Permuting rows and columns */
/* will not change the condition numbers (in exact arithmetic) but */
/* diagonal scaling will. For further explanation of balancing, see */
/* section 4.11.1.2 of LAPACK Users' Guide. */
/* An approximate error bound on the chordal distance between the i-th */
/* computed generalized eigenvalue w and the corresponding exact */
/* eigenvalue lambda is */
/* chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) */
/* An approximate error bound for the angle between the i-th computed */
/* eigenvector VL(i) or VR(i) is given by */
/* EPS * norm(ABNRM, BBNRM) / DIF(i). */
/* For further explanation of the reciprocal condition numbers RCONDE */
/* and RCONDV, see section 4.11 of LAPACK User's Guide. */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. 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;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--lscale;
--rscale;
--rconde;
--rcondv;
--work;
--rwork;
--iwork;
--bwork;
/* Function Body */
if (lsame_(jobvl, "N")) {
ijobvl = 1;
ilvl = FALSE_;
} else if (lsame_(jobvl, "V")) {
ijobvl = 2;
ilvl = TRUE_;
} else {
ijobvl = -1;
ilvl = FALSE_;
}
if (lsame_(jobvr, "N")) {
ijobvr = 1;
ilvr = FALSE_;
} else if (lsame_(jobvr, "V")) {
ijobvr = 2;
ilvr = TRUE_;
} else {
ijobvr = -1;
ilvr = FALSE_;
}
ilv = ilvl || ilvr;
noscl = lsame_(balanc, "N") || lsame_(balanc, "P");
wantsn = lsame_(sense, "N");
wantse = lsame_(sense, "E");
wantsv = lsame_(sense, "V");
wantsb = lsame_(sense, "B");
/* Test the input arguments */
*info = 0;
lquery = *lwork == -1;
if (! (noscl || lsame_(balanc, "S") || lsame_(
balanc, "B"))) {
*info = -1;
} else if (ijobvl <= 0) {
*info = -2;
} else if (ijobvr <= 0) {
*info = -3;
} else if (! (wantsn || wantse || wantsb || wantsv)) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else if (*lda < max(1,*n)) {
*info = -7;
} else if (*ldb < max(1,*n)) {
*info = -9;
} else if (*ldvl < 1 || ilvl && *ldvl < *n) {
*info = -13;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -15;
}
/* 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. The workspace is */
/* computed assuming ILO = 1 and IHI = N, the worst case.) */
if (*info == 0) {
if (*n == 0) {
minwrk = 1;
maxwrk = 1;
} else {
minwrk = *n << 1;
if (wantse) {
minwrk = *n << 2;
} else if (wantsv || wantsb) {
minwrk = (*n << 1) * (*n + 1);
}
maxwrk = minwrk;
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "CGEQRF", " ", n, &
c__1, n, &c__0);
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "CUNMQR", " ", n, &
c__1, n, &c__0);
maxwrk = max(i__1,i__2);
if (ilvl) {
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "CUNGQR",
" ", n, &c__1, n, &c__0);
maxwrk = max(i__1,i__2);
}
}
work[1].r = (real) maxwrk, work[1].i = 0.f;
if (*lwork < minwrk && ! lquery) {
*info = -25;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGGEVX", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S");
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1.f / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = 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 and/or balance the matrix pair (A,B) */
/* (Real Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise) */
cggbal_(balanc, n, &a[a_offset], lda, &b[b_offset], ldb, ilo, ihi, &
lscale[1], &rscale[1], &rwork[1], &ierr);
/* Compute ABNRM and BBNRM */
*abnrm = clange_("1", n, n, &a[a_offset], lda, &rwork[1]);
if (ilascl) {
rwork[1] = *abnrm;
slascl_("G", &c__0, &c__0, &anrmto, &anrm, &c__1, &c__1, &rwork[1], &
c__1, &ierr);
*abnrm = rwork[1];
}
*bbnrm = clange_("1", n, n, &b[b_offset], ldb, &rwork[1]);
if (ilbscl) {
rwork[1] = *bbnrm;
slascl_("G", &c__0, &c__0, &bnrmto, &bnrm, &c__1, &c__1, &rwork[1], &
c__1, &ierr);
*bbnrm = rwork[1];
}
/* Reduce B to triangular form (QR decomposition of B) */
/* (Complex Workspace: need N, prefer N*NB ) */
irows = *ihi + 1 - *ilo;
if (ilv || ! wantsn) {
icols = *n + 1 - *ilo;
} else {
icols = irows;
}
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 unitary transformation to 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 VL and/or VR */
/* (Workspace: need N, prefer N*NB) */
if (ilvl) {
claset_("Full", n, n, &c_b1, &c_b2, &vl[vl_offset], ldvl);
if (irows > 1) {
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b[*ilo + 1 + *ilo * b_dim1], ldb, &vl[
*ilo + 1 + *ilo * vl_dim1], ldvl);
}
i__1 = *lwork + 1 - iwrk;
cungqr_(&irows, &irows, &irows, &vl[*ilo + *ilo * vl_dim1], ldvl, &
work[itau], &work[iwrk], &i__1, &ierr);
}
if (ilvr) {
claset_("Full", n, n, &c_b1, &c_b2, &vr[vr_offset], ldvr);
}
/* Reduce to generalized Hessenberg form */
/* (Workspace: none needed) */
if (ilv || ! wantsn) {
/* Eigenvectors requested -- work on whole matrix. */
cgghrd_(jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr);
} else {
cgghrd_("N", "N", &irows, &c__1, &irows, &a[*ilo + *ilo * a_dim1],
lda, &b[*ilo + *ilo * b_dim1], ldb, &vl[vl_offset], ldvl, &vr[
vr_offset], ldvr, &ierr);
}
/* Perform QZ algorithm (Compute eigenvalues, and optionally, the */
/* Schur forms and Schur vectors) */
/* (Complex Workspace: need N) */
/* (Real Workspace: need N) */
iwrk = itau;
if (ilv || ! wantsn) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwrk;
chgeqz_(chtemp, jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset]
, ldb, &alpha[1], &beta[1], &vl[vl_offset], ldvl, &vr[vr_offset],
ldvr, &work[iwrk], &i__1, &rwork[1], &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 L90;
}
/* Compute Eigenvectors and estimate condition numbers if desired */
/* CTGEVC: (Complex Workspace: need 2*N ) */
/* (Real Workspace: need 2*N ) */
/* CTGSNA: (Complex Workspace: need 2*N*N if SENSE='V' or 'B') */
/* (Integer Workspace: need N+2 ) */
if (ilv || ! wantsn) {
if (ilv) {
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
ctgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &
work[iwrk], &rwork[1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L90;
}
}
if (! wantsn) {
/* compute eigenvectors (STGEVC) and estimate condition */
/* numbers (STGSNA). Note that the definition of the condition */
/* number is not invariant under transformation (u,v) to */
/* (Q*u, Z*v), where (u,v) are eigenvectors of the generalized */
/* Schur form (S,T), Q and Z are orthogonal matrices. In order */
/* to avoid using extra 2*N*N workspace, we have to */
/* re-calculate eigenvectors and estimate the condition numbers */
/* one at a time. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
bwork[j] = FALSE_;
/* L10: */
}
bwork[i__] = TRUE_;
iwrk = *n + 1;
iwrk1 = iwrk + *n;
if (wantse || wantsb) {
ctgevc_("B", "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &
c__1, &m, &work[iwrk1], &rwork[1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L90;
}
}
i__2 = *lwork - iwrk1 + 1;
ctgsna_(sense, "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &rconde[
i__], &rcondv[i__], &c__1, &m, &work[iwrk1], &i__2, &
iwork[1], &ierr);
/* L20: */
}
}
}
/* Undo balancing on VL and VR and normalization */
/* (Workspace: none needed) */
if (ilvl) {
cggbak_(balanc, "L", n, ilo, ihi, &lscale[1], &rscale[1], n, &vl[
vl_offset], ldvl, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = jr + jc * vl_dim1;
r__3 = temp, r__4 = (r__1 = vl[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&vl[jr + jc * vl_dim1]), dabs(r__2));
temp = dmax(r__3,r__4);
/* L30: */
}
if (temp < smlnum) {
goto L50;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = jr + jc * vl_dim1;
i__4 = jr + jc * vl_dim1;
q__1.r = temp * vl[i__4].r, q__1.i = temp * vl[i__4].i;
vl[i__3].r = q__1.r, vl[i__3].i = q__1.i;
/* L40: */
}
L50:
;
}
}
if (ilvr) {
cggbak_(balanc, "R", n, ilo, ihi, &lscale[1], &rscale[1], n, &vr[
vr_offset], ldvr, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = jr + jc * vr_dim1;
r__3 = temp, r__4 = (r__1 = vr[i__3].r, dabs(r__1)) + (r__2 =
r_imag(&vr[jr + jc * vr_dim1]), dabs(r__2));
temp = dmax(r__3,r__4);
/* L60: */
}
if (temp < smlnum) {
goto L80;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = jr + jc * vr_dim1;
i__4 = jr + jc * vr_dim1;
q__1.r = temp * vr[i__4].r, q__1.i = temp * vr[i__4].i;
vr[i__3].r = q__1.r, vr[i__3].i = q__1.i;
/* L70: */
}
L80:
;
}
}
/* Undo scaling if necessary */
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);
}
L90:
work[1].r = (real) maxwrk, work[1].i = 0.f;
return 0;
/* End of CGGEVX */
} /* cggevx_ */
/* Subroutine */ int cgegv_(char *jobvl, char *jobvr, integer *n, complex *a,
integer *lda, complex *b, integer *ldb, complex *alpha, complex *beta,
complex *vl, integer *ldvl, complex *vr, integer *ldvr, complex *
work, integer *lwork, real *rwork, 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
=======
This routine is deprecated and has been replaced by routine CGGEV.
CGEGV computes for a pair of N-by-N complex nonsymmetric matrices A
and B, the generalized eigenvalues (alpha, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, 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 good beginning reference is the book, "Matrix
Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized
eigenvalue w for a pair of matrices (A,B) is a vector r such
that (A - w B) r = 0 . A left generalized eigenvector is a vector
l such that l**H * (A - w B) = 0, where l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B -- see
"Further Details", below.
Arguments
=========
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the first of the pair of matrices whose
generalized eigenvalues and (optionally) generalized
eigenvectors are to be computed.
On exit, the contents will have been destroyed. (For a
description of the contents of A on exit, see "Further
Details", below.)
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 whose
generalized eigenvalues and (optionally) generalized
eigenvectors are to be computed.
On exit, the contents will have been destroyed. (For a
description of the contents of B on exit, see "Further
Details", below.)
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
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.
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).
VL (output) COMPLEX array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.)
Each eigenvector will be scaled so the largest component
will have abs(real part) + abs(imag. part) = 1, *except*
that for eigenvalues with alpha=beta=0, a zero vector will
be returned as the corresponding eigenvector.
Not referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.
VR (output) COMPLEX array, dimension (LDVR,N)
If JOBVR = 'V', the right generalized eigenvectors. (See
"Purpose", above.)
Each eigenvector will be scaled so the largest component
will have abs(real part) + abs(imag. part) = 1, *except*
that for eigenvalues with alpha=beta=0, a zero vector will
be returned as the corresponding eigenvector.
Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = 'V', LDVR >= N.
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 >= max(1,2*N).
For good performance, LWORK must generally be larger.
To compute the optimal value of LWORK, call ILAENV to get
blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
The optimal LWORK is MAX( 2*N, N*(NB+1) ).
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/output) REAL array, dimension (8*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. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be
correct for j=INFO+1,...,N.
> N: errors that usually indicate LAPACK problems:
=N+1: error return from CGGBAL
=N+2: error return from CGEQRF
=N+3: error return from CUNMQR
=N+4: error return from CUNGQR
=N+5: error return from CGGHRD
=N+6: error return from CHGEQZ (other than failed
iteration)
=N+7: error return from CTGEVC
=N+8: error return from CGGBAK (computing VL)
=N+9: error return from CGGBAK (computing VR)
=N+10: error return from CLASCL (various calls)
Further Details
===============
Balancing
---------
This driver calls CGGBAL to both permute and scale rows and columns
of A and B. The permutations PL and PR are chosen so that PL*A*PR
and PL*B*R will be upper triangular except for the diagonal blocks
A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
possible. The diagonal scaling matrices DL and DR are chosen so
that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
one (except for the elements that start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices
have been computed, CGGBAK transforms the eigenvectors back to what
they would have been (in perfect arithmetic) if they had not been
balanced.
Contents of A and B on Exit
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
both), then on exit the arrays A and B will contain the complex Schur
form[*] of the "balanced" versions of A and B. If no eigenvectors
are computed, then only the diagonal blocks will be correct.
[*] In other words, upper triangular form.
=====================================================================
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_n1 = -1;
static real c_b29 = 1.f;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2, i__3, i__4;
real r__1, r__2, r__3, r__4;
complex q__1, q__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
static real absb, anrm, bnrm;
static integer itau;
static real temp;
static logical ilvl, ilvr;
static integer lopt;
static real anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta;
extern logical lsame_(char *, char *);
static integer ileft, iinfo, icols, iwork, irows, jc;
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 *);
static integer nb, in;
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
static integer jr;
extern /* Subroutine */ int cgghrd_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, integer *, complex *,
integer *, complex *, integer *, integer *);
static real salfai;
extern /* Subroutine */ int clascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, complex *, integer *, integer *), cgeqrf_(integer *, integer *, complex *, integer *,
complex *, complex *, integer *, integer *);
static real salfar;
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;
extern /* Subroutine */ int ctgevc_(char *, char *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, integer *,
complex *, integer *, integer *, integer *, complex *, real *,
integer *);
static real safmax;
static char chtemp[1];
static logical ldumma[1];
extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *,
integer *, integer *, complex *, integer *, complex *, integer *,
complex *, complex *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ijobvl, iright;
static logical ilimit;
static integer ijobvr;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
static integer lwkmin, nb1, nb2, nb3;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
static integer irwork, lwkopt;
static logical lquery;
static integer ihi, ilo;
static real eps;
static logical ilv;
#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 vl_subscr(a_1,a_2) (a_2)*vl_dim1 + a_1
#define vl_ref(a_1,a_2) vl[vl_subscr(a_1,a_2)]
#define vr_subscr(a_1,a_2) (a_2)*vr_dim1 + a_1
#define vr_ref(a_1,a_2) vr[vr_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;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1 * 1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1 * 1;
vr -= vr_offset;
--work;
--rwork;
/* Function Body */
if (lsame_(jobvl, "N")) {
ijobvl = 1;
ilvl = FALSE_;
} else if (lsame_(jobvl, "V")) {
ijobvl = 2;
ilvl = TRUE_;
} else {
ijobvl = -1;
ilvl = FALSE_;
}
if (lsame_(jobvr, "N")) {
ijobvr = 1;
ilvr = FALSE_;
} else if (lsame_(jobvr, "V")) {
ijobvr = 2;
ilvr = TRUE_;
} else {
ijobvr = -1;
ilvr = FALSE_;
}
ilv = ilvl || ilvr;
/* Test the input arguments
Computing MAX */
i__1 = *n << 1;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1].r = (real) lwkopt, work[1].i = 0.f;
lquery = *lwork == -1;
*info = 0;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -7;
} else if (*ldvl < 1 || ilvl && *ldvl < *n) {
*info = -11;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -13;
} else if (*lwork < lwkmin && ! lquery) {
*info = -15;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "CGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb2 = ilaenv_(&c__1, "CUNMQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
nb3 = ilaenv_(&c__1, "CUNGQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
/* Computing MAX */
i__1 = *n << 1, i__2 = *n * (nb + 1);
lopt = max(i__1,i__2);
work[1].r = (real) lopt, work[1].i = 0.f;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEGV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("E") * slamch_("B");
safmin = slamch_("S");
safmin += safmin;
safmax = 1.f / safmin;
/* Scale A */
anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]);
anrm1 = anrm;
anrm2 = 1.f;
if (anrm < 1.f) {
if (safmax * anrm < 1.f) {
anrm1 = safmin;
anrm2 = safmax * anrm;
}
}
if (anrm > 0.f) {
clascl_("G", &c_n1, &c_n1, &anrm, &c_b29, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Scale B */
bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]);
bnrm1 = bnrm;
bnrm2 = 1.f;
if (bnrm < 1.f) {
if (safmax * bnrm < 1.f) {
bnrm1 = safmin;
bnrm2 = safmax * bnrm;
}
}
if (bnrm > 0.f) {
clascl_("G", &c_n1, &c_n1, &bnrm, &c_b29, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular
Also "balance" the matrix. */
ileft = 1;
iright = *n + 1;
irwork = iright + *n;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L80;
}
/* Reduce B to triangular form, and initialize VL and/or VR */
irows = ihi + 1 - ilo;
if (ilv) {
icols = *n + 1 - ilo;
} else {
icols = irows;
}
itau = 1;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
cgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwork],
&i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L80;
}
i__1 = *lwork + 1 - iwork;
cunmqr_("L", "C", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[
itau], &a_ref(ilo, ilo), lda, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L80;
}
if (ilvl) {
claset_("Full", n, n, &c_b1, &c_b2, &vl[vl_offset], ldvl);
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vl_ref(ilo + 1,
ilo), ldvl);
i__1 = *lwork + 1 - iwork;
cungqr_(&irows, &irows, &irows, &vl_ref(ilo, ilo), ldvl, &work[itau],
&work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L80;
}
}
if (ilvr) {
claset_("Full", n, n, &c_b1, &c_b2, &vr[vr_offset], ldvr);
}
/* Reduce to generalized Hessenberg form */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
cgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo);
} else {
cgghrd_("N", "N", &irows, &c__1, &irows, &a_ref(ilo, ilo), lda, &
b_ref(ilo, ilo), ldb, &vl[vl_offset], ldvl, &vr[vr_offset],
ldvr, &iinfo);
}
if (iinfo != 0) {
*info = *n + 5;
goto L80;
}
/* Perform QZ algorithm */
iwork = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwork;
chgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vl[vl_offset], ldvl, &vr[
vr_offset], ldvr, &work[iwork], &i__1, &rwork[irwork], &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
if (iinfo > 0 && iinfo <= *n) {
*info = iinfo;
} else if (iinfo > *n && iinfo <= *n << 1) {
*info = iinfo - *n;
} else {
*info = *n + 6;
}
goto L80;
}
if (ilv) {
/* Compute Eigenvectors */
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
ctgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwork], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L80;
}
/* Undo balancing on VL and VR, rescale */
if (ilvl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vl[vl_offset], ldvl, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L80;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = vl_subscr(jr, jc);
r__3 = temp, r__4 = (r__1 = vl[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vl_ref(jr, jc)), dabs(r__2));
temp = dmax(r__3,r__4);
/* L10: */
}
if (temp < safmin) {
goto L30;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = vl_subscr(jr, jc);
i__4 = vl_subscr(jr, jc);
q__1.r = temp * vl[i__4].r, q__1.i = temp * vl[i__4].i;
vl[i__3].r = q__1.r, vl[i__3].i = q__1.i;
/* L20: */
}
L30:
;
}
}
if (ilvr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vr[vr_offset], ldvr, &iinfo);
if (iinfo != 0) {
*info = *n + 9;
goto L80;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = vr_subscr(jr, jc);
r__3 = temp, r__4 = (r__1 = vr[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vr_ref(jr, jc)), dabs(r__2));
temp = dmax(r__3,r__4);
/* L40: */
}
if (temp < safmin) {
goto L60;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = vr_subscr(jr, jc);
i__4 = vr_subscr(jr, jc);
q__1.r = temp * vr[i__4].r, q__1.i = temp * vr[i__4].i;
vr[i__3].r = q__1.r, vr[i__3].i = q__1.i;
/* L50: */
}
L60:
;
}
}
/* End of eigenvector calculation */
}
/* Undo scaling in alpha, beta
Note: this does not give the alpha and beta for the unscaled
problem.
Un-scaling is limited to avoid underflow in alpha and beta
if they are significant. */
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
i__2 = jc;
absar = (r__1 = alpha[i__2].r, dabs(r__1));
absai = (r__1 = r_imag(&alpha[jc]), dabs(r__1));
i__2 = jc;
absb = (r__1 = beta[i__2].r, dabs(r__1));
i__2 = jc;
salfar = anrm * alpha[i__2].r;
salfai = anrm * r_imag(&alpha[jc]);
i__2 = jc;
sbeta = bnrm * beta[i__2].r;
ilimit = FALSE_;
scale = 1.f;
/* Check for significant underflow in imaginary part of ALPHA
Computing MAX */
r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps *
absb;
if (dabs(salfai) < safmin && absai >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX */
r__1 = safmin, r__2 = anrm2 * absai;
scale = safmin / anrm1 / dmax(r__1,r__2);
}
/* Check for significant underflow in real part of ALPHA
Computing MAX */
r__1 = safmin, r__2 = eps * absai, r__1 = max(r__1,r__2), r__2 = eps *
absb;
if (dabs(salfar) < safmin && absar >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX
Computing MAX */
r__3 = safmin, r__4 = anrm2 * absar;
r__1 = scale, r__2 = safmin / anrm1 / dmax(r__3,r__4);
scale = dmax(r__1,r__2);
}
/* Check for significant underflow in BETA
Computing MAX */
r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps *
absai;
if (dabs(sbeta) < safmin && absb >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX
Computing MAX */
r__3 = safmin, r__4 = bnrm2 * absb;
r__1 = scale, r__2 = safmin / bnrm1 / dmax(r__3,r__4);
scale = dmax(r__1,r__2);
}
/* Check for possible overflow when limiting scaling */
if (ilimit) {
/* Computing MAX */
r__1 = dabs(salfar), r__2 = dabs(salfai), r__1 = max(r__1,r__2),
r__2 = dabs(sbeta);
temp = scale * safmin * dmax(r__1,r__2);
if (temp > 1.f) {
scale /= temp;
}
if (scale < 1.f) {
ilimit = FALSE_;
}
}
/* Recompute un-scaled ALPHA, BETA if necessary. */
if (ilimit) {
i__2 = jc;
salfar = scale * alpha[i__2].r * anrm;
salfai = scale * r_imag(&alpha[jc]) * anrm;
i__2 = jc;
q__2.r = scale * beta[i__2].r, q__2.i = scale * beta[i__2].i;
q__1.r = bnrm * q__2.r, q__1.i = bnrm * q__2.i;
sbeta = q__1.r;
}
i__2 = jc;
q__1.r = salfar, q__1.i = salfai;
alpha[i__2].r = q__1.r, alpha[i__2].i = q__1.i;
i__2 = jc;
beta[i__2].r = sbeta, beta[i__2].i = 0.f;
/* L70: */
}
L80:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGEGV */
} /* cgegv_ */
/* Subroutine */ int cgegv_(char *jobvl, char *jobvr, integer *n, complex *a,
integer *lda, complex *b, integer *ldb, complex *alpha, complex *beta,
complex *vl, integer *ldvl, complex *vr, integer *ldvr, complex *
work, integer *lwork, real *rwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2, i__3, i__4;
real r__1, r__2, r__3, r__4;
complex q__1, q__2;
/* Builtin functions */
double r_imag(complex *);
/* Local variables */
integer jc, nb, in, jr, nb1, nb2, nb3, ihi, ilo;
real eps;
logical ilv;
real absb, anrm, bnrm;
integer itau;
real temp;
logical ilvl, ilvr;
integer lopt;
real anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta;
extern logical lsame_(char *, char *);
integer ileft, iinfo, icols, iwork, 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 *);
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 *);
real salfai;
extern /* Subroutine */ int clascl_(char *, integer *, integer *, real *,
real *, integer *, integer *, complex *, integer *, integer *), cgeqrf_(integer *, integer *, complex *, integer *,
complex *, complex *, integer *, integer *);
real salfar;
extern doublereal slamch_(char *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), claset_(char *,
integer *, integer *, complex *, complex *, complex *, integer *);
real safmin;
extern /* Subroutine */ int ctgevc_(char *, char *, logical *, integer *,
complex *, integer *, complex *, integer *, complex *, integer *,
complex *, integer *, integer *, integer *, complex *, real *,
integer *);
real safmax;
char chtemp[1];
logical ldumma[1];
extern /* Subroutine */ int chgeqz_(char *, char *, char *, integer *,
integer *, integer *, complex *, integer *, complex *, integer *,
complex *, complex *, complex *, integer *, complex *, integer *,
complex *, integer *, real *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
integer ijobvl, iright;
logical ilimit;
integer ijobvr;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
integer lwkmin;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
integer irwork, lwkopt;
logical lquery;
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine CGGEV. */
/* CGEGV computes the eigenvalues and, optionally, the left and/or right */
/* eigenvectors of a complex matrix pair (A,B). */
/* Given two square matrices A and B, */
/* the generalized nonsymmetric eigenvalue problem (GNEP) is to find the */
/* eigenvalues lambda and corresponding (non-zero) eigenvectors x such */
/* that */
/* A*x = lambda*B*x. */
/* An alternate form is to find the eigenvalues mu and corresponding */
/* eigenvectors y such that */
/* mu*A*y = B*y. */
/* These two forms are equivalent with mu = 1/lambda and x = y if */
/* neither lambda nor mu is zero. In order to deal with the case that */
/* lambda or mu is zero or small, two values alpha and beta are returned */
/* for each eigenvalue, such that lambda = alpha/beta and */
/* mu = beta/alpha. */
/* The vectors x and y in the above equations are right eigenvectors of */
/* the matrix pair (A,B). Vectors u and v satisfying */
/* u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B */
/* are left eigenvectors of (A,B). */
/* Note: this routine performs "full balancing" on A and B -- see */
/* "Further Details", below. */
/* Arguments */
/* ========= */
/* JOBVL (input) CHARACTER*1 */
/* = 'N': do not compute the left generalized eigenvectors; */
/* = 'V': compute the left generalized eigenvectors (returned */
/* in VL). */
/* JOBVR (input) CHARACTER*1 */
/* = 'N': do not compute the right generalized eigenvectors; */
/* = 'V': compute the right generalized eigenvectors (returned */
/* in VR). */
/* N (input) INTEGER */
/* The order of the matrices A, B, VL, and VR. N >= 0. */
/* A (input/output) COMPLEX array, dimension (LDA, N) */
/* On entry, the matrix A. */
/* If JOBVL = 'V' or JOBVR = 'V', then on exit A */
/* contains the Schur form of A from the generalized Schur */
/* factorization of the pair (A,B) after balancing. If no */
/* eigenvectors were computed, then only the diagonal elements */
/* of the Schur form will be correct. See CGGHRD and CHGEQZ */
/* for details. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) COMPLEX array, dimension (LDB, N) */
/* On entry, the matrix B. */
/* If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the */
/* upper triangular matrix obtained from B in the generalized */
/* Schur factorization of the pair (A,B) after balancing. */
/* If no eigenvectors were computed, then only the diagonal */
/* elements of B will be correct. See CGGHRD and CHGEQZ for */
/* details. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* ALPHA (output) COMPLEX array, dimension (N) */
/* The complex scalars alpha that define the eigenvalues of */
/* GNEP. */
/* BETA (output) COMPLEX array, dimension (N) */
/* The complex scalars beta that define the eigenvalues of GNEP. */
/* Together, the quantities alpha = ALPHA(j) and beta = BETA(j) */
/* represent the j-th eigenvalue of the matrix pair (A,B), in */
/* one of the forms lambda = alpha/beta or mu = beta/alpha. */
/* Since either lambda or mu may overflow, they should not, */
/* in general, be computed. */
/* VL (output) COMPLEX array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left eigenvectors u(j) are stored */
/* in the columns of VL, in the same order as their eigenvalues. */
/* Each eigenvector is scaled so that its largest component has */
/* abs(real part) + abs(imag. part) = 1, except for eigenvectors */
/* corresponding to an eigenvalue with alpha = beta = 0, which */
/* are set to zero. */
/* Not referenced if JOBVL = 'N'. */
/* LDVL (input) INTEGER */
/* The leading dimension of the matrix VL. LDVL >= 1, and */
/* if JOBVL = 'V', LDVL >= N. */
/* VR (output) COMPLEX array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right eigenvectors x(j) are stored */
/* in the columns of VR, in the same order as their eigenvalues. */
/* Each eigenvector is scaled so that its largest component has */
/* abs(real part) + abs(imag. part) = 1, except for eigenvectors */
/* corresponding to an eigenvalue with alpha = beta = 0, which */
/* are set to zero. */
/* Not referenced if JOBVR = 'N'. */
/* LDVR (input) INTEGER */
/* The leading dimension of the matrix VR. LDVR >= 1, and */
/* if JOBVR = 'V', LDVR >= 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. */
/* To compute the optimal value of LWORK, call ILAENV to get */
/* blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute: */
/* NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR; */
/* The optimal LWORK is MAX( 2*N, N*(NB+1) ). */
/* 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/output) REAL array, dimension (8*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. No eigenvectors have been */
/* calculated, but ALPHA(j) and BETA(j) should be */
/* correct for j=INFO+1,...,N. */
/* > N: errors that usually indicate LAPACK problems: */
/* =N+1: error return from CGGBAL */
/* =N+2: error return from CGEQRF */
/* =N+3: error return from CUNMQR */
/* =N+4: error return from CUNGQR */
/* =N+5: error return from CGGHRD */
/* =N+6: error return from CHGEQZ (other than failed */
/* iteration) */
/* =N+7: error return from CTGEVC */
/* =N+8: error return from CGGBAK (computing VL) */
/* =N+9: error return from CGGBAK (computing VR) */
/* =N+10: error return from CLASCL (various calls) */
/* Further Details */
/* =============== */
/* Balancing */
/* --------- */
/* This driver calls CGGBAL to both permute and scale rows and columns */
/* of A and B. The permutations PL and PR are chosen so that PL*A*PR */
/* and PL*B*R will be upper triangular except for the diagonal blocks */
/* A(i:j,i:j) and B(i:j,i:j), with i and j as close together as */
/* possible. The diagonal scaling matrices DL and DR are chosen so */
/* that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to */
/* one (except for the elements that start out zero.) */
/* After the eigenvalues and eigenvectors of the balanced matrices */
/* have been computed, CGGBAK transforms the eigenvectors back to what */
/* they would have been (in perfect arithmetic) if they had not been */
/* balanced. */
/* Contents of A and B on Exit */
/* -------- -- - --- - -- ---- */
/* If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or */
/* both), then on exit the arrays A and B will contain the complex Schur */
/* form[*] of the "balanced" versions of A and B. If no eigenvectors */
/* are computed, then only the diagonal blocks will be correct. */
/* [*] In other words, upper triangular form. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Statement Functions .. */
/* .. */
/* .. Statement Function definitions .. */
/* .. */
/* .. 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;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
--rwork;
/* Function Body */
if (lsame_(jobvl, "N")) {
ijobvl = 1;
ilvl = FALSE_;
} else if (lsame_(jobvl, "V")) {
ijobvl = 2;
ilvl = TRUE_;
} else {
ijobvl = -1;
ilvl = FALSE_;
}
if (lsame_(jobvr, "N")) {
ijobvr = 1;
ilvr = FALSE_;
} else if (lsame_(jobvr, "V")) {
ijobvr = 2;
ilvr = TRUE_;
} else {
ijobvr = -1;
ilvr = FALSE_;
}
ilv = ilvl || ilvr;
/* Test the input arguments */
/* Computing MAX */
i__1 = *n << 1;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1].r = (real) lwkopt, work[1].i = 0.f;
lquery = *lwork == -1;
*info = 0;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -7;
} else if (*ldvl < 1 || ilvl && *ldvl < *n) {
*info = -11;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -13;
} else if (*lwork < lwkmin && ! lquery) {
*info = -15;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "CGEQRF", " ", n, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "CUNMQR", " ", n, n, n, &c_n1);
nb3 = ilaenv_(&c__1, "CUNGQR", " ", n, n, n, &c_n1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
/* Computing MAX */
i__1 = *n << 1, i__2 = *n * (nb + 1);
lopt = max(i__1,i__2);
work[1].r = (real) lopt, work[1].i = 0.f;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEGV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("E") * slamch_("B");
safmin = slamch_("S");
safmin += safmin;
safmax = 1.f / safmin;
/* Scale A */
anrm = clange_("M", n, n, &a[a_offset], lda, &rwork[1]);
anrm1 = anrm;
anrm2 = 1.f;
if (anrm < 1.f) {
if (safmax * anrm < 1.f) {
anrm1 = safmin;
anrm2 = safmax * anrm;
}
}
if (anrm > 0.f) {
clascl_("G", &c_n1, &c_n1, &anrm, &c_b29, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Scale B */
bnrm = clange_("M", n, n, &b[b_offset], ldb, &rwork[1]);
bnrm1 = bnrm;
bnrm2 = 1.f;
if (bnrm < 1.f) {
if (safmax * bnrm < 1.f) {
bnrm1 = safmin;
bnrm2 = safmax * bnrm;
}
}
if (bnrm > 0.f) {
clascl_("G", &c_n1, &c_n1, &bnrm, &c_b29, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular */
/* Also "balance" the matrix. */
ileft = 1;
iright = *n + 1;
irwork = iright + *n;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L80;
}
/* Reduce B to triangular form, and initialize VL and/or VR */
irows = ihi + 1 - ilo;
if (ilv) {
icols = *n + 1 - ilo;
} else {
icols = irows;
}
itau = 1;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
cgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L80;
}
i__1 = *lwork + 1 - iwork;
cunmqr_("L", "C", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwork], &i__1, &
iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L80;
}
if (ilvl) {
claset_("Full", n, n, &c_b1, &c_b2, &vl[vl_offset], ldvl);
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo +
1 + ilo * vl_dim1], ldvl);
i__1 = *lwork + 1 - iwork;
cungqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[
itau], &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L80;
}
}
if (ilvr) {
claset_("Full", n, n, &c_b1, &c_b2, &vr[vr_offset], ldvr);
}
/* Reduce to generalized Hessenberg form */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
cgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo);
} else {
cgghrd_("N", "N", &irows, &c__1, &irows, &a[ilo + ilo * a_dim1], lda,
&b[ilo + ilo * b_dim1], ldb, &vl[vl_offset], ldvl, &vr[
vr_offset], ldvr, &iinfo);
}
if (iinfo != 0) {
*info = *n + 5;
goto L80;
}
/* Perform QZ algorithm */
iwork = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwork;
chgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vl[vl_offset], ldvl, &vr[
vr_offset], ldvr, &work[iwork], &i__1, &rwork[irwork], &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
if (iinfo > 0 && iinfo <= *n) {
*info = iinfo;
} else if (iinfo > *n && iinfo <= *n << 1) {
*info = iinfo - *n;
} else {
*info = *n + 6;
}
goto L80;
}
if (ilv) {
/* Compute Eigenvectors */
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
ctgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwork], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L80;
}
/* Undo balancing on VL and VR, rescale */
if (ilvl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vl[vl_offset], ldvl, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L80;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = jr + jc * vl_dim1;
r__3 = temp, r__4 = (r__1 = vl[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vl[jr + jc * vl_dim1]), dabs(r__2))
;
temp = dmax(r__3,r__4);
/* L10: */
}
if (temp < safmin) {
goto L30;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = jr + jc * vl_dim1;
i__4 = jr + jc * vl_dim1;
q__1.r = temp * vl[i__4].r, q__1.i = temp * vl[i__4].i;
vl[i__3].r = q__1.r, vl[i__3].i = q__1.i;
/* L20: */
}
L30:
;
}
}
if (ilvr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vr[vr_offset], ldvr, &iinfo);
if (iinfo != 0) {
*info = *n + 9;
goto L80;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = jr + jc * vr_dim1;
r__3 = temp, r__4 = (r__1 = vr[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vr[jr + jc * vr_dim1]), dabs(r__2))
;
temp = dmax(r__3,r__4);
/* L40: */
}
if (temp < safmin) {
goto L60;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = jr + jc * vr_dim1;
i__4 = jr + jc * vr_dim1;
q__1.r = temp * vr[i__4].r, q__1.i = temp * vr[i__4].i;
vr[i__3].r = q__1.r, vr[i__3].i = q__1.i;
/* L50: */
}
L60:
;
}
}
/* End of eigenvector calculation */
}
/* Undo scaling in alpha, beta */
/* Note: this does not give the alpha and beta for the unscaled */
/* problem. */
/* Un-scaling is limited to avoid underflow in alpha and beta */
/* if they are significant. */
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
i__2 = jc;
absar = (r__1 = alpha[i__2].r, dabs(r__1));
absai = (r__1 = r_imag(&alpha[jc]), dabs(r__1));
i__2 = jc;
absb = (r__1 = beta[i__2].r, dabs(r__1));
i__2 = jc;
salfar = anrm * alpha[i__2].r;
salfai = anrm * r_imag(&alpha[jc]);
i__2 = jc;
sbeta = bnrm * beta[i__2].r;
ilimit = FALSE_;
scale = 1.f;
/* Check for significant underflow in imaginary part of ALPHA */
/* Computing MAX */
r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps *
absb;
if (dabs(salfai) < safmin && absai >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX */
r__1 = safmin, r__2 = anrm2 * absai;
scale = safmin / anrm1 / dmax(r__1,r__2);
}
/* Check for significant underflow in real part of ALPHA */
/* Computing MAX */
r__1 = safmin, r__2 = eps * absai, r__1 = max(r__1,r__2), r__2 = eps *
absb;
if (dabs(salfar) < safmin && absar >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
r__3 = safmin, r__4 = anrm2 * absar;
r__1 = scale, r__2 = safmin / anrm1 / dmax(r__3,r__4);
scale = dmax(r__1,r__2);
}
/* Check for significant underflow in BETA */
/* Computing MAX */
r__1 = safmin, r__2 = eps * absar, r__1 = max(r__1,r__2), r__2 = eps *
absai;
if (dabs(sbeta) < safmin && absb >= dmax(r__1,r__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
r__3 = safmin, r__4 = bnrm2 * absb;
r__1 = scale, r__2 = safmin / bnrm1 / dmax(r__3,r__4);
scale = dmax(r__1,r__2);
}
/* Check for possible overflow when limiting scaling */
if (ilimit) {
/* Computing MAX */
r__1 = dabs(salfar), r__2 = dabs(salfai), r__1 = max(r__1,r__2),
r__2 = dabs(sbeta);
temp = scale * safmin * dmax(r__1,r__2);
if (temp > 1.f) {
scale /= temp;
}
if (scale < 1.f) {
ilimit = FALSE_;
}
}
/* Recompute un-scaled ALPHA, BETA if necessary. */
if (ilimit) {
i__2 = jc;
salfar = scale * alpha[i__2].r * anrm;
salfai = scale * r_imag(&alpha[jc]) * anrm;
i__2 = jc;
q__2.r = scale * beta[i__2].r, q__2.i = scale * beta[i__2].i;
q__1.r = bnrm * q__2.r, q__1.i = bnrm * q__2.i;
sbeta = q__1.r;
}
i__2 = jc;
q__1.r = salfar, q__1.i = salfai;
alpha[i__2].r = q__1.r, alpha[i__2].i = q__1.i;
i__2 = jc;
beta[i__2].r = sbeta, beta[i__2].i = 0.f;
/* L70: */
}
L80:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGEGV */
} /* cgegv_ */
/* Subroutine */ int cgegs_(char *jobvsl, char *jobvsr, integer *n, complex *
a, integer *lda, complex *b, integer *ldb, complex *alpha, complex *
beta, complex *vsl, integer *ldvsl, complex *vsr, integer *ldvsr,
complex *work, integer *lwork, real *rwork, 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
=======
This routine is deprecated and has been replaced by routine CGGES.
CGEGS computes for a pair of N-by-N complex nonsymmetric matrices A,
B: the generalized eigenvalues (alpha, beta), the complex Schur
form (A, B), and optionally left and/or right Schur vectors
(VSL and VSR).
(If only the generalized eigenvalues are needed, use the driver CGEGV
instead.)
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, 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 good beginning reference is the book, "Matrix
Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press)
The (generalized) Schur form of a pair of matrices is the result of
multiplying both matrices on the left by one unitary matrix and
both on the right by another unitary matrix, these two unitary
matrices being chosen so as to bring the pair of matrices into
upper triangular form with the diagonal elements of B being
non-negative real numbers (this is also called complex Schur form.)
The left and right Schur vectors are the columns of VSL and VSR,
respectively, where VSL and VSR are the unitary matrices
which reduce A and B to Schur form:
Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) )
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.
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 whose generalized
eigenvalues and (optionally) Schur vectors are to be
computed.
On exit, the generalized Schur form of A.
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 whose
generalized eigenvalues and (optionally) Schur vectors are
to be computed.
On exit, the generalized Schur form of B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
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 CGEGS. 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.
(See "Purpose", above.)
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.
(See "Purpose", above.)
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 (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.
To compute the optimal value of LWORK, call ILAENV to get
blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
the optimal LWORK is N*(NB+1).
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 (3*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: errors that usually indicate LAPACK problems:
=N+1: error return from CGGBAL
=N+2: error return from CGEQRF
=N+3: error return from CUNMQR
=N+4: error return from CUNGQR
=N+5: error return from CGGHRD
=N+6: error return from CHGEQZ (other than failed
iteration)
=N+7: error return from CGGBAK (computing VSL)
=N+8: error return from CGGBAK (computing VSR)
=N+9: error return from CLASCL (various places)
=====================================================================
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_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, i__3;
/* Local variables */
static real anrm, bnrm;
static integer itau, lopt;
extern logical lsame_(char *, char *);
static integer ileft, iinfo, icols;
static logical ilvsl;
static integer iwork;
static logical ilvsr;
static integer 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 *);
static integer nb;
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 *);
static 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 *);
static real safmin;
extern /* Subroutine */ int xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
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 *);
static integer ijobvl, iright, ijobvr;
static real anrmto;
static integer lwkmin, nb1, nb2, nb3;
static 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 *);
static real smlnum;
static integer irwork, lwkopt;
static logical lquery;
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;
--work;
--rwork;
/* 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_;
}
/* Test the input arguments
Computing MAX */
i__1 = *n << 1;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1].r = (real) lwkopt, work[1].i = 0.f;
lquery = *lwork == -1;
*info = 0;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -7;
} else if (*ldvsl < 1 || ilvsl && *ldvsl < *n) {
*info = -11;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*info = -13;
} else if (*lwork < lwkmin && ! lquery) {
*info = -15;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "CGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb2 = ilaenv_(&c__1, "CUNMQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
nb3 = ilaenv_(&c__1, "CUNGQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
lopt = *n * (nb + 1);
work[1].r = (real) lopt, work[1].i = 0.f;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGEGS ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("E") * slamch_("B");
safmin = slamch_("S");
smlnum = *n * safmin / 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_n1, &c_n1, &anrm, &anrmto, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
/* 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_n1, &c_n1, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular */
ileft = 1;
iright = *n + 1;
irwork = iright + *n;
iwork = 1;
cggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &rwork[
ileft], &rwork[iright], &rwork[irwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L10;
}
/* Reduce B to triangular form, and initialize VSL and/or VSR */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
cgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwork],
&i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L10;
}
i__1 = *lwork + 1 - iwork;
cunmqr_("L", "C", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[
itau], &a_ref(ilo, ilo), lda, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L10;
}
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 - iwork;
cungqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau]
, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L10;
}
}
if (ilvsr) {
claset_("Full", n, n, &c_b1, &c_b2, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
cgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 5;
goto L10;
}
/* Perform QZ algorithm, computing Schur vectors if desired */
iwork = itau;
i__1 = *lwork + 1 - iwork;
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[iwork], &i__1, &rwork[irwork], &
iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__3 = iwork;
i__1 = lwkopt, i__2 = (integer) work[i__3].r + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
if (iinfo > 0 && iinfo <= *n) {
*info = iinfo;
} else if (iinfo > *n && iinfo <= *n << 1) {
*info = iinfo - *n;
} else {
*info = *n + 6;
}
goto L10;
}
/* Apply permutation to VSL and VSR */
if (ilvsl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsl[vsl_offset], ldvsl, &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L10;
}
}
if (ilvsr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n, &
vsr[vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L10;
}
}
/* Undo scaling */
if (ilascl) {
clascl_("U", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
clascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alpha[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
if (ilbscl) {
clascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
clascl_("G", &c_n1, &c_n1, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
L10:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGEGS */
} /* cgegs_ */
/* Subroutine */ int cggev_(char *jobvl, char *jobvr, integer *n, complex *a,
integer *lda, complex *b, integer *ldb, complex *alpha, complex *beta,
complex *vl, integer *ldvl, complex *vr, integer *ldvr, complex *
work, integer *lwork, real *rwork, 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
=======
CGGEV computes for a pair of N-by-N complex nonsymmetric matrices
(A,B), the generalized eigenvalues, and optionally, the left and/or
right generalized eigenvectors.
A generalized eigenvalue for a pair of matrices (A,B) is a scalar
lambda or a ratio alpha/beta = lambda, such that A - lambda*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.
The right generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies
A * v(j) = lambda(j) * B * v(j).
The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues lambda(j) of (A,B) satisfies
u(j)**H * A = lambda(j) * u(j)**H * B
where u(j)**H is the conjugate-transpose of u(j).
Arguments
=========
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) COMPLEX array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
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.
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).
VL (output) COMPLEX array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors u(j) are
stored one after another in the columns of VL, in the same
order as their eigenvalues.
Each eigenvector will be scaled so the largest component
will have abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.
VR (output) COMPLEX array, dimension (LDVR,N)
If JOBVR = 'V', the right generalized eigenvectors v(j) are
stored one after another in the columns of VR, in the same
order as their eigenvalues.
Each eigenvector will be scaled so the largest component
will have abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = 'V', LDVR >= N.
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 >= 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/output) REAL array, dimension (8*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. No eigenvectors have been
calculated, but ALPHA(j) and BETA(j) should be
correct for j=INFO+1,...,N.
> N: =N+1: other then QZ iteration failed in SHGEQZ,
=N+2: error return from STGEVC.
=====================================================================
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;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2, i__3, i__4;
real r__1, r__2, r__3, r__4;
complex q__1;
/* Builtin functions */
double sqrt(doublereal), r_imag(complex *);
/* Local variables */
static real anrm, bnrm;
static integer ierr, itau;
static real temp;
static logical ilvl, ilvr;
static integer iwrk;
extern logical lsame_(char *, char *);
static integer ileft, icols, irwrk, irows, jc;
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 *);
static integer in;
extern doublereal clange_(char *, integer *, integer *, complex *,
integer *, real *);
static integer jr;
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 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 *), ctgevc_(char *, char
*, logical *, integer *, complex *, integer *, complex *, integer
*, complex *, integer *, complex *, integer *, integer *, integer
*, complex *, real *, integer *), xerbla_(char *,
integer *);
static logical ldumma[1];
static char chtemp[1];
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 *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
extern doublereal slamch_(char *);
static integer ijobvl, iright, ijobvr;
extern /* Subroutine */ int cungqr_(integer *, integer *, integer *,
complex *, integer *, complex *, complex *, integer *, integer *);
static real anrmto;
static integer lwkmin;
static real bnrmto;
extern /* Subroutine */ int cunmqr_(char *, char *, integer *, integer *,
integer *, complex *, integer *, complex *, complex *, integer *,
complex *, integer *, integer *);
static real smlnum;
static integer lwkopt;
static logical lquery;
static integer ihi, ilo;
static real eps;
static logical ilv;
#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 vl_subscr(a_1,a_2) (a_2)*vl_dim1 + a_1
#define vl_ref(a_1,a_2) vl[vl_subscr(a_1,a_2)]
#define vr_subscr(a_1,a_2) (a_2)*vr_dim1 + a_1
#define vr_ref(a_1,a_2) vr[vr_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;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1 * 1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1 * 1;
vr -= vr_offset;
--work;
--rwork;
/* Function Body */
if (lsame_(jobvl, "N")) {
ijobvl = 1;
ilvl = FALSE_;
} else if (lsame_(jobvl, "V")) {
ijobvl = 2;
ilvl = TRUE_;
} else {
ijobvl = -1;
ilvl = FALSE_;
}
if (lsame_(jobvr, "N")) {
ijobvr = 1;
ilvr = FALSE_;
} else if (lsame_(jobvr, "V")) {
ijobvr = 2;
ilvr = TRUE_;
} else {
ijobvr = -1;
ilvr = FALSE_;
}
ilv = ilvl || ilvr;
/* Test the input arguments */
*info = 0;
lquery = *lwork == -1;
if (ijobvl <= 0) {
*info = -1;
} else if (ijobvr <= 0) {
*info = -2;
} else if (*n < 0) {
*info = -3;
} else if (*lda < max(1,*n)) {
*info = -5;
} else if (*ldb < max(1,*n)) {
*info = -7;
} else if (*ldvl < 1 || ilvl && *ldvl < *n) {
*info = -11;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -13;
}
/* Compute workspace
(Note: Comments in the code beginning "Workspace:" describe the
minimal amount of workspace needed at that point in the code,
as well as the preferred amount for good performance.
NB refers to the optimal block size for the immediately
following subroutine, as returned by ILAENV. The workspace is
computed assuming ILO = 1 and IHI = N, the worst case.) */
lwkmin = 1;
if (*info == 0 && (*lwork >= 1 || lquery)) {
lwkopt = *n + *n * ilaenv_(&c__1, "CGEQRF", " ", n, &c__1, n, &c__0, (
ftnlen)6, (ftnlen)1);
/* Computing MAX */
i__1 = 1, i__2 = *n << 1;
lwkmin = max(i__1,i__2);
work[1].r = (real) lwkopt, work[1].i = 0.f;
}
if (*lwork < lwkmin && ! lquery) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CGGEV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
work[1].r = (real) lwkopt, work[1].i = 0.f;
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("E") * slamch_("B");
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 matrices A, B to isolate eigenvalues if possible
(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;
if (ilv) {
icols = *n + 1 - ilo;
} else {
icols = irows;
}
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 orthogonal 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 VL
(Complex Workspace: need N, prefer N*NB) */
if (ilvl) {
claset_("Full", n, n, &c_b1, &c_b2, &vl[vl_offset], ldvl);
i__1 = irows - 1;
i__2 = irows - 1;
clacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vl_ref(ilo + 1,
ilo), ldvl);
i__1 = *lwork + 1 - iwrk;
cungqr_(&irows, &irows, &irows, &vl_ref(ilo, ilo), ldvl, &work[itau],
&work[iwrk], &i__1, &ierr);
}
/* Initialize VR */
if (ilvr) {
claset_("Full", n, n, &c_b1, &c_b2, &vr[vr_offset], ldvr);
}
/* Reduce to generalized Hessenberg form */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
cgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr);
} else {
cgghrd_("N", "N", &irows, &c__1, &irows, &a_ref(ilo, ilo), lda, &
b_ref(ilo, ilo), ldb, &vl[vl_offset], ldvl, &vr[vr_offset],
ldvr, &ierr);
}
/* Perform QZ algorithm (Compute eigenvalues, and optionally, the
Schur form and Schur vectors)
(Complex Workspace: need N)
(Real Workspace: need N) */
iwrk = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwrk;
chgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alpha[1], &beta[1], &vl[vl_offset], ldvl, &vr[
vr_offset], ldvr, &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 L70;
}
/* Compute Eigenvectors
(Real Workspace: need 2*N)
(Complex Workspace: need 2*N) */
if (ilv) {
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
ctgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwrk], &rwork[irwrk], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L70;
}
/* Undo balancing on VL and VR and normalization
(Workspace: none needed) */
if (ilvl) {
cggbak_("P", "L", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vl[vl_offset], ldvl, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = vl_subscr(jr, jc);
r__3 = temp, r__4 = (r__1 = vl[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vl_ref(jr, jc)), dabs(r__2));
temp = dmax(r__3,r__4);
/* L10: */
}
if (temp < smlnum) {
goto L30;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = vl_subscr(jr, jc);
i__4 = vl_subscr(jr, jc);
q__1.r = temp * vl[i__4].r, q__1.i = temp * vl[i__4].i;
vl[i__3].r = q__1.r, vl[i__3].i = q__1.i;
/* L20: */
}
L30:
;
}
}
if (ilvr) {
cggbak_("P", "R", n, &ilo, &ihi, &rwork[ileft], &rwork[iright], n,
&vr[vr_offset], ldvr, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
temp = 0.f;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
i__3 = vr_subscr(jr, jc);
r__3 = temp, r__4 = (r__1 = vr[i__3].r, dabs(r__1)) + (
r__2 = r_imag(&vr_ref(jr, jc)), dabs(r__2));
temp = dmax(r__3,r__4);
/* L40: */
}
if (temp < smlnum) {
goto L60;
}
temp = 1.f / temp;
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
i__3 = vr_subscr(jr, jc);
i__4 = vr_subscr(jr, jc);
q__1.r = temp * vr[i__4].r, q__1.i = temp * vr[i__4].i;
vr[i__3].r = q__1.r, vr[i__3].i = q__1.i;
/* L50: */
}
L60:
;
}
}
}
/* Undo scaling if necessary */
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);
}
L70:
work[1].r = (real) lwkopt, work[1].i = 0.f;
return 0;
/* End of CGGEV */
} /* cggev_ */