/* Subroutine */ int dggevx_(char *balanc, char *jobvl, char *jobvr, char *
sense, integer *n, doublereal *a, integer *lda, doublereal *b,
integer *ldb, doublereal *alphar, doublereal *alphai, doublereal *
beta, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr,
integer *ilo, integer *ihi, doublereal *lscale, doublereal *rscale,
doublereal *abnrm, doublereal *bbnrm, doublereal *rconde, doublereal *
rcondv, doublereal *work, integer *lwork, integer *iwork, 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
=======
DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right
generalized eigenvectors.
Optionally also, it 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) DOUBLE PRECISION 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 real 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) DOUBLE PRECISION 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 real Schur
form of the "balanced" versions of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. If ALPHAI(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector will be scaled so the largest component 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) DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector will be scaled so the largest component 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,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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION
The one-norm of the balanced matrix A.
BBNRM (output) DOUBLE PRECISION
The one-norm of the balanced matrix B.
RCONDE (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of
the selected eigenvalues, stored in consecutive elements of
the array. For a complex conjugate pair of eigenvalues two
consecutive elements of RCONDE are set to the same value.
Thus RCONDE(j), RCONDV(j), and the j-th columns of VL and VR
all correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If SENSE = 'V', RCONDE is not referenced.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of RCONDV are set to the same value. 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 = 'E', RCONDV is not referenced.
WORK (workspace/output) DOUBLE PRECISION 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(1,6*N).
If SENSE = 'E', LWORK >= 12*N.
If SENSE = 'V' or 'B', LWORK >= 2*N*N+12*N+16.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
IWORK (workspace) INTEGER array, dimension (N+6)
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 ALPHAR(j), ALPHAI(j), and BETA(j)
should be correct for j=INFO+1,...,N.
> N: =N+1: other than QZ iteration failed in DHGEQZ.
=N+2: error return from DTGEVC.
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.
=====================================================================
Decode the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c__0 = 0;
static doublereal c_b47 = 0.;
static doublereal c_b48 = 1.;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static logical pair;
static doublereal anrm, bnrm;
static integer ierr, itau;
static doublereal temp;
static logical ilvl, ilvr;
static integer iwrk, iwrk1, i__, j, m;
extern logical lsame_(char *, char *);
static integer icols, irows;
extern /* Subroutine */ int dlabad_(doublereal *, doublereal *);
static integer jc;
extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
static integer in;
extern doublereal dlamch_(char *);
static integer mm;
extern doublereal dlange_(char *, integer *, integer *, doublereal *,
integer *, doublereal *);
static integer jr;
extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
static logical ilascl, ilbscl;
extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *);
static logical ldumma[1];
static char chtemp[1];
static doublereal bignum;
extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *), dlaset_(char *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
static integer ijobvl;
extern /* Subroutine */ int dtgevc_(char *, char *, logical *, integer *,
doublereal *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, integer *, integer *, integer *,
doublereal *, integer *), dtgsna_(char *, char *,
logical *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, doublereal *, integer *, integer *, doublereal *,
integer *, integer *, integer *), xerbla_(char *,
integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ijobvr;
static logical wantsb;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *);
static doublereal anrmto;
static logical wantse;
static doublereal bnrmto;
extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
static integer minwrk, maxwrk;
static logical wantsn;
static doublereal smlnum;
static logical lquery, wantsv;
static doublereal eps;
static logical ilv;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define vl_ref(a_1,a_2) vl[(a_2)*vl_dim1 + a_1]
#define vr_ref(a_1,a_2) vr[(a_2)*vr_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--alphar;
--alphai;
--beta;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1 * 1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1 * 1;
vr -= vr_offset;
--lscale;
--rscale;
--rconde;
--rcondv;
--work;
--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;
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 (! (lsame_(balanc, "N") || lsame_(balanc, "S") || lsame_(balanc, "P")
|| 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 = -14;
} else if (*ldvr < 1 || ilvr && *ldvr < *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. The workspace is
computed assuming ILO = 1 and IHI = N, the worst case.) */
minwrk = 1;
if (*info == 0 && (*lwork >= 1 || lquery)) {
maxwrk = *n * 5 + *n * ilaenv_(&c__1, "DGEQRF", " ", n, &c__1, n, &
c__0, (ftnlen)6, (ftnlen)1);
/* Computing MAX */
i__1 = 1, i__2 = *n * 6;
minwrk = max(i__1,i__2);
if (wantse) {
/* Computing MAX */
i__1 = 1, i__2 = *n * 12;
minwrk = max(i__1,i__2);
} else if (wantsv || wantsb) {
minwrk = (*n << 1) * *n + *n * 12 + 16;
/* Computing MAX */
i__1 = maxwrk, i__2 = (*n << 1) * *n + *n * 12 + 16;
maxwrk = max(i__1,i__2);
}
work[1] = (doublereal) maxwrk;
}
if (*lwork < minwrk && ! lquery) {
*info = -26;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGGEVX", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S");
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute and/or balance the matrix pair (A,B)
(Workspace: need 6*N) */
dggbal_(balanc, n, &a[a_offset], lda, &b[b_offset], ldb, ilo, ihi, &
lscale[1], &rscale[1], &work[1], &ierr);
/* Compute ABNRM and BBNRM */
*abnrm = dlange_("1", n, n, &a[a_offset], lda, &work[1]);
if (ilascl) {
work[1] = *abnrm;
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, &c__1, &c__1, &work[1], &
c__1, &ierr);
*abnrm = work[1];
}
*bbnrm = dlange_("1", n, n, &b[b_offset], ldb, &work[1]);
if (ilbscl) {
work[1] = *bbnrm;
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, &c__1, &c__1, &work[1], &
c__1, &ierr);
*bbnrm = work[1];
}
/* Reduce B to triangular form (QR decomposition of B)
(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;
dgeqrf_(&irows, &icols, &b_ref(*ilo, *ilo), ldb, &work[itau], &work[iwrk],
&i__1, &ierr);
/* Apply the orthogonal transformation to A
(Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
dormqr_("L", "T", &irows, &icols, &irows, &b_ref(*ilo, *ilo), ldb, &work[
itau], &a_ref(*ilo, *ilo), lda, &work[iwrk], &i__1, &ierr);
/* Initialize VL and/or VR
(Workspace: need N, prefer N*NB) */
if (ilvl) {
dlaset_("Full", n, n, &c_b47, &c_b48, &vl[vl_offset], ldvl)
;
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b_ref(*ilo + 1, *ilo), ldb, &vl_ref(*ilo
+ 1, *ilo), ldvl);
i__1 = *lwork + 1 - iwrk;
dorgqr_(&irows, &irows, &irows, &vl_ref(*ilo, *ilo), ldvl, &work[itau]
, &work[iwrk], &i__1, &ierr);
}
if (ilvr) {
dlaset_("Full", n, n, &c_b47, &c_b48, &vr[vr_offset], ldvr)
;
}
/* Reduce to generalized Hessenberg form
(Workspace: none needed) */
if (ilv || ! wantsn) {
/* Eigenvectors requested -- work on whole matrix. */
dgghrd_(jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr);
} else {
dgghrd_("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 forms and Schur vectors)
(Workspace: need N) */
if (ilv || ! wantsn) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
dhgeqz_(chtemp, jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset]
, ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset], ldvl, &
vr[vr_offset], ldvr, &work[1], lwork, &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 L130;
}
/* Compute Eigenvectors and estimate condition numbers if desired
(Workspace: DTGEVC: need 6*N
DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
need N otherwise ) */
if (ilv || ! wantsn) {
if (ilv) {
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &
work[1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L130;
}
}
if (! wantsn) {
/* compute eigenvectors (DTGEVC) and estimate condition
numbers (DTGSNA). 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 recalculate
eigenvectors and estimate one condition numbers at a time. */
pair = FALSE_;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (pair) {
pair = FALSE_;
goto L20;
}
mm = 1;
if (i__ < *n) {
if (a_ref(i__ + 1, i__) != 0.) {
pair = TRUE_;
mm = 2;
}
}
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
bwork[j] = FALSE_;
/* L10: */
}
if (mm == 1) {
bwork[i__] = TRUE_;
} else if (mm == 2) {
bwork[i__] = TRUE_;
bwork[i__ + 1] = TRUE_;
}
iwrk = mm * *n + 1;
iwrk1 = iwrk + mm * *n;
/* Compute a pair of left and right eigenvectors.
(compute workspace: need up to 4*N + 6*N) */
if (wantse || wantsb) {
dtgevc_("B", "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &mm,
&m, &work[iwrk1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L130;
}
}
i__2 = *lwork - iwrk1 + 1;
dtgsna_(sense, "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &rconde[
i__], &rcondv[i__], &mm, &m, &work[iwrk1], &i__2, &
iwork[1], &ierr);
L20:
;
}
}
}
/* Undo balancing on VL and VR and normalization
(Workspace: none needed) */
if (ilvl) {
dggbak_(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) {
if (alphai[jc] < 0.) {
goto L70;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vl_ref(jr, jc), abs(d__1));
temp = max(d__2,d__3);
/* L30: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vl_ref(jr, jc), abs(d__1)) + (
d__2 = vl_ref(jr, jc + 1), abs(d__2));
temp = max(d__3,d__4);
/* L40: */
}
}
if (temp < smlnum) {
goto L70;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl_ref(jr, jc) = vl_ref(jr, jc) * temp;
/* L50: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl_ref(jr, jc) = vl_ref(jr, jc) * temp;
vl_ref(jr, jc + 1) = vl_ref(jr, jc + 1) * temp;
/* L60: */
}
}
L70:
;
}
}
if (ilvr) {
dggbak_(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) {
if (alphai[jc] < 0.) {
goto L120;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vr_ref(jr, jc), abs(d__1));
temp = max(d__2,d__3);
/* L80: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vr_ref(jr, jc), abs(d__1)) + (
d__2 = vr_ref(jr, jc + 1), abs(d__2));
temp = max(d__3,d__4);
/* L90: */
}
}
if (temp < smlnum) {
goto L120;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr_ref(jr, jc) = vr_ref(jr, jc) * temp;
/* L100: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr_ref(jr, jc) = vr_ref(jr, jc) * temp;
vr_ref(jr, jc + 1) = vr_ref(jr, jc + 1) * temp;
/* L110: */
}
}
L120:
;
}
}
/* Undo scaling if necessary */
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
ierr);
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
L130:
work[1] = (doublereal) maxwrk;
return 0;
/* End of DGGEVX */
} /* dggevx_ */
/* Subroutine */ int dgegv_(char *jobvl, char *jobvr, integer *n, doublereal *
a, integer *lda, doublereal *b, integer *ldb, doublereal *alphar,
doublereal *alphai, doublereal *beta, doublereal *vl, integer *ldvl,
doublereal *vr, integer *ldvr, doublereal *work, integer *lwork,
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;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
integer jc, nb, in, jr, nb1, nb2, nb3, ihi, ilo;
doublereal eps;
logical ilv;
doublereal absb, anrm, bnrm;
integer itau;
doublereal temp;
logical ilvl, ilvr;
integer lopt;
doublereal anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta;
integer ileft, iinfo, icols, iwork, irows;
doublereal salfai;
doublereal salfar;
doublereal safmin;
doublereal safmax;
char chtemp[1];
logical ldumma[1];
integer ijobvl, iright;
logical ilimit;
integer ijobvr;
doublereal onepls;
integer lwkmin;
integer lwkopt;
logical lquery;
/* -- LAPACK driver routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine DGGEV. */
/* DGEGV computes the eigenvalues and, optionally, the left and/or right */
/* eigenvectors of a real 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) DOUBLE PRECISION array, dimension (LDA, N) */
/* On entry, the matrix A. */
/* If JOBVL = 'V' or JOBVR = 'V', then on exit A */
/* contains the real 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 */
/* blocks from the Schur form will be correct. See DGGHRD and */
/* DHGEQZ for details. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) DOUBLE PRECISION 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 those elements of */
/* B corresponding to the diagonal blocks from the Schur form of */
/* A will be correct. See DGGHRD and DHGEQZ for details. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* The real parts of each scalar alpha defining an eigenvalue of */
/* GNEP. */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* The imaginary parts of each scalar alpha defining an */
/* eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th */
/* eigenvalue is real; if positive, then the j-th and */
/* (j+1)-st eigenvalues are a complex conjugate pair, with */
/* ALPHAI(j+1) = -ALPHAI(j). */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* The scalars beta that define the eigenvalues of GNEP. */
/* Together, the quantities alpha = (ALPHAR(j),ALPHAI(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) DOUBLE PRECISION 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. */
/* If the j-th eigenvalue is real, then u(j) = VL(:,j). */
/* If the j-th and (j+1)-st eigenvalues form a complex conjugate */
/* pair, then */
/* u(j) = VL(:,j) + i*VL(:,j+1) */
/* and */
/* u(j+1) = VL(:,j) - i*VL(:,j+1). */
/* 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) DOUBLE PRECISION 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. */
/* If the j-th eigenvalue is real, then x(j) = VR(:,j). */
/* If the j-th and (j+1)-st eigenvalues form a complex conjugate */
/* pair, then */
/* x(j) = VR(:,j) + i*VR(:,j+1) */
/* and */
/* x(j+1) = VR(:,j) - i*VR(:,j+1). */
/* Each eigenvector is scaled so that its largest component has */
/* abs(real part) + abs(imag. part) = 1, except for eigenvalues */
/* 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) DOUBLE PRECISION 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(1,8*N). */
/* For good performance, LWORK must generally be larger. */
/* To compute the optimal value of LWORK, call ILAENV to get */
/* blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: */
/* NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; */
/* The optimal LWORK is: */
/* 2*N + MAX( 6*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. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* The QZ iteration failed. No eigenvectors have been */
/* calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) */
/* > N: errors that usually indicate LAPACK problems: */
/* =N+1: error return from DGGBAL */
/* =N+2: error return from DGEQRF */
/* =N+3: error return from DORMQR */
/* =N+4: error return from DORGQR */
/* =N+5: error return from DGGHRD */
/* =N+6: error return from DHGEQZ (other than failed */
/* iteration) */
/* =N+7: error return from DTGEVC */
/* =N+8: error return from DGGBAK (computing VL) */
/* =N+9: error return from DGGBAK (computing VR) */
/* =N+10: error return from DLASCL (various calls) */
/* Further Details */
/* =============== */
/* Balancing */
/* --------- */
/* This driver calls DGGBAL 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, DGGBAK 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 real Schur */
/* form[*] of the "balanced" versions of A and B. If no eigenvectors */
/* are computed, then only the diagonal blocks will be correct. */
/* [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", */
/* by Golub & van Loan, pub. by Johns Hopkins U. Press. */
/* ===================================================================== */
/* 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;
--alphar;
--alphai;
--beta;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
/* Function Body */
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 << 3;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1] = (doublereal) lwkopt;
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 = -12;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -14;
} else if (*lwork < lwkmin && ! lquery) {
*info = -16;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1);
nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
/* Computing MAX */
i__1 = *n * 6, i__2 = *n * (nb + 1);
lopt = (*n << 1) + max(i__1,i__2);
work[1] = (doublereal) lopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGEGV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("E") * dlamch_("B");
safmin = dlamch_("S");
safmin += safmin;
safmax = 1. / safmin;
onepls = eps * 4 + 1.;
/* Scale A */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
anrm1 = anrm;
anrm2 = 1.;
if (anrm < 1.) {
if (safmax * anrm < 1.) {
anrm1 = safmin;
anrm2 = safmax * anrm;
}
}
if (anrm > 0.) {
dlascl_("G", &c_n1, &c_n1, &anrm, &c_b27, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Scale B */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
bnrm1 = bnrm;
bnrm2 = 1.;
if (bnrm < 1.) {
if (safmax * bnrm < 1.) {
bnrm1 = safmin;
bnrm2 = safmax * bnrm;
}
}
if (bnrm > 0.) {
dlascl_("G", &c_n1, &c_n1, &bnrm, &c_b27, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular */
/* Workspace layout: (8*N words -- "work" requires 6*N words) */
ileft = 1;
iright = *n + 1;
iwork = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L120;
}
/* 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 = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L120;
}
i__1 = *lwork + 1 - iwork;
dormqr_("L", "T", &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__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L120;
}
if (ilvl) {
dlaset_("Full", n, n, &c_b38, &c_b27, &vl[vl_offset], ldvl)
;
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo +
1 + ilo * vl_dim1], ldvl);
i__1 = *lwork + 1 - iwork;
dorgqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[
itau], &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L120;
}
}
if (ilvr) {
dlaset_("Full", n, n, &c_b38, &c_b27, &vr[vr_offset], ldvr)
;
}
/* Reduce to generalized Hessenberg form */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
dgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo);
} else {
dgghrd_("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 L120;
}
/* Perform QZ algorithm */
iwork = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwork;
dhgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset],
ldvl, &vr[vr_offset], ldvr, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + 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 L120;
}
if (ilv) {
/* Compute Eigenvectors (DTGEVC requires 6*N words of workspace) */
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwork], &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L120;
}
/* Undo balancing on VL and VR, rescale */
if (ilvl) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vl[vl_offset], ldvl, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L120;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L50;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vl[jr + jc * vl_dim1],
abs(d__1));
temp = max(d__2,d__3);
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vl[jr + jc * vl_dim1],
abs(d__1)) + (d__2 = vl[jr + (jc + 1) *
vl_dim1], abs(d__2));
temp = max(d__3,d__4);
}
}
if (temp < safmin) {
goto L50;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
vl[jr + (jc + 1) * vl_dim1] *= temp;
}
}
L50:
;
}
}
if (ilvr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vr[vr_offset], ldvr, &iinfo);
if (iinfo != 0) {
*info = *n + 9;
goto L120;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L100;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vr[jr + jc * vr_dim1],
abs(d__1));
temp = max(d__2,d__3);
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vr[jr + jc * vr_dim1],
abs(d__1)) + (d__2 = vr[jr + (jc + 1) *
vr_dim1], abs(d__2));
temp = max(d__3,d__4);
}
}
if (temp < safmin) {
goto L100;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
vr[jr + (jc + 1) * vr_dim1] *= temp;
}
}
L100:
;
}
}
/* 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) {
absar = (d__1 = alphar[jc], abs(d__1));
absai = (d__1 = alphai[jc], abs(d__1));
absb = (d__1 = beta[jc], abs(d__1));
salfar = anrm * alphar[jc];
salfai = anrm * alphai[jc];
sbeta = bnrm * beta[jc];
ilimit = FALSE_;
scale = 1.;
/* Check for significant underflow in ALPHAI */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absar, d__1 = max(d__1,d__2), d__2 = eps *
absb;
if (abs(salfai) < safmin && absai >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
d__1 = onepls * safmin, d__2 = anrm2 * absai;
scale = onepls * safmin / anrm1 / max(d__1,d__2);
} else if (salfai == 0.) {
/* If insignificant underflow in ALPHAI, then make the */
/* conjugate eigenvalue real. */
if (alphai[jc] < 0. && jc > 1) {
alphai[jc - 1] = 0.;
} else if (alphai[jc] > 0. && jc < *n) {
alphai[jc + 1] = 0.;
}
}
/* Check for significant underflow in ALPHAR */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absai, d__1 = max(d__1,d__2), d__2 = eps *
absb;
if (abs(salfar) < safmin && absar >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
d__3 = onepls * safmin, d__4 = anrm2 * absar;
d__1 = scale, d__2 = onepls * safmin / anrm1 / max(d__3,d__4);
scale = max(d__1,d__2);
}
/* Check for significant underflow in BETA */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absar, d__1 = max(d__1,d__2), d__2 = eps *
absai;
if (abs(sbeta) < safmin && absb >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
d__3 = onepls * safmin, d__4 = bnrm2 * absb;
d__1 = scale, d__2 = onepls * safmin / bnrm1 / max(d__3,d__4);
scale = max(d__1,d__2);
}
/* Check for possible overflow when limiting scaling */
if (ilimit) {
/* Computing MAX */
d__1 = abs(salfar), d__2 = abs(salfai), d__1 = max(d__1,d__2),
d__2 = abs(sbeta);
temp = scale * safmin * max(d__1,d__2);
if (temp > 1.) {
scale /= temp;
}
if (scale < 1.) {
ilimit = FALSE_;
}
}
/* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary. */
if (ilimit) {
salfar = scale * alphar[jc] * anrm;
salfai = scale * alphai[jc] * anrm;
sbeta = scale * beta[jc] * bnrm;
}
alphar[jc] = salfar;
alphai[jc] = salfai;
beta[jc] = sbeta;
}
L120:
work[1] = (doublereal) lwkopt;
return 0;
/* End of DGEGV */
} /* dgegv_ */
/* Subroutine */ int dggevx_(char *balanc, char *jobvl, char *jobvr, char *
sense, integer *n, doublereal *a, integer *lda, doublereal *b,
integer *ldb, doublereal *alphar, doublereal *alphai, doublereal *
beta, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr,
integer *ilo, integer *ihi, doublereal *lscale, doublereal *rscale,
doublereal *abnrm, doublereal *bbnrm, doublereal *rconde, doublereal *
rcondv, doublereal *work, integer *lwork, 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;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
integer i__, j, m, jc, in, mm, jr;
doublereal eps;
logical ilv, pair;
doublereal anrm, bnrm;
integer ierr, itau;
doublereal temp;
logical ilvl, ilvr;
integer iwrk, iwrk1;
integer icols;
logical noscl;
integer irows;
logical ilascl, ilbscl;
logical ldumma[1];
char chtemp[1];
doublereal bignum;
integer ijobvl;
integer ijobvr;
logical wantsb;
doublereal anrmto;
logical wantse;
doublereal bnrmto;
integer minwrk, maxwrk;
logical wantsn;
doublereal smlnum;
logical lquery, wantsv;
/* -- LAPACK driver routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) */
/* the generalized eigenvalues, and optionally, the left and/or right */
/* generalized eigenvectors. */
/* Optionally also, it 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) DOUBLE PRECISION 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 real 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) DOUBLE PRECISION 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 real Schur */
/* form of the "balanced" versions of the input A and B. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* be the generalized eigenvalues. If ALPHAI(j) is zero, then */
/* the j-th eigenvalue is real; if positive, then the j-th and */
/* (j+1)-st eigenvalues are a complex conjugate pair, with */
/* ALPHAI(j+1) negative. */
/* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left eigenvectors u(j) are stored one */
/* after another in the columns of VL, in the same order as */
/* their eigenvalues. If the j-th eigenvalue is real, then */
/* u(j) = VL(:,j), the j-th column of VL. If the j-th and */
/* (j+1)-th eigenvalues form a complex conjugate pair, then */
/* u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). */
/* Each eigenvector will be scaled so the largest component 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) DOUBLE PRECISION array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right eigenvectors v(j) are stored one */
/* after another in the columns of VR, in the same order as */
/* their eigenvalues. If the j-th eigenvalue is real, then */
/* v(j) = VR(:,j), the j-th column of VR. If the j-th and */
/* (j+1)-th eigenvalues form a complex conjugate pair, then */
/* v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). */
/* Each eigenvector will be scaled so the largest component 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 */
/* If BALANC = 'N' or 'S', ILO = 1 and IHI = N. */
/* LSCALE (output) DOUBLE PRECISION array, dimension (N) */
/* Details of the permutations and scaling factors applied */
/* to the left side of A and B. If PL(j) is the index of the */
/* row interchanged with row j, and DL(j) is the scaling */
/* factor applied to row j, then */
/* The order in which the interchanges are made is N to IHI+1, */
/* then 1 to ILO-1. */
/* RSCALE (output) DOUBLE PRECISION array, dimension (N) */
/* Details of the permutations and scaling factors applied */
/* to the right side of A and B. If PR(j) is the index of the */
/* column interchanged with column j, and DR(j) is the scaling */
/* factor applied to column j, then */
/* The order in which the interchanges are made is N to IHI+1, */
/* then 1 to ILO-1. */
/* ABNRM (output) DOUBLE PRECISION */
/* The one-norm of the balanced matrix A. */
/* BBNRM (output) DOUBLE PRECISION */
/* The one-norm of the balanced matrix B. */
/* RCONDE (output) DOUBLE PRECISION array, dimension (N) */
/* If SENSE = 'E' or 'B', the reciprocal condition numbers of */
/* the eigenvalues, stored in consecutive elements of the array. */
/* For a complex conjugate pair of eigenvalues two consecutive */
/* elements of RCONDE are set to the same value. Thus RCONDE(j), */
/* RCONDV(j), and the j-th columns of VL and VR all correspond */
/* to the j-th eigenpair. */
/* If SENSE = 'N or 'V', RCONDE is not referenced. */
/* RCONDV (output) DOUBLE PRECISION array, dimension (N) */
/* If SENSE = 'V' or 'B', the estimated reciprocal condition */
/* numbers of the eigenvectors, stored in consecutive elements */
/* of the array. For a complex eigenvector two consecutive */
/* elements of RCONDV are set to the same value. 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) DOUBLE PRECISION 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 BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', */
/* LWORK >= max(1,6*N). */
/* If SENSE = 'E' or 'B', LWORK >= max(1,10*N). */
/* If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace) INTEGER array, dimension (N+6) */
/* 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. */
/* The QZ iteration failed. No eigenvectors have been */
/* calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) */
/* > N: =N+1: other than QZ iteration failed in DHGEQZ. */
/* =N+2: error return from DTGEVC. */
/* 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. */
/* ===================================================================== */
/* 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;
--alphar;
--alphai;
--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;
--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 (! (lsame_(balanc, "N") || lsame_(balanc, "S") || lsame_(balanc, "P")
|| 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 = -14;
} else if (*ldvr < 1 || ilvr && *ldvr < *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. The workspace is */
/* computed assuming ILO = 1 and IHI = N, the worst case.) */
if (*info == 0) {
if (*n == 0) {
minwrk = 1;
maxwrk = 1;
} else {
if (noscl && ! ilv) {
minwrk = *n << 1;
} else {
minwrk = *n * 6;
}
if (wantse || wantsb) {
minwrk = *n * 10;
}
if (wantsv || wantsb) {
/* Computing MAX */
i__1 = minwrk, i__2 = (*n << 1) * (*n + 4) + 16;
minwrk = max(i__1,i__2);
}
maxwrk = minwrk;
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "DGEQRF", " ", n, &
c__1, n, &c__0);
maxwrk = max(i__1,i__2);
/* Computing MAX */
i__1 = maxwrk, i__2 = *n + *n * ilaenv_(&c__1, "DORMQR", " ", 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, "DORGQR",
" ", n, &c__1, n, &c__0);
maxwrk = max(i__1,i__2);
}
}
work[1] = (doublereal) maxwrk;
if (*lwork < minwrk && ! lquery) {
*info = -26;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGGEVX", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S");
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute and/or balance the matrix pair (A,B) */
/* (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise) */
dggbal_(balanc, n, &a[a_offset], lda, &b[b_offset], ldb, ilo, ihi, &
lscale[1], &rscale[1], &work[1], &ierr);
/* Compute ABNRM and BBNRM */
*abnrm = dlange_("1", n, n, &a[a_offset], lda, &work[1]);
if (ilascl) {
work[1] = *abnrm;
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, &c__1, &c__1, &work[1], &
c__1, &ierr);
*abnrm = work[1];
}
*bbnrm = dlange_("1", n, n, &b[b_offset], ldb, &work[1]);
if (ilbscl) {
work[1] = *bbnrm;
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, &c__1, &c__1, &work[1], &
c__1, &ierr);
*bbnrm = work[1];
}
/* Reduce B to triangular form (QR decomposition of B) */
/* (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;
dgeqrf_(&irows, &icols, &b[*ilo + *ilo * b_dim1], ldb, &work[itau], &work[
iwrk], &i__1, &ierr);
/* Apply the orthogonal transformation to A */
/* (Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
dormqr_("L", "T", &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) {
dlaset_("Full", n, n, &c_b59, &c_b60, &vl[vl_offset], ldvl)
;
if (irows > 1) {
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b[*ilo + 1 + *ilo * b_dim1], ldb, &vl[
*ilo + 1 + *ilo * vl_dim1], ldvl);
}
i__1 = *lwork + 1 - iwrk;
dorgqr_(&irows, &irows, &irows, &vl[*ilo + *ilo * vl_dim1], ldvl, &
work[itau], &work[iwrk], &i__1, &ierr);
}
if (ilvr) {
dlaset_("Full", n, n, &c_b59, &c_b60, &vr[vr_offset], ldvr)
;
}
/* Reduce to generalized Hessenberg form */
/* (Workspace: none needed) */
if (ilv || ! wantsn) {
/* Eigenvectors requested -- work on whole matrix. */
dgghrd_(jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr);
} else {
dgghrd_("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) */
/* (Workspace: need N) */
if (ilv || ! wantsn) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
dhgeqz_(chtemp, jobvl, jobvr, n, ilo, ihi, &a[a_offset], lda, &b[b_offset]
, ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset], ldvl, &
vr[vr_offset], ldvr, &work[1], lwork, &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 L130;
}
/* Compute Eigenvectors and estimate condition numbers if desired */
/* (Workspace: DTGEVC: need 6*N */
/* DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B', */
/* need N otherwise ) */
if (ilv || ! wantsn) {
if (ilv) {
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &
work[1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L130;
}
}
if (! wantsn) {
/* compute eigenvectors (DTGEVC) and estimate condition */
/* numbers (DTGSNA). 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 recalculate */
/* eigenvectors and estimate one condition numbers at a time. */
pair = FALSE_;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (pair) {
pair = FALSE_;
goto L20;
}
mm = 1;
if (i__ < *n) {
if (a[i__ + 1 + i__ * a_dim1] != 0.) {
pair = TRUE_;
mm = 2;
}
}
i__2 = *n;
for (j = 1; j <= i__2; ++j) {
bwork[j] = FALSE_;
}
if (mm == 1) {
bwork[i__] = TRUE_;
} else if (mm == 2) {
bwork[i__] = TRUE_;
bwork[i__ + 1] = TRUE_;
}
iwrk = mm * *n + 1;
iwrk1 = iwrk + mm * *n;
/* Compute a pair of left and right eigenvectors. */
/* (compute workspace: need up to 4*N + 6*N) */
if (wantse || wantsb) {
dtgevc_("B", "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &mm,
&m, &work[iwrk1], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L130;
}
}
i__2 = *lwork - iwrk1 + 1;
dtgsna_(sense, "S", &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &work[1], n, &work[iwrk], n, &rconde[
i__], &rcondv[i__], &mm, &m, &work[iwrk1], &i__2, &
iwork[1], &ierr);
L20:
;
}
}
}
/* Undo balancing on VL and VR and normalization */
/* (Workspace: none needed) */
if (ilvl) {
dggbak_(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) {
if (alphai[jc] < 0.) {
goto L70;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vl[jr + jc * vl_dim1], abs(
d__1));
temp = max(d__2,d__3);
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vl[jr + jc * vl_dim1], abs(
d__1)) + (d__2 = vl[jr + (jc + 1) * vl_dim1], abs(
d__2));
temp = max(d__3,d__4);
}
}
if (temp < smlnum) {
goto L70;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
vl[jr + (jc + 1) * vl_dim1] *= temp;
}
}
L70:
;
}
}
if (ilvr) {
dggbak_(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) {
if (alphai[jc] < 0.) {
goto L120;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vr[jr + jc * vr_dim1], abs(
d__1));
temp = max(d__2,d__3);
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vr[jr + jc * vr_dim1], abs(
d__1)) + (d__2 = vr[jr + (jc + 1) * vr_dim1], abs(
d__2));
temp = max(d__3,d__4);
}
}
if (temp < smlnum) {
goto L120;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
vr[jr + (jc + 1) * vr_dim1] *= temp;
}
}
L120:
;
}
}
/* Undo scaling if necessary */
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
ierr);
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
L130:
work[1] = (doublereal) maxwrk;
return 0;
/* End of DGGEVX */
} /* dggevx_ */
/* Subroutine */ int dgegs_(char *jobvsl, char *jobvsr, integer *n,
doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *
alphar, doublereal *alphai, doublereal *beta, doublereal *vsl,
integer *ldvsl, doublereal *vsr, integer *ldvsr, doublereal *work,
integer *lwork, 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;
/* Local variables */
integer nb, nb1, nb2, nb3, ihi, ilo;
doublereal 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 dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
extern doublereal dlamch_(char *), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *);
extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
logical ilascl, ilbscl;
extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *);
doublereal safmin;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *);
doublereal bignum;
extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *);
integer ijobvl, iright, ijobvr;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *);
doublereal anrmto;
integer lwkmin;
doublereal bnrmto;
extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
doublereal smlnum;
integer 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 DGGES. */
/* DGEGS computes the eigenvalues, real Schur form, and, optionally, */
/* left and or/right Schur vectors of a real matrix pair (A,B). */
/* Given two square matrices A and B, the generalized real Schur */
/* factorization has the form */
/* A = Q*S*Z**T, B = Q*T*Z**T */
/* where Q and Z are orthogonal matrices, T is upper triangular, and S */
/* is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal */
/* blocks, the 2-by-2 blocks corresponding to complex conjugate pairs */
/* of eigenvalues of (A,B). 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 */
/* DGEGV should be used instead. See DGEGV 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) DOUBLE PRECISION array, dimension (LDA, N) */
/* On entry, the matrix A. */
/* On exit, the upper quasi-triangular matrix S from the */
/* generalized real Schur factorization. */
/* LDA (input) INTEGER */
/* The leading dimension of A. LDA >= max(1,N). */
/* B (input/output) DOUBLE PRECISION array, dimension (LDB, N) */
/* On entry, the matrix B. */
/* On exit, the upper triangular matrix T from the generalized */
/* real Schur factorization. */
/* LDB (input) INTEGER */
/* The leading dimension of B. LDB >= max(1,N). */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* The real parts of each scalar alpha defining an eigenvalue */
/* of GNEP. */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* The imaginary parts of each scalar alpha defining an */
/* eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th */
/* eigenvalue is real; if positive, then the j-th and (j+1)-st */
/* eigenvalues are a complex conjugate pair, with */
/* ALPHAI(j+1) = -ALPHAI(j). */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* The scalars beta that define the eigenvalues of GNEP. */
/* Together, the quantities alpha = (ALPHAR(j),ALPHAI(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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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(1,4*N). */
/* For good performance, LWORK must generally be larger. */
/* To compute the optimal value of LWORK, call ILAENV to get */
/* blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: */
/* NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR */
/* The optimal LWORK is 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. */
/* 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 ALPHAR(j), ALPHAI(j), and BETA(j) should */
/* be correct for j=INFO+1,...,N. */
/* > N: errors that usually indicate LAPACK problems: */
/* =N+1: error return from DGGBAL */
/* =N+2: error return from DGEQRF */
/* =N+3: error return from DORMQR */
/* =N+4: error return from DORGQR */
/* =N+5: error return from DGGHRD */
/* =N+6: error return from DHGEQZ (other than failed */
/* iteration) */
/* =N+7: error return from DGGBAK (computing VSL) */
/* =N+8: error return from DGGBAK (computing VSR) */
/* =N+9: error return from DLASCL (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;
--alphar;
--alphai;
--beta;
vsl_dim1 = *ldvsl;
vsl_offset = 1 + vsl_dim1;
vsl -= vsl_offset;
vsr_dim1 = *ldvsr;
vsr_offset = 1 + vsr_dim1;
vsr -= vsr_offset;
--work;
/* 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 << 2;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1] = (doublereal) lwkopt;
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 = -12;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*info = -14;
} else if (*lwork < lwkmin && ! lquery) {
*info = -16;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1);
nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1);
nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
lopt = (*n << 1) + *n * (nb + 1);
work[1] = (doublereal) lopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGEGS ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("E") * dlamch_("B");
safmin = dlamch_("S");
smlnum = *n * safmin / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("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 = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("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 */
/* Workspace layout: (2*N words -- "work..." not actually used) */
/* left_permutation, right_permutation, work... */
ileft = 1;
iright = *n + 1;
iwork = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L10;
}
/* Reduce B to triangular form, and initialize VSL and/or VSR */
/* Workspace layout: ("work..." must have at least N words) */
/* left_permutation, right_permutation, tau, work... */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L10;
}
i__1 = *lwork + 1 - iwork;
dormqr_("L", "T", &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__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L10;
}
if (ilvsl) {
dlaset_("Full", n, n, &c_b36, &c_b37, &vsl[vsl_offset], ldvsl);
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vsl[ilo
+ 1 + ilo * vsl_dim1], ldvsl);
i__1 = *lwork + 1 - iwork;
dorgqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, &
work[itau], &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L10;
}
}
if (ilvsr) {
dlaset_("Full", n, n, &c_b36, &c_b37, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
dgghrd_(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 */
/* Workspace layout: ("work..." must have at least 1 word) */
/* left_permutation, right_permutation, work... */
iwork = itau;
i__1 = *lwork + 1 - iwork;
dhgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[vsl_offset]
, ldvsl, &vsr[vsr_offset], ldvsr, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + 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) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsl[
vsl_offset], ldvsl, &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L10;
}
}
if (ilvsr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsr[
vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L10;
}
}
/* Undo scaling */
if (ilascl) {
dlascl_("H", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
if (ilbscl) {
dlascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("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] = (doublereal) lwkopt;
return 0;
/* End of DGEGS */
} /* dgegs_ */
/* Subroutine */ int dgegs_(char *jobvsl, char *jobvsr, integer *n,
doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *
alphar, doublereal *alphai, doublereal *beta, doublereal *vsl,
integer *ldvsl, doublereal *vsr, integer *ldvsr, doublereal *work,
integer *lwork, 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 DGGES.
DGEGS computes for a pair of N-by-N real nonsymmetric matrices A, B:
the generalized eigenvalues (alphar +/- alphai*i, beta), the real
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 DGEGV
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 orthogonal matrix and
both on the right by another orthogonal matrix, these two orthogonal
matrices being chosen so as to bring the pair of matrices into
(real) Schur form.
A pair of matrices A, B is in generalized real Schur form if B is
upper triangular with non-negative diagonal and A is block upper
triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of A will be
"standardized" by making the corresponding elements of B have the
form:
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2-by-2 blocks in A and B will
have a complex conjugate pair of generalized eigenvalues.
The left and right Schur vectors are the columns of VSL and VSR,
respectively, where VSL and VSR are the orthogonal matrices
which reduce A and B to Schur form:
Schur form of (A,B) = ( (VSL)**T A (VSR), (VSL)**T 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) DOUBLE PRECISION 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.
Note: to avoid overflow, the Frobenius norm of the matrix
A should be less than the overflow threshold.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION 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.
Note: to avoid overflow, the Frobenius norm of the matrix
B should be less than the overflow threshold.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i,
j=1,...,N and BETA(j),j=1,...,N are the diagonals of the
complex Schur form (A,B) that would result if the 2-by-2
diagonal blocks of the real Schur form of (A,B) were further
reduced to triangular form using 2-by-2 complex unitary
transformations. If ALPHAI(j) is zero, then the j-th
eigenvalue is real; if positive, then the j-th and (j+1)-st
eigenvalues are a complex conjugate pair, with ALPHAI(j+1)
negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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(1,4*N).
For good performance, LWORK must generally be larger.
To compute the optimal value of LWORK, call ILAENV to get
blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
The optimal LWORK is 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.
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 ALPHAR(j), ALPHAI(j), and BETA(j) should
be correct for j=INFO+1,...,N.
> N: errors that usually indicate LAPACK problems:
=N+1: error return from DGGBAL
=N+2: error return from DGEQRF
=N+3: error return from DORMQR
=N+4: error return from DORGQR
=N+5: error return from DGGHRD
=N+6: error return from DHGEQZ (other than failed
iteration)
=N+7: error return from DGGBAK (computing VSL)
=N+8: error return from DGGBAK (computing VSR)
=N+9: error return from DLASCL (various places)
=====================================================================
Decode the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c_n1 = -1;
static doublereal c_b36 = 0.;
static doublereal c_b37 = 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;
/* Local variables */
static doublereal 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 dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *);
static integer nb;
extern /* Subroutine */ int dggbal_(char *, integer *, doublereal *,
integer *, doublereal *, integer *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *);
extern doublereal dlamch_(char *), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *);
extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
static logical ilascl, ilbscl;
extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *);
static doublereal safmin;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *),
xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static doublereal bignum;
extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *);
static integer ijobvl, iright, ijobvr;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *);
static doublereal anrmto;
static integer lwkmin, nb1, nb2, nb3;
static doublereal bnrmto;
extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
static doublereal smlnum;
static integer lwkopt;
static logical lquery;
static integer ihi, ilo;
static doublereal eps;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define vsl_ref(a_1,a_2) vsl[(a_2)*vsl_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--alphar;
--alphai;
--beta;
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;
/* 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 << 2;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1] = (doublereal) lwkopt;
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 = -12;
} else if (*ldvsr < 1 || ilvsr && *ldvsr < *n) {
*info = -14;
} else if (*lwork < lwkmin && ! lquery) {
*info = -16;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
nb3 = ilaenv_(&c__1, "DORGQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
/* Computing MAX */
i__1 = max(nb1,nb2);
nb = max(i__1,nb3);
lopt = (*n << 1) + *n * (nb + 1);
work[1] = (doublereal) lopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGEGS ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("E") * dlamch_("B");
safmin = dlamch_("S");
smlnum = *n * safmin / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("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 = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("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
Workspace layout: (2*N words -- "work..." not actually used)
left_permutation, right_permutation, work... */
ileft = 1;
iright = *n + 1;
iwork = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwork], &iinfo);
if (iinfo != 0) {
*info = *n + 1;
goto L10;
}
/* Reduce B to triangular form, and initialize VSL and/or VSR
Workspace layout: ("work..." must have at least N words)
left_permutation, right_permutation, tau, work... */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
dgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwork],
&i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L10;
}
i__1 = *lwork + 1 - iwork;
dormqr_("L", "T", &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__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L10;
}
if (ilvsl) {
dlaset_("Full", n, n, &c_b36, &c_b37, &vsl[vsl_offset], ldvsl);
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vsl_ref(ilo +
1, ilo), ldvsl);
i__1 = *lwork + 1 - iwork;
dorgqr_(&irows, &irows, &irows, &vsl_ref(ilo, ilo), ldvsl, &work[itau]
, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L10;
}
}
if (ilvsr) {
dlaset_("Full", n, n, &c_b36, &c_b37, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
dgghrd_(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
Workspace layout: ("work..." must have at least 1 word)
left_permutation, right_permutation, work... */
iwork = itau;
i__1 = *lwork + 1 - iwork;
dhgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[vsl_offset]
, ldvsl, &vsr[vsr_offset], ldvsr, &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + 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) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsl[
vsl_offset], ldvsl, &iinfo);
if (iinfo != 0) {
*info = *n + 7;
goto L10;
}
}
if (ilvsr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsr[
vsr_offset], ldvsr, &iinfo);
if (iinfo != 0) {
*info = *n + 8;
goto L10;
}
}
/* Undo scaling */
if (ilascl) {
dlascl_("H", &c_n1, &c_n1, &anrmto, &anrm, n, n, &a[a_offset], lda, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("G", &c_n1, &c_n1, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
}
if (ilbscl) {
dlascl_("U", &c_n1, &c_n1, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
iinfo);
if (iinfo != 0) {
*info = *n + 9;
return 0;
}
dlascl_("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] = (doublereal) lwkopt;
return 0;
/* End of DGEGS */
} /* dgegs_ */
/* Subroutine */ int dgges_(char *jobvsl, char *jobvsr, char *sort, L_fp
selctg, integer *n, doublereal *a, integer *lda, doublereal *b,
integer *ldb, integer *sdim, doublereal *alphar, doublereal *alphai,
doublereal *beta, doublereal *vsl, integer *ldvsl, doublereal *vsr,
integer *ldvsr, doublereal *work, integer *lwork, 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;
doublereal d__1;
/* Local variables */
integer i__, ip;
doublereal dif[2];
integer ihi, ilo;
doublereal eps, anrm, bnrm;
integer idum[1], ierr, itau, iwrk;
doublereal pvsl, pvsr;
integer ileft, icols;
logical cursl, ilvsl, ilvsr;
integer irows;
logical lst2sl;
logical ilascl, ilbscl;
doublereal safmin;
doublereal safmax;
doublereal bignum;
integer ijobvl, iright;
integer ijobvr;
doublereal anrmto, bnrmto;
logical lastsl;
integer minwrk, maxwrk;
doublereal smlnum;
logical wantst, lquery;
/* -- LAPACK driver routine (version 3.2) -- */
/* November 2006 */
/* Purpose */
/* ======= */
/* DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), */
/* the generalized eigenvalues, the generalized real Schur form (S,T), */
/* optionally, the left and/or right matrices of Schur vectors (VSL and */
/* VSR). This gives the generalized Schur factorization */
/* (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T ) */
/* Optionally, it also orders the eigenvalues so that a selected cluster */
/* of eigenvalues appears in the leading diagonal blocks of the upper */
/* quasi-triangular matrix S and the upper triangular matrix T.The */
/* leading columns of VSL and VSR then form an orthonormal basis for the */
/* corresponding left and right eigenspaces (deflating subspaces). */
/* (If only the generalized eigenvalues are needed, use the driver */
/* DGGEV 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 or both being zero. */
/* A pair of matrices (S,T) is in generalized real Schur form if T is */
/* upper triangular with non-negative diagonal and S is block upper */
/* triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond */
/* to real generalized eigenvalues, while 2-by-2 blocks of S will be */
/* "standardized" by making the corresponding elements of T have the */
/* form: */
/* [ a 0 ] */
/* [ 0 b ] */
/* and the pair of corresponding 2-by-2 blocks in S and T will have a */
/* complex conjugate pair of generalized eigenvalues. */
/* 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 three DOUBLE PRECISION 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 (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if */
/* SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either */
/* one of a complex conjugate pair of eigenvalues is selected, */
/* then both complex eigenvalues are selected. */
/* Note that in the ill-conditioned case, a selected complex */
/* eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), */
/* BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 */
/* in this case. */
/* N (input) INTEGER */
/* The order of the matrices A, B, VSL, and VSR. N >= 0. */
/* A (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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. (Complex conjugate pairs for which */
/* SELCTG is true for either eigenvalue count as 2.) */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, */
/* form (S,T) that would result if the 2-by-2 diagonal blocks of */
/* the real Schur form of (A,B) were further reduced to */
/* triangular form using 2-by-2 complex unitary transformations. */
/* If ALPHAI(j) is zero, then the j-th eigenvalue is real; if */
/* positive, then the j-th and (j+1)-st eigenvalues are a */
/* complex conjugate pair, with ALPHAI(j+1) negative. */
/* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) */
/* may easily over- or underflow, and BETA(j) may even be zero. */
/* Thus, the user should avoid naively computing the ratio. */
/* However, ALPHAR and ALPHAI 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
/* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. */
/* If N = 0, LWORK >= 1, else LWORK >= 8*N+16. */
/* 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. */
/* 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. */
/* The QZ iteration failed. (A,B) are not in Schur */
/* form, but ALPHAR(j), ALPHAI(j), and BETA(j) should */
/* > N: =N+1: other than QZ iteration failed in DHGEQZ. */
/* =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 DTGSEN. */
/* ===================================================================== */
/* 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;
--alphar;
--alphai;
--beta;
vsl_dim1 = *ldvsl;
vsl_offset = 1 + vsl_dim1;
vsl -= vsl_offset;
vsr_dim1 = *ldvsr;
vsr_offset = 1 + vsr_dim1;
vsr -= vsr_offset;
--work;
--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 = -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.) */
if (*info == 0) {
if (*n > 0) {
/* Computing MAX */
i__1 = *n << 3, i__2 = *n * 6 + 16;
minwrk = max(i__1,i__2);
maxwrk = minwrk - *n + *n * ilaenv_(&c__1, "DGEQRF", " ", n, &
c__1, n, &c__0);
/* Computing MAX */
i__1 = maxwrk, i__2 = minwrk - *n + *n * ilaenv_(&c__1, "DORMQR",
" ", n, &c__1, n, &c_n1);
maxwrk = max(i__1,i__2);
if (ilvsl) {
/* Computing MAX */
i__1 = maxwrk, i__2 = minwrk - *n + *n * ilaenv_(&c__1, "DOR"
"GQR", " ", n, &c__1, n, &c_n1);
maxwrk = max(i__1,i__2);
}
} else {
minwrk = 1;
maxwrk = 1;
}
work[1] = (doublereal) maxwrk;
if (*lwork < minwrk && ! lquery) {
*info = -19;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGGES ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
*sdim = 0;
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
safmin = dlamch_("S");
safmax = 1. / safmin;
dlabad_(&safmin, &safmax);
smlnum = sqrt(safmin) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute the matrix to make it more nearly triangular */
/* (Workspace: need 6*N + 2*N space for storing balancing factors) */
ileft = 1;
iright = *n + 1;
iwrk = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwrk], &ierr);
/* Reduce B to triangular form (QR decomposition of B) */
/* (Workspace: need N, prefer N*NB) */
irows = ihi + 1 - ilo;
icols = *n + 1 - ilo;
itau = iwrk;
iwrk = itau + irows;
i__1 = *lwork + 1 - iwrk;
dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwrk], &i__1, &ierr);
/* Apply the orthogonal transformation to matrix A */
/* (Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
dormqr_("L", "T", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwrk], &i__1, &
ierr);
/* Initialize VSL */
/* (Workspace: need N, prefer N*NB) */
if (ilvsl) {
dlaset_("Full", n, n, &c_b38, &c_b39, &vsl[vsl_offset], ldvsl);
if (irows > 1) {
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vsl[
ilo + 1 + ilo * vsl_dim1], ldvsl);
}
i__1 = *lwork + 1 - iwrk;
dorgqr_(&irows, &irows, &irows, &vsl[ilo + ilo * vsl_dim1], ldvsl, &
work[itau], &work[iwrk], &i__1, &ierr);
}
/* Initialize VSR */
if (ilvsr) {
dlaset_("Full", n, n, &c_b38, &c_b39, &vsr[vsr_offset], ldvsr);
}
/* Reduce to generalized Hessenberg form */
/* (Workspace: none needed) */
dgghrd_(jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vsl[vsl_offset], ldvsl, &vsr[vsr_offset], ldvsr, &ierr);
/* Perform QZ algorithm, computing Schur vectors if desired */
/* (Workspace: need N) */
iwrk = itau;
i__1 = *lwork + 1 - iwrk;
dhgeqz_("S", jobvsl, jobvsr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vsl[vsl_offset]
, ldvsl, &vsr[vsr_offset], ldvsr, &work[iwrk], &i__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 L50;
}
/* Sort eigenvalues ALPHA/BETA if desired */
/* (Workspace: need 4*N+16 ) */
*sdim = 0;
if (wantst) {
/* Undo scaling on eigenvalues before SELCTGing */
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1],
n, &ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1],
n, &ierr);
}
if (ilbscl) {
dlascl_("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)(&alphar[i__], &alphai[i__], &beta[i__]);
}
i__1 = *lwork - iwrk + 1;
dtgsen_(&c__0, &ilvsl, &ilvsr, &bwork[1], n, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[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) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsl[
vsl_offset], ldvsl, &ierr);
}
if (ilvsr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &vsr[
vsr_offset], ldvsr, &ierr);
}
/* Check if unscaling would cause over/underflow, if so, rescale */
/* (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of */
/* B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I) */
if (ilascl) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (alphai[i__] != 0.) {
if (alphar[i__] / safmax > anrmto / anrm || safmin / alphar[
i__] > anrm / anrmto) {
work[1] = (d__1 = a[i__ + i__ * a_dim1] / alphar[i__],
abs(d__1));
beta[i__] *= work[1];
alphar[i__] *= work[1];
alphai[i__] *= work[1];
} else if (alphai[i__] / safmax > anrmto / anrm || safmin /
alphai[i__] > anrm / anrmto) {
work[1] = (d__1 = a[i__ + (i__ + 1) * a_dim1] / alphai[
i__], abs(d__1));
beta[i__] *= work[1];
alphar[i__] *= work[1];
alphai[i__] *= work[1];
}
}
}
}
if (ilbscl) {
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
if (alphai[i__] != 0.) {
if (beta[i__] / safmax > bnrmto / bnrm || safmin / beta[i__]
> bnrm / bnrmto) {
work[1] = (d__1 = b[i__ + i__ * b_dim1] / beta[i__], abs(
d__1));
beta[i__] *= work[1];
alphar[i__] *= work[1];
alphai[i__] *= work[1];
}
}
}
}
/* Undo scaling */
if (ilascl) {
dlascl_("H", &c__0, &c__0, &anrmto, &anrm, n, n, &a[a_offset], lda, &
ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
ierr);
}
if (ilbscl) {
dlascl_("U", &c__0, &c__0, &bnrmto, &bnrm, n, n, &b[b_offset], ldb, &
ierr);
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
if (wantst) {
/* Check if reordering is correct */
lastsl = TRUE_;
lst2sl = TRUE_;
*sdim = 0;
ip = 0;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
cursl = (*selctg)(&alphar[i__], &alphai[i__], &beta[i__]);
if (alphai[i__] == 0.) {
if (cursl) {
++(*sdim);
}
ip = 0;
if (cursl && ! lastsl) {
*info = *n + 2;
}
} else {
if (ip == 1) {
/* Last eigenvalue of conjugate pair */
cursl = cursl || lastsl;
lastsl = cursl;
if (cursl) {
*sdim += 2;
}
ip = -1;
if (cursl && ! lst2sl) {
*info = *n + 2;
}
} else {
/* First eigenvalue of conjugate pair */
ip = 1;
}
}
lst2sl = lastsl;
lastsl = cursl;
}
}
L50:
work[1] = (doublereal) maxwrk;
return 0;
/* End of DGGES */
} /* dgges_ */
//Attention: A,B are overwritten !!
void LaEigNSSolve(int hn, double * A, double * B, std::complex<double> * lami, int evecs_bool, double * evecs_re, double * evecs_im, char balance_type )
{
integer n = hn;
char jobvr , jobvl= 'N';
bool balancing = 0;
if ( balance_type == 'B' || balance_type == 'S' || balance_type == 'P' )
balancing =1;
integer info;
double * wi= new double[n], * wr = new double[n];
double * beta = new double[n];
double vl=0;
integer nvl = 1;
integer nvr ;
if (evecs_bool)
{
jobvr = 'V';
nvr = n;
}
else
{
nvr=1;
jobvr = 'N';
}
double *vr = new double[nvr*nvr];
double* work = new double[16*n];
integer lwork = 16*n;
int i,j;
char job=balance_type; // Permute and Scale in Balancing
integer ihi,ilo;
double * lscale, *rscale;
lscale = new double[n];
rscale = new double[n];
char side = 'R';
if(balancing)
dggbal_(&job,&n, A,&n , B, &n, &ilo, &ihi, lscale, rscale, work, &info) ;
else info =0;
if(info ==0 )
{
dggev_(&jobvl, &jobvr, &n, A, &n, B, &n, wr, wi, beta, &vl, &nvl, vr, &nvr,
work , &lwork, &info);
if(info==0)
{
if(jobvr == 'V' && balancing)
{
dggbak_(&job, &side, &n, &ilo, &ihi, lscale, rscale, &n, vr, &n,&info) ;
if(info!=0)
{
cout << " Error in dggbak_ :: info " << info << endl;
return;
}
}
}
else
{
cout << " Error in dggev_ :: info " << info << endl;
return;
}
}
else
{
cout << " Error in dggbal_ :: info " << info << endl;
return;
}
delete[] lscale;
delete[] rscale;
for(i=0;i<n;i++)
{
if (fabs(beta[i])>1e-30) // not infinite eigenvalue
lami[i]=std::complex<double>(wr[i]/beta[i],wi[i]/beta[i]);
else
{
lami[i]=std::complex<double>(100.,100.);
}
}
if(evecs_bool)
{
for(i=0;i<n;i++)
{
if( imag(lami[i])== 0. || beta[i] == 0.) //real eigenvalue -> real eigenvector in i-th line
{
for(j=0;j<n;j++)
{
// evecs[i*n+j]= std::complex<double>(vr[i*n + j],0.);
evecs_re[i*n+j]= vr[i*n+j];
evecs_im[i*n+j]= 0.;
}
}
else // conjugate complex eigenvectors
{
for(j=0;j<n;j++)
{
// evecs[i*n+j]= std::complex<double>(vr[i*n+j],vr[(i+1)*n+j]);
// evecs[(i+1)*n+j]=std::complex<double>(vr[i*n+j],-vr[(i+1)*n+j]);
evecs_re[i*n+j]= vr[i*n+j];
evecs_re[(i+1)*n+j]=vr[i*n+j];
evecs_im[i*n+j]= vr[(i+1)*n+j];
evecs_im[(i+1)*n+j]=-vr[(i+1)*n+j];
}
i++;
}
}
}
delete[] wi;
delete[] wr;
delete[] beta;
delete[] work;
delete[] vr;
}
/* Subroutine */ int dgegv_(char *jobvl, char *jobvr, integer *n, doublereal *
a, integer *lda, doublereal *b, integer *ldb, doublereal *alphar,
doublereal *alphai, doublereal *beta, doublereal *vl, integer *ldvl,
doublereal *vr, integer *ldvr, doublereal *work, integer *lwork,
integer *info, ftnlen jobvl_len, ftnlen jobvr_len)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Local variables */
static integer jc, nb, in, jr, nb1, nb2, nb3, ihi, ilo;
static doublereal eps;
static logical ilv;
static doublereal absb, anrm, bnrm;
static integer itau;
static doublereal temp;
static logical ilvl, ilvr;
static integer lopt;
static doublereal anrm1, anrm2, bnrm1, bnrm2, absai, scale, absar, sbeta;
extern logical lsame_(char *, char *, ftnlen, ftnlen);
static integer ileft, iinfo, icols, iwork, irows;
extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *, ftnlen, ftnlen), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *,
ftnlen);
extern doublereal dlamch_(char *, ftnlen), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *, ftnlen);
static doublereal salfai;
extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
ftnlen, ftnlen), dlascl_(char *, integer *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, integer *,
integer *, ftnlen);
static doublereal salfar;
extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *, ftnlen);
static doublereal safmin;
extern /* Subroutine */ int dlaset_(char *, integer *, integer *,
doublereal *, doublereal *, doublereal *, integer *, ftnlen);
static doublereal safmax;
static char chtemp[1];
static logical ldumma[1];
extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *, ftnlen, ftnlen, ftnlen), dtgevc_(char *, char *,
logical *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *, integer *, doublereal *, integer *, ftnlen, ftnlen),
xerbla_(char *, integer *, ftnlen);
static integer ijobvl, iright;
static logical ilimit;
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ijobvr;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *);
static doublereal onepls;
static integer lwkmin;
extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *, ftnlen, ftnlen);
static integer lwkopt;
static logical lquery;
/* -- 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 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* This routine is deprecated and has been replaced by routine DGGEV. */
/* DGEGV computes for a pair of n-by-n real nonsymmetric matrices A and */
/* B, the generalized eigenvalues (alphar +/- alphai*i, 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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). */
/* ALPHAR (output) DOUBLE PRECISION array, dimension (N) */
/* ALPHAI (output) DOUBLE PRECISION array, dimension (N) */
/* BETA (output) DOUBLE PRECISION array, dimension (N) */
/* On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will */
/* be the generalized eigenvalues. If ALPHAI(j) is zero, then */
/* the j-th eigenvalue is real; if positive, then the j-th and */
/* (j+1)-st eigenvalues are a complex conjugate pair, with */
/* ALPHAI(j+1) negative. */
/* Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N) */
/* If JOBVL = 'V', the left generalized eigenvectors. (See */
/* "Purpose", above.) Real eigenvectors take one column, */
/* complex take two columns, the first for the real part and */
/* the second for the imaginary part. Complex eigenvectors */
/* correspond to an eigenvalue with positive imaginary part. */
/* 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) DOUBLE PRECISION array, dimension (LDVR,N) */
/* If JOBVR = 'V', the right generalized eigenvectors. (See */
/* "Purpose", above.) Real eigenvectors take one column, */
/* complex take two columns, the first for the real part and */
/* the second for the imaginary part. Complex eigenvectors */
/* correspond to an eigenvalue with positive imaginary part. */
/* 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) DOUBLE PRECISION 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(1,8*N). */
/* For good performance, LWORK must generally be larger. */
/* To compute the optimal value of LWORK, call ILAENV to get */
/* blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute: */
/* NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR; */
/* The optimal LWORK is: */
/* 2*N + MAX( 6*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. */
/* 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 ALPHAR(j), ALPHAI(j), and BETA(j) */
/* should be correct for j=INFO+1,...,N. */
/* > N: errors that usually indicate LAPACK problems: */
/* =N+1: error return from DGGBAL */
/* =N+2: error return from DGEQRF */
/* =N+3: error return from DORMQR */
/* =N+4: error return from DORGQR */
/* =N+5: error return from DGGHRD */
/* =N+6: error return from DHGEQZ (other than failed */
/* iteration) */
/* =N+7: error return from DTGEVC */
/* =N+8: error return from DGGBAK (computing VL) */
/* =N+9: error return from DGGBAK (computing VR) */
/* =N+10: error return from DLASCL (various calls) */
/* Further Details */
/* =============== */
/* Balancing */
/* --------- */
/* This driver calls DGGBAL 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, DGGBAK 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 real Schur */
/* form[*] of the "balanced" versions of A and B. If no eigenvectors */
/* are computed, then only the diagonal blocks will be correct. */
/* [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations", */
/* by Golub & van Loan, pub. by Johns Hopkins U. Press. */
/* ===================================================================== */
/* .. 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;
--alphar;
--alphai;
--beta;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
vr -= vr_offset;
--work;
/* Function Body */
if (lsame_(jobvl, "N", (ftnlen)1, (ftnlen)1)) {
ijobvl = 1;
ilvl = FALSE_;
} else if (lsame_(jobvl, "V", (ftnlen)1, (ftnlen)1)) {
ijobvl = 2;
ilvl = TRUE_;
} else {
ijobvl = -1;
ilvl = FALSE_;
}
if (lsame_(jobvr, "N", (ftnlen)1, (ftnlen)1)) {
ijobvr = 1;
ilvr = FALSE_;
} else if (lsame_(jobvr, "V", (ftnlen)1, (ftnlen)1)) {
ijobvr = 2;
ilvr = TRUE_;
} else {
ijobvr = -1;
ilvr = FALSE_;
}
ilv = ilvl || ilvr;
/* Test the input arguments */
/* Computing MAX */
i__1 = *n << 3;
lwkmin = max(i__1,1);
lwkopt = lwkmin;
work[1] = (doublereal) lwkopt;
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 = -12;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -14;
} else if (*lwork < lwkmin && ! lquery) {
*info = -16;
}
if (*info == 0) {
nb1 = ilaenv_(&c__1, "DGEQRF", " ", n, n, &c_n1, &c_n1, (ftnlen)6, (
ftnlen)1);
nb2 = ilaenv_(&c__1, "DORMQR", " ", n, n, n, &c_n1, (ftnlen)6, (
ftnlen)1);
nb3 = ilaenv_(&c__1, "DORGQR", " ", 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 * 6, i__2 = *n * (nb + 1);
lopt = (*n << 1) + max(i__1,i__2);
work[1] = (doublereal) lopt;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGEGV ", &i__1, (ftnlen)6);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("E", (ftnlen)1) * dlamch_("B", (ftnlen)1);
safmin = dlamch_("S", (ftnlen)1);
safmin += safmin;
safmax = 1. / safmin;
onepls = eps * 4 + 1.;
/* Scale A */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1], (ftnlen)1);
anrm1 = anrm;
anrm2 = 1.;
if (anrm < 1.) {
if (safmax * anrm < 1.) {
anrm1 = safmin;
anrm2 = safmax * anrm;
}
}
if (anrm > 0.) {
dlascl_("G", &c_n1, &c_n1, &anrm, &c_b27, n, n, &a[a_offset], lda, &
iinfo, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Scale B */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1], (ftnlen)1);
bnrm1 = bnrm;
bnrm2 = 1.;
if (bnrm < 1.) {
if (safmax * bnrm < 1.) {
bnrm1 = safmin;
bnrm2 = safmax * bnrm;
}
}
if (bnrm > 0.) {
dlascl_("G", &c_n1, &c_n1, &bnrm, &c_b27, n, n, &b[b_offset], ldb, &
iinfo, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 10;
return 0;
}
}
/* Permute the matrix to make it more nearly triangular */
/* Workspace layout: (8*N words -- "work" requires 6*N words) */
/* left_permutation, right_permutation, work... */
ileft = 1;
iright = *n + 1;
iwork = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwork], &iinfo, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 1;
goto L120;
}
/* Reduce B to triangular form, and initialize VL and/or VR */
/* Workspace layout: ("work..." must have at least N words) */
/* left_permutation, right_permutation, tau, work... */
irows = ihi + 1 - ilo;
if (ilv) {
icols = *n + 1 - ilo;
} else {
icols = irows;
}
itau = iwork;
iwork = itau + irows;
i__1 = *lwork + 1 - iwork;
dgeqrf_(&irows, &icols, &b[ilo + ilo * b_dim1], ldb, &work[itau], &work[
iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 2;
goto L120;
}
i__1 = *lwork + 1 - iwork;
dormqr_("L", "T", &irows, &icols, &irows, &b[ilo + ilo * b_dim1], ldb, &
work[itau], &a[ilo + ilo * a_dim1], lda, &work[iwork], &i__1, &
iinfo, (ftnlen)1, (ftnlen)1);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 3;
goto L120;
}
if (ilvl) {
dlaset_("Full", n, n, &c_b38, &c_b27, &vl[vl_offset], ldvl, (ftnlen)4)
;
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b[ilo + 1 + ilo * b_dim1], ldb, &vl[ilo +
1 + ilo * vl_dim1], ldvl, (ftnlen)1);
i__1 = *lwork + 1 - iwork;
dorgqr_(&irows, &irows, &irows, &vl[ilo + ilo * vl_dim1], ldvl, &work[
itau], &work[iwork], &i__1, &iinfo);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + iwork - 1;
lwkopt = max(i__1,i__2);
}
if (iinfo != 0) {
*info = *n + 4;
goto L120;
}
}
if (ilvr) {
dlaset_("Full", n, n, &c_b38, &c_b27, &vr[vr_offset], ldvr, (ftnlen)4)
;
}
/* Reduce to generalized Hessenberg form */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
dgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &iinfo, (
ftnlen)1, (ftnlen)1);
} else {
dgghrd_("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, (ftnlen)1, (ftnlen)1);
}
if (iinfo != 0) {
*info = *n + 5;
goto L120;
}
/* Perform QZ algorithm */
/* Workspace layout: ("work..." must have at least 1 word) */
/* left_permutation, right_permutation, work... */
iwork = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwork;
dhgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset],
ldvl, &vr[vr_offset], ldvr, &work[iwork], &i__1, &iinfo, (ftnlen)
1, (ftnlen)1, (ftnlen)1);
if (iinfo >= 0) {
/* Computing MAX */
i__1 = lwkopt, i__2 = (integer) work[iwork] + 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 L120;
}
if (ilv) {
/* Compute Eigenvectors (DTGEVC requires 6*N words of workspace) */
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwork], &iinfo, (ftnlen)1, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 7;
goto L120;
}
/* Undo balancing on VL and VR, rescale */
if (ilvl) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vl[vl_offset], ldvl, &iinfo, (ftnlen)1, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 8;
goto L120;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L50;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vl[jr + jc * vl_dim1],
abs(d__1));
temp = max(d__2,d__3);
/* L10: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vl[jr + jc * vl_dim1],
abs(d__1)) + (d__2 = vl[jr + (jc + 1) *
vl_dim1], abs(d__2));
temp = max(d__3,d__4);
/* L20: */
}
}
if (temp < safmin) {
goto L50;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
/* L30: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl[jr + jc * vl_dim1] *= temp;
vl[jr + (jc + 1) * vl_dim1] *= temp;
/* L40: */
}
}
L50:
;
}
}
if (ilvr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vr[vr_offset], ldvr, &iinfo, (ftnlen)1, (ftnlen)1);
if (iinfo != 0) {
*info = *n + 9;
goto L120;
}
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L100;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vr[jr + jc * vr_dim1],
abs(d__1));
temp = max(d__2,d__3);
/* L60: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vr[jr + jc * vr_dim1],
abs(d__1)) + (d__2 = vr[jr + (jc + 1) *
vr_dim1], abs(d__2));
temp = max(d__3,d__4);
/* L70: */
}
}
if (temp < safmin) {
goto L100;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
/* L80: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr[jr + jc * vr_dim1] *= temp;
vr[jr + (jc + 1) * vr_dim1] *= temp;
/* L90: */
}
}
L100:
;
}
}
/* 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) {
absar = (d__1 = alphar[jc], abs(d__1));
absai = (d__1 = alphai[jc], abs(d__1));
absb = (d__1 = beta[jc], abs(d__1));
salfar = anrm * alphar[jc];
salfai = anrm * alphai[jc];
sbeta = bnrm * beta[jc];
ilimit = FALSE_;
scale = 1.;
/* Check for significant underflow in ALPHAI */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absar, d__1 = max(d__1,d__2), d__2 = eps *
absb;
if (abs(salfai) < safmin && absai >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
d__1 = onepls * safmin, d__2 = anrm2 * absai;
scale = onepls * safmin / anrm1 / max(d__1,d__2);
} else if (salfai == 0.) {
/* If insignificant underflow in ALPHAI, then make the */
/* conjugate eigenvalue real. */
if (alphai[jc] < 0. && jc > 1) {
alphai[jc - 1] = 0.;
} else if (alphai[jc] > 0. && jc < *n) {
alphai[jc + 1] = 0.;
}
}
/* Check for significant underflow in ALPHAR */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absai, d__1 = max(d__1,d__2), d__2 = eps *
absb;
if (abs(salfar) < safmin && absar >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
d__3 = onepls * safmin, d__4 = anrm2 * absar;
d__1 = scale, d__2 = onepls * safmin / anrm1 / max(d__3,d__4);
scale = max(d__1,d__2);
}
/* Check for significant underflow in BETA */
/* Computing MAX */
d__1 = safmin, d__2 = eps * absar, d__1 = max(d__1,d__2), d__2 = eps *
absai;
if (abs(sbeta) < safmin && absb >= max(d__1,d__2)) {
ilimit = TRUE_;
/* Computing MAX */
/* Computing MAX */
d__3 = onepls * safmin, d__4 = bnrm2 * absb;
d__1 = scale, d__2 = onepls * safmin / bnrm1 / max(d__3,d__4);
scale = max(d__1,d__2);
}
/* Check for possible overflow when limiting scaling */
if (ilimit) {
/* Computing MAX */
d__1 = abs(salfar), d__2 = abs(salfai), d__1 = max(d__1,d__2),
d__2 = abs(sbeta);
temp = scale * safmin * max(d__1,d__2);
if (temp > 1.) {
scale /= temp;
}
if (scale < 1.) {
ilimit = FALSE_;
}
}
/* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary. */
if (ilimit) {
salfar = scale * alphar[jc] * anrm;
salfai = scale * alphai[jc] * anrm;
sbeta = scale * beta[jc] * bnrm;
}
alphar[jc] = salfar;
alphai[jc] = salfai;
beta[jc] = sbeta;
/* L110: */
}
L120:
work[1] = (doublereal) lwkopt;
return 0;
/* End of DGEGV */
} /* dgegv_ */
/* Subroutine */ int dggev_(char *jobvl, char *jobvr, integer *n, doublereal *
a, integer *lda, doublereal *b, integer *ldb, doublereal *alphar,
doublereal *alphai, doublereal *beta, doublereal *vl, integer *ldvl,
doublereal *vr, integer *ldvr, doublereal *work, integer *lwork,
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
=======
DGGEV computes for a pair of N-by-N real 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 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
=========
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) DOUBLE PRECISION 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) DOUBLE PRECISION 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).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. If ALPHAI(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector will be scaled so the largest component 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) DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector will be scaled so the largest component 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) DOUBLE PRECISION 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(1,8*N).
For good performance, LWORK must generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. No eigenvectors have been
calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
should be correct for j=INFO+1,...,N.
> N: =N+1: other than QZ iteration failed in DHGEQZ.
=N+2: error return from DTGEVC.
=====================================================================
Decode the input arguments
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static integer c__0 = 0;
static doublereal c_b26 = 0.;
static doublereal c_b27 = 1.;
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static doublereal anrm, bnrm;
static integer ierr, itau;
static doublereal temp;
static logical ilvl, ilvr;
static integer iwrk;
extern logical lsame_(char *, char *);
static integer ileft, icols, irows;
extern /* Subroutine */ int dlabad_(doublereal *, doublereal *);
static integer jc;
extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
static integer in;
extern doublereal dlamch_(char *), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *);
static integer jr;
extern /* Subroutine */ int dgghrd_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal
*, doublereal *, integer *, integer *, doublereal *, integer *,
integer *);
static logical ilascl, ilbscl;
extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *, integer *),
dlacpy_(char *, integer *, integer *, doublereal *, integer *,
doublereal *, integer *), dlaset_(char *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *), dtgevc_(char *, char *, logical *, integer *, doublereal
*, integer *, doublereal *, integer *, doublereal *, integer *,
doublereal *, integer *, integer *, integer *, doublereal *,
integer *);
static logical ldumma[1];
static char chtemp[1];
static doublereal bignum;
extern /* Subroutine */ int dhgeqz_(char *, char *, char *, integer *,
integer *, integer *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
integer *, doublereal *, integer *, doublereal *, integer *,
integer *), xerbla_(char *, integer *);
extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
integer *, integer *, ftnlen, ftnlen);
static integer ijobvl, iright, ijobvr;
extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *,
doublereal *, integer *, doublereal *, doublereal *, integer *,
integer *);
static doublereal anrmto, bnrmto;
extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
integer *, doublereal *, integer *, doublereal *, doublereal *,
integer *, doublereal *, integer *, integer *);
static integer minwrk, maxwrk;
static doublereal smlnum;
static logical lquery;
static integer ihi, ilo;
static doublereal eps;
static logical ilv;
#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
#define vl_ref(a_1,a_2) vl[(a_2)*vl_dim1 + a_1]
#define vr_ref(a_1,a_2) vr[(a_2)*vr_dim1 + a_1]
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
--alphar;
--alphai;
--beta;
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;
/* 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 = -12;
} else if (*ldvr < 1 || ilvr && *ldvr < *n) {
*info = -14;
}
/* 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.) */
minwrk = 1;
if (*info == 0 && (*lwork >= 1 || lquery)) {
maxwrk = *n * 7 + *n * ilaenv_(&c__1, "DGEQRF", " ", n, &c__1, n, &
c__0, (ftnlen)6, (ftnlen)1);
/* Computing MAX */
i__1 = 1, i__2 = *n << 3;
minwrk = max(i__1,i__2);
work[1] = (doublereal) maxwrk;
}
if (*lwork < minwrk && ! lquery) {
*info = -16;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("DGGEV ", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = dlamch_("P");
smlnum = dlamch_("S");
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
smlnum = sqrt(smlnum) / eps;
bignum = 1. / smlnum;
/* Scale A if max element outside range [SMLNUM,BIGNUM] */
anrm = dlange_("M", n, n, &a[a_offset], lda, &work[1]);
ilascl = FALSE_;
if (anrm > 0. && anrm < smlnum) {
anrmto = smlnum;
ilascl = TRUE_;
} else if (anrm > bignum) {
anrmto = bignum;
ilascl = TRUE_;
}
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrm, &anrmto, n, n, &a[a_offset], lda, &
ierr);
}
/* Scale B if max element outside range [SMLNUM,BIGNUM] */
bnrm = dlange_("M", n, n, &b[b_offset], ldb, &work[1]);
ilbscl = FALSE_;
if (bnrm > 0. && bnrm < smlnum) {
bnrmto = smlnum;
ilbscl = TRUE_;
} else if (bnrm > bignum) {
bnrmto = bignum;
ilbscl = TRUE_;
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrm, &bnrmto, n, n, &b[b_offset], ldb, &
ierr);
}
/* Permute the matrices A, B to isolate eigenvalues if possible
(Workspace: need 6*N) */
ileft = 1;
iright = *n + 1;
iwrk = iright + *n;
dggbal_("P", n, &a[a_offset], lda, &b[b_offset], ldb, &ilo, &ihi, &work[
ileft], &work[iright], &work[iwrk], &ierr);
/* Reduce B to triangular form (QR decomposition of B)
(Workspace: need N, prefer N*NB) */
irows = ihi + 1 - ilo;
if (ilv) {
icols = *n + 1 - ilo;
} else {
icols = irows;
}
itau = iwrk;
iwrk = itau + irows;
i__1 = *lwork + 1 - iwrk;
dgeqrf_(&irows, &icols, &b_ref(ilo, ilo), ldb, &work[itau], &work[iwrk], &
i__1, &ierr);
/* Apply the orthogonal transformation to matrix A
(Workspace: need N, prefer N*NB) */
i__1 = *lwork + 1 - iwrk;
dormqr_("L", "T", &irows, &icols, &irows, &b_ref(ilo, ilo), ldb, &work[
itau], &a_ref(ilo, ilo), lda, &work[iwrk], &i__1, &ierr);
/* Initialize VL
(Workspace: need N, prefer N*NB) */
if (ilvl) {
dlaset_("Full", n, n, &c_b26, &c_b27, &vl[vl_offset], ldvl)
;
i__1 = irows - 1;
i__2 = irows - 1;
dlacpy_("L", &i__1, &i__2, &b_ref(ilo + 1, ilo), ldb, &vl_ref(ilo + 1,
ilo), ldvl);
i__1 = *lwork + 1 - iwrk;
dorgqr_(&irows, &irows, &irows, &vl_ref(ilo, ilo), ldvl, &work[itau],
&work[iwrk], &i__1, &ierr);
}
/* Initialize VR */
if (ilvr) {
dlaset_("Full", n, n, &c_b26, &c_b27, &vr[vr_offset], ldvr)
;
}
/* Reduce to generalized Hessenberg form
(Workspace: none needed) */
if (ilv) {
/* Eigenvectors requested -- work on whole matrix. */
dgghrd_(jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[b_offset],
ldb, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, &ierr);
} else {
dgghrd_("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 forms and Schur vectors)
(Workspace: need N) */
iwrk = itau;
if (ilv) {
*(unsigned char *)chtemp = 'S';
} else {
*(unsigned char *)chtemp = 'E';
}
i__1 = *lwork + 1 - iwrk;
dhgeqz_(chtemp, jobvl, jobvr, n, &ilo, &ihi, &a[a_offset], lda, &b[
b_offset], ldb, &alphar[1], &alphai[1], &beta[1], &vl[vl_offset],
ldvl, &vr[vr_offset], ldvr, &work[iwrk], &i__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 L110;
}
/* Compute Eigenvectors
(Workspace: need 6*N) */
if (ilv) {
if (ilvl) {
if (ilvr) {
*(unsigned char *)chtemp = 'B';
} else {
*(unsigned char *)chtemp = 'L';
}
} else {
*(unsigned char *)chtemp = 'R';
}
dtgevc_(chtemp, "B", ldumma, n, &a[a_offset], lda, &b[b_offset], ldb,
&vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &in, &work[
iwrk], &ierr);
if (ierr != 0) {
*info = *n + 2;
goto L110;
}
/* Undo balancing on VL and VR and normalization
(Workspace: none needed) */
if (ilvl) {
dggbak_("P", "L", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vl[vl_offset], ldvl, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L50;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vl_ref(jr, jc), abs(d__1))
;
temp = max(d__2,d__3);
/* L10: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vl_ref(jr, jc), abs(d__1))
+ (d__2 = vl_ref(jr, jc + 1), abs(d__2));
temp = max(d__3,d__4);
/* L20: */
}
}
if (temp < smlnum) {
goto L50;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl_ref(jr, jc) = vl_ref(jr, jc) * temp;
/* L30: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vl_ref(jr, jc) = vl_ref(jr, jc) * temp;
vl_ref(jr, jc + 1) = vl_ref(jr, jc + 1) * temp;
/* L40: */
}
}
L50:
;
}
}
if (ilvr) {
dggbak_("P", "R", n, &ilo, &ihi, &work[ileft], &work[iright], n, &
vr[vr_offset], ldvr, &ierr);
i__1 = *n;
for (jc = 1; jc <= i__1; ++jc) {
if (alphai[jc] < 0.) {
goto L100;
}
temp = 0.;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = vr_ref(jr, jc), abs(d__1))
;
temp = max(d__2,d__3);
/* L60: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
/* Computing MAX */
d__3 = temp, d__4 = (d__1 = vr_ref(jr, jc), abs(d__1))
+ (d__2 = vr_ref(jr, jc + 1), abs(d__2));
temp = max(d__3,d__4);
/* L70: */
}
}
if (temp < smlnum) {
goto L100;
}
temp = 1. / temp;
if (alphai[jc] == 0.) {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr_ref(jr, jc) = vr_ref(jr, jc) * temp;
/* L80: */
}
} else {
i__2 = *n;
for (jr = 1; jr <= i__2; ++jr) {
vr_ref(jr, jc) = vr_ref(jr, jc) * temp;
vr_ref(jr, jc + 1) = vr_ref(jr, jc + 1) * temp;
/* L90: */
}
}
L100:
;
}
}
/* End of eigenvector calculation */
}
/* Undo scaling if necessary */
if (ilascl) {
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphar[1], n, &
ierr);
dlascl_("G", &c__0, &c__0, &anrmto, &anrm, n, &c__1, &alphai[1], n, &
ierr);
}
if (ilbscl) {
dlascl_("G", &c__0, &c__0, &bnrmto, &bnrm, n, &c__1, &beta[1], n, &
ierr);
}
L110:
work[1] = (doublereal) maxwrk;
return 0;
/* End of DGGEV */
} /* dggev_ */
/* Subroutine */ int dchkgk_(integer *nin, integer *nout)
{
/* Format strings */
static char fmt_9999[] = "(1x,\002.. test output of DGGBAK .. \002)";
static char fmt_9998[] = "(\002 value of largest test error "
" =\002,d12.3)";
static char fmt_9997[] = "(\002 example number where DGGBAL info is not "
"0 =\002,i4)";
static char fmt_9996[] = "(\002 example number where DGGBAK(L) info is n"
"ot 0 =\002,i4)";
static char fmt_9995[] = "(\002 example number where DGGBAK(R) info is n"
"ot 0 =\002,i4)";
static char fmt_9994[] = "(\002 example number having largest error "
" =\002,i4)";
static char fmt_9993[] = "(\002 number of examples where info is not 0 "
" =\002,i4)";
static char fmt_9992[] = "(\002 total number of examples tested "
" =\002,i4)";
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
integer s_rsle(cilist *), do_lio(integer *, integer *, char *, ftnlen),
e_rsle(void), s_wsfe(cilist *), e_wsfe(void), do_fio(integer *,
char *, ftnlen);
/* Local variables */
static integer info, lmax[4];
static doublereal rmax, vmax, work[2500] /* was [50][50] */, a[2500]
/* was [50][50] */, b[2500] /* was [50][50] */, e[2500] /*
was [50][50] */, f[2500] /* was [50][50] */;
static integer i__, j, m, n;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
static integer ninfo;
static doublereal anorm, bnorm, af[2500] /* was [50][50] */, bf[2500]
/* was [50][50] */;
extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
extern doublereal dlamch_(char *), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *);
static doublereal vl[2500] /* was [50][50] */, lscale[50], vr[2500]
/* was [50][50] */, rscale[50];
extern /* Subroutine */ int dlacpy_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, integer *);
static integer ihi, ilo;
static doublereal eps, vlf[2500] /* was [50][50] */;
static integer knt;
static doublereal vrf[2500] /* was [50][50] */;
/* Fortran I/O blocks */
static cilist io___6 = { 0, 0, 0, 0, 0 };
static cilist io___10 = { 0, 0, 0, 0, 0 };
static cilist io___13 = { 0, 0, 0, 0, 0 };
static cilist io___15 = { 0, 0, 0, 0, 0 };
static cilist io___17 = { 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 };
static cilist io___41 = { 0, 0, 0, fmt_9992, 0 };
#define a_ref(a_1,a_2) a[(a_2)*50 + a_1 - 51]
#define b_ref(a_1,a_2) b[(a_2)*50 + a_1 - 51]
#define e_ref(a_1,a_2) e[(a_2)*50 + a_1 - 51]
#define f_ref(a_1,a_2) f[(a_2)*50 + a_1 - 51]
#define vl_ref(a_1,a_2) vl[(a_2)*50 + a_1 - 51]
#define vr_ref(a_1,a_2) vr[(a_2)*50 + a_1 - 51]
/* -- LAPACK test routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
Purpose
=======
DCHKGK tests DGGBAK, a routine for backward balancing of
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.
=====================================================================
Initialization */
lmax[0] = 0;
lmax[1] = 0;
lmax[2] = 0;
lmax[3] = 0;
ninfo = 0;
knt = 0;
rmax = 0.;
eps = dlamch_("Precision");
L10:
io___6.ciunit = *nin;
s_rsle(&io___6);
do_lio(&c__3, &c__1, (char *)&n, (ftnlen)sizeof(integer));
do_lio(&c__3, &c__1, (char *)&m, (ftnlen)sizeof(integer));
e_rsle();
if (n == 0) {
goto L100;
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___10.ciunit = *nin;
s_rsle(&io___10);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&a_ref(i__, j), (ftnlen)sizeof(
doublereal));
}
e_rsle();
/* L20: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___13.ciunit = *nin;
s_rsle(&io___13);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&b_ref(i__, j), (ftnlen)sizeof(
doublereal));
}
e_rsle();
/* L30: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___15.ciunit = *nin;
s_rsle(&io___15);
i__2 = m;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&vl_ref(i__, j), (ftnlen)sizeof(
doublereal));
}
e_rsle();
/* L40: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___17.ciunit = *nin;
s_rsle(&io___17);
i__2 = m;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&vr_ref(i__, j), (ftnlen)sizeof(
doublereal));
}
e_rsle();
/* L50: */
}
++knt;
anorm = dlange_("M", &n, &n, a, &c__50, work);
bnorm = dlange_("M", &n, &n, b, &c__50, work);
dlacpy_("FULL", &n, &n, a, &c__50, af, &c__50);
dlacpy_("FULL", &n, &n, b, &c__50, bf, &c__50);
dggbal_("B", &n, a, &c__50, b, &c__50, &ilo, &ihi, lscale, rscale, work, &
info);
if (info != 0) {
++ninfo;
lmax[0] = knt;
}
dlacpy_("FULL", &n, &m, vl, &c__50, vlf, &c__50);
dlacpy_("FULL", &n, &m, vr, &c__50, vrf, &c__50);
dggbak_("B", "L", &n, &ilo, &ihi, lscale, rscale, &m, vl, &c__50, &info);
if (info != 0) {
++ninfo;
lmax[1] = knt;
}
dggbak_("B", "R", &n, &ilo, &ihi, lscale, rscale, &m, vr, &c__50, &info);
if (info != 0) {
++ninfo;
lmax[2] = knt;
}
/* Test of DGGBAK
Check tilde(VL)'*A*tilde(VR) - VL'*tilde(A)*VR
where tilde(A) denotes the transformed matrix. */
dgemm_("N", "N", &n, &m, &n, &c_b52, af, &c__50, vr, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vl, &c__50, work, &c__50, &c_b55, e,
&c__50);
dgemm_("N", "N", &n, &m, &n, &c_b52, a, &c__50, vrf, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vlf, &c__50, work, &c__50, &c_b55, f,
&c__50);
vmax = 0.;
i__1 = m;
for (j = 1; j <= i__1; ++j) {
i__2 = m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = vmax, d__3 = (d__1 = e_ref(i__, j) - f_ref(i__, j), abs(
d__1));
vmax = max(d__2,d__3);
/* L60: */
}
/* L70: */
}
vmax /= eps * max(anorm,bnorm);
if (vmax > rmax) {
lmax[3] = knt;
rmax = vmax;
}
/* Check tilde(VL)'*B*tilde(VR) - VL'*tilde(B)*VR */
dgemm_("N", "N", &n, &m, &n, &c_b52, bf, &c__50, vr, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vl, &c__50, work, &c__50, &c_b55, e,
&c__50);
dgemm_("N", "N", &n, &m, &n, &c_b52, b, &c__50, vrf, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vlf, &c__50, work, &c__50, &c_b55, f,
&c__50);
vmax = 0.;
i__1 = m;
for (j = 1; j <= i__1; ++j) {
i__2 = m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = vmax, d__3 = (d__1 = e_ref(i__, j) - f_ref(i__, j), abs(
d__1));
vmax = max(d__2,d__3);
/* L80: */
}
/* L90: */
}
vmax /= eps * max(anorm,bnorm);
if (vmax > rmax) {
lmax[3] = knt;
rmax = vmax;
}
goto L10;
L100:
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(doublereal));
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 *)&lmax[3], (ftnlen)sizeof(integer));
e_wsfe();
io___40.ciunit = *nout;
s_wsfe(&io___40);
do_fio(&c__1, (char *)&ninfo, (ftnlen)sizeof(integer));
e_wsfe();
io___41.ciunit = *nout;
s_wsfe(&io___41);
do_fio(&c__1, (char *)&knt, (ftnlen)sizeof(integer));
e_wsfe();
return 0;
/* End of DCHKGK */
} /* dchkgk_ */
/* Subroutine */ int dchkgk_(integer *nin, integer *nout)
{
/* Format strings */
static char fmt_9999[] = "(1x,\002.. test output of DGGBAK .. \002)";
static char fmt_9998[] = "(\002 value of largest test error "
" =\002,d12.3)";
static char fmt_9997[] = "(\002 example number where DGGBAL info is not "
"0 =\002,i4)";
static char fmt_9996[] = "(\002 example number where DGGBAK(L) info is n"
"ot 0 =\002,i4)";
static char fmt_9995[] = "(\002 example number where DGGBAK(R) info is n"
"ot 0 =\002,i4)";
static char fmt_9994[] = "(\002 example number having largest error "
" =\002,i4)";
static char fmt_9993[] = "(\002 number of examples where info is not 0 "
" =\002,i4)";
static char fmt_9992[] = "(\002 total number of examples tested "
" =\002,i4)";
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3;
/* Local variables */
doublereal a[2500] /* was [50][50] */, b[2500] /* was [50][50] */, e[
2500] /* was [50][50] */, f[2500] /* was [50][50] */;
integer i__, j, m, n;
doublereal af[2500] /* was [50][50] */, bf[2500] /* was [50][50] */,
vl[2500] /* was [50][50] */, vr[2500] /* was [50][50] */;
integer ihi, ilo;
doublereal eps, vlf[2500] /* was [50][50] */;
integer knt;
doublereal vrf[2500] /* was [50][50] */;
integer info, lmax[4];
doublereal rmax, vmax, work[2500] /* was [50][50] */;
extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, doublereal *, doublereal *, integer *);
integer ninfo;
doublereal anorm, bnorm;
extern /* Subroutine */ int dggbak_(char *, char *, integer *, integer *,
integer *, doublereal *, doublereal *, integer *, doublereal *,
integer *, integer *), dggbal_(char *, integer *,
doublereal *, integer *, doublereal *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, integer *);
extern doublereal dlamch_(char *), dlange_(char *, integer *,
integer *, doublereal *, integer *, doublereal *);
doublereal lscale[50], rscale[50];
extern /* Subroutine */ int dlacpy_(char *, integer *, integer *,
doublereal *, integer *, doublereal *, integer *);
/* Fortran I/O blocks */
static cilist io___6 = { 0, 0, 0, 0, 0 };
static cilist io___10 = { 0, 0, 0, 0, 0 };
static cilist io___13 = { 0, 0, 0, 0, 0 };
static cilist io___15 = { 0, 0, 0, 0, 0 };
static cilist io___17 = { 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 };
static cilist io___41 = { 0, 0, 0, fmt_9992, 0 };
/* -- LAPACK test routine (version 3.1) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* DCHKGK tests DGGBAK, a routine for backward balancing of */
/* 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 .. */
/* Initialization */
lmax[0] = 0;
lmax[1] = 0;
lmax[2] = 0;
lmax[3] = 0;
ninfo = 0;
knt = 0;
rmax = 0.;
eps = dlamch_("Precision");
L10:
io___6.ciunit = *nin;
s_rsle(&io___6);
do_lio(&c__3, &c__1, (char *)&n, (ftnlen)sizeof(integer));
do_lio(&c__3, &c__1, (char *)&m, (ftnlen)sizeof(integer));
e_rsle();
if (n == 0) {
goto L100;
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___10.ciunit = *nin;
s_rsle(&io___10);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&a[i__ + j * 50 - 51], (ftnlen)
sizeof(doublereal));
}
e_rsle();
/* L20: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___13.ciunit = *nin;
s_rsle(&io___13);
i__2 = n;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&b[i__ + j * 50 - 51], (ftnlen)
sizeof(doublereal));
}
e_rsle();
/* L30: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___15.ciunit = *nin;
s_rsle(&io___15);
i__2 = m;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&vl[i__ + j * 50 - 51], (ftnlen)
sizeof(doublereal));
}
e_rsle();
/* L40: */
}
i__1 = n;
for (i__ = 1; i__ <= i__1; ++i__) {
io___17.ciunit = *nin;
s_rsle(&io___17);
i__2 = m;
for (j = 1; j <= i__2; ++j) {
do_lio(&c__5, &c__1, (char *)&vr[i__ + j * 50 - 51], (ftnlen)
sizeof(doublereal));
}
e_rsle();
/* L50: */
}
++knt;
anorm = dlange_("M", &n, &n, a, &c__50, work);
bnorm = dlange_("M", &n, &n, b, &c__50, work);
dlacpy_("FULL", &n, &n, a, &c__50, af, &c__50);
dlacpy_("FULL", &n, &n, b, &c__50, bf, &c__50);
dggbal_("B", &n, a, &c__50, b, &c__50, &ilo, &ihi, lscale, rscale, work, &
info);
if (info != 0) {
++ninfo;
lmax[0] = knt;
}
dlacpy_("FULL", &n, &m, vl, &c__50, vlf, &c__50);
dlacpy_("FULL", &n, &m, vr, &c__50, vrf, &c__50);
dggbak_("B", "L", &n, &ilo, &ihi, lscale, rscale, &m, vl, &c__50, &info);
if (info != 0) {
++ninfo;
lmax[1] = knt;
}
dggbak_("B", "R", &n, &ilo, &ihi, lscale, rscale, &m, vr, &c__50, &info);
if (info != 0) {
++ninfo;
lmax[2] = knt;
}
/* Test of DGGBAK */
/* Check tilde(VL)'*A*tilde(VR) - VL'*tilde(A)*VR */
/* where tilde(A) denotes the transformed matrix. */
dgemm_("N", "N", &n, &m, &n, &c_b52, af, &c__50, vr, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vl, &c__50, work, &c__50, &c_b55, e,
&c__50);
dgemm_("N", "N", &n, &m, &n, &c_b52, a, &c__50, vrf, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vlf, &c__50, work, &c__50, &c_b55, f,
&c__50);
vmax = 0.;
i__1 = m;
for (j = 1; j <= i__1; ++j) {
i__2 = m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = vmax, d__3 = (d__1 = e[i__ + j * 50 - 51] - f[i__ + j * 50
- 51], abs(d__1));
vmax = max(d__2,d__3);
/* L60: */
}
/* L70: */
}
vmax /= eps * max(anorm,bnorm);
if (vmax > rmax) {
lmax[3] = knt;
rmax = vmax;
}
/* Check tilde(VL)'*B*tilde(VR) - VL'*tilde(B)*VR */
dgemm_("N", "N", &n, &m, &n, &c_b52, bf, &c__50, vr, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vl, &c__50, work, &c__50, &c_b55, e,
&c__50);
dgemm_("N", "N", &n, &m, &n, &c_b52, b, &c__50, vrf, &c__50, &c_b55, work,
&c__50);
dgemm_("T", "N", &m, &m, &n, &c_b52, vlf, &c__50, work, &c__50, &c_b55, f,
&c__50);
vmax = 0.;
i__1 = m;
for (j = 1; j <= i__1; ++j) {
i__2 = m;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = vmax, d__3 = (d__1 = e[i__ + j * 50 - 51] - f[i__ + j * 50
- 51], abs(d__1));
vmax = max(d__2,d__3);
/* L80: */
}
/* L90: */
}
vmax /= eps * max(anorm,bnorm);
if (vmax > rmax) {
lmax[3] = knt;
rmax = vmax;
}
goto L10;
L100:
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(doublereal));
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 *)&lmax[3], (ftnlen)sizeof(integer));
e_wsfe();
io___40.ciunit = *nout;
s_wsfe(&io___40);
do_fio(&c__1, (char *)&ninfo, (ftnlen)sizeof(integer));
e_wsfe();
io___41.ciunit = *nout;
s_wsfe(&io___41);
do_fio(&c__1, (char *)&knt, (ftnlen)sizeof(integer));
e_wsfe();
return 0;
/* End of DCHKGK */
} /* dchkgk_ */