/* Subroutine */ int ztgsna_(char *job, char *howmny, logical *select,
integer *n, doublecomplex *a, integer *lda, doublecomplex *b, integer
*ldb, doublecomplex *vl, integer *ldvl, doublecomplex *vr, integer *
ldvr, doublereal *s, doublereal *dif, integer *mm, integer *m,
doublecomplex *work, integer *lwork, integer *iwork, integer *info)
{
/* -- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
June 30, 1999
Purpose
=======
ZTGSNA estimates reciprocal condition numbers for specified
eigenvalues and/or eigenvectors of a matrix pair (A, B).
(A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.
Arguments
=========
JOB (input) CHARACTER*1
Specifies whether condition numbers are required for
eigenvalues (S) or eigenvectors (DIF):
= 'E': for eigenvalues only (S);
= 'V': for eigenvectors only (DIF);
= 'B': for both eigenvalues and eigenvectors (S and DIF).
HOWMNY (input) CHARACTER*1
= 'A': compute condition numbers for all eigenpairs;
= 'S': compute condition numbers for selected eigenpairs
specified by the array SELECT.
SELECT (input) LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenpairs for which
condition numbers are required. To select condition numbers
for the corresponding j-th eigenvalue and/or eigenvector,
SELECT(j) must be set to .TRUE..
If HOWMNY = 'A', SELECT is not referenced.
N (input) INTEGER
The order of the square matrix pair (A, B). N >= 0.
A (input) COMPLEX*16 array, dimension (LDA,N)
The upper triangular matrix A in the pair (A,B).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input) COMPLEX*16 array, dimension (LDB,N)
The upper triangular matrix B in the pair (A, B).
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
VL (input) COMPLEX*16 array, dimension (LDVL,M)
IF JOB = 'E' or 'B', VL must contain left eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VL, as returned by ZTGEVC.
If JOB = 'V', VL is not referenced.
LDVL (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; and
If JOB = 'E' or 'B', LDVL >= N.
VR (input) COMPLEX*16 array, dimension (LDVR,M)
IF JOB = 'E' or 'B', VR must contain right eigenvectors of
(A, B), corresponding to the eigenpairs specified by HOWMNY
and SELECT. The eigenvectors must be stored in consecutive
columns of VR, as returned by ZTGEVC.
If JOB = 'V', VR is not referenced.
LDVR (input) INTEGER
The leading dimension of the array VR. LDVR >= 1;
If JOB = 'E' or 'B', LDVR >= N.
S (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array.
If JOB = 'V', S is not referenced.
DIF (output) DOUBLE PRECISION array, dimension (MM)
If JOB = 'V' or 'B', the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array.
If the eigenvalues cannot be reordered to compute DIF(j),
DIF(j) is set to 0; this can only occur when the true value
would be very small anyway.
For each eigenvalue/vector specified by SELECT, DIF stores
a Frobenius norm-based estimate of Difl.
If JOB = 'E', DIF is not referenced.
MM (input) INTEGER
The number of elements in the arrays S and DIF. MM >= M.
M (output) INTEGER
The number of elements of the arrays S and DIF used to store
the specified condition numbers; for each selected eigenvalue
one element is used. If HOWMNY = 'A', M is set to N.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
If JOB = 'E', WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1.
If JOB = 'V' or 'B', LWORK >= 2*N*N.
IWORK (workspace) INTEGER array, dimension (N+2)
If JOB = 'E', IWORK is not referenced.
INFO (output) INTEGER
= 0: Successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
Further Details
===============
The reciprocal of the condition number of the i-th generalized
eigenvalue w = (a, b) is defined as
S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
where u and v are the right and left eigenvectors of (A, B)
corresponding to w; |z| denotes the absolute value of the complex
number, and norm(u) denotes the 2-norm of the vector u. The pair
(a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
matrix pair (A, B). If both a and b equal zero, then (A,B) is
singular and S(I) = -1 is returned.
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(A, B) / S(I),
where EPS is the machine precision.
The reciprocal of the condition number of the right eigenvector u
and left eigenvector v corresponding to the generalized eigenvalue w
is defined as follows. Suppose
(A, B) = ( a * ) ( b * ) 1
( 0 A22 ),( 0 B22 ) n-1
1 n-1 1 n-1
Then the reciprocal condition number DIF(I) is
Difl[(a, b), (A22, B22)] = sigma-min( Zl )
where sigma-min(Zl) denotes the smallest singular value of
Zl = [ kron(a, In-1) -kron(1, A22) ]
[ kron(b, In-1) -kron(1, B22) ].
Here In-1 is the identity matrix of size n-1 and X' is the conjugate
transpose of X. kron(X, Y) is the Kronecker product between the
matrices X and Y.
We approximate the smallest singular value of Zl with an upper
bound. This is done by ZLATDF.
An approximate error bound for a computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(A, B) / DIF(i).
See ref. [2-3] for more details and further references.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software, Report
UMINF - 94.04, Department of Computing Science, Umea University,
S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75.
To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
=====================================================================
Decode and test the input parameters
Parameter adjustments */
/* Table of constant values */
static integer c__1 = 1;
static doublecomplex c_b19 = {1.,0.};
static doublecomplex c_b20 = {0.,0.};
static logical c_false = FALSE_;
static integer c__3 = 3;
/* 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;
doublecomplex z__1;
/* Builtin functions */
double z_abs(doublecomplex *);
/* Local variables */
static doublereal cond;
static integer ierr, ifst;
static doublereal lnrm;
static doublecomplex yhax, yhbx;
static integer ilst;
static doublereal rnrm;
static integer i__, k;
static doublereal scale;
extern logical lsame_(char *, char *);
extern /* Double Complex */ VOID zdotc_(doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *);
static integer lwmin;
extern /* Subroutine */ int zgemv_(char *, integer *, integer *,
doublecomplex *, doublecomplex *, integer *, doublecomplex *,
integer *, doublecomplex *, doublecomplex *, integer *);
static logical wants;
static integer llwrk, n1, n2;
static doublecomplex dummy[1];
extern doublereal dlapy2_(doublereal *, doublereal *);
extern /* Subroutine */ int dlabad_(doublereal *, doublereal *);
static doublecomplex dummy1[1];
extern doublereal dznrm2_(integer *, doublecomplex *, integer *), dlamch_(
char *);
static integer ks;
extern /* Subroutine */ int xerbla_(char *, integer *);
static doublereal bignum;
static logical wantbh, wantdf, somcon;
extern /* Subroutine */ int zlacpy_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *),
ztgexc_(logical *, logical *, integer *, doublecomplex *, integer
*, doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, integer *, integer *, integer *);
static doublereal smlnum;
static logical lquery;
extern /* Subroutine */ int ztgsyl_(char *, integer *, integer *, integer
*, doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *,
doublereal *, doublereal *, doublecomplex *, integer *, integer *,
integer *);
static doublereal eps;
#define a_subscr(a_1,a_2) (a_2)*a_dim1 + a_1
#define a_ref(a_1,a_2) a[a_subscr(a_1,a_2)]
#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]
#define vl_subscr(a_1,a_2) (a_2)*vl_dim1 + a_1
#define vl_ref(a_1,a_2) vl[vl_subscr(a_1,a_2)]
#define vr_subscr(a_1,a_2) (a_2)*vr_dim1 + a_1
#define vr_ref(a_1,a_2) vr[vr_subscr(a_1,a_2)]
--select;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1 * 1;
b -= b_offset;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1 * 1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1 * 1;
vr -= vr_offset;
--s;
--dif;
--work;
--iwork;
/* Function Body */
wantbh = lsame_(job, "B");
wants = lsame_(job, "E") || wantbh;
wantdf = lsame_(job, "V") || wantbh;
somcon = lsame_(howmny, "S");
*info = 0;
lquery = *lwork == -1;
if (lsame_(job, "V") || lsame_(job, "B")) {
/* Computing MAX */
i__1 = 1, i__2 = (*n << 1) * *n;
lwmin = max(i__1,i__2);
} else {
lwmin = 1;
}
if (! wants && ! wantdf) {
*info = -1;
} else if (! lsame_(howmny, "A") && ! somcon) {
*info = -2;
} else if (*n < 0) {
*info = -4;
} else if (*lda < max(1,*n)) {
*info = -6;
} else if (*ldb < max(1,*n)) {
*info = -8;
} else if (wants && *ldvl < *n) {
*info = -10;
} else if (wants && *ldvr < *n) {
*info = -12;
} else {
/* Set M to the number of eigenpairs for which condition numbers
are required, and test MM. */
if (somcon) {
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
if (select[k]) {
++(*m);
}
/* L10: */
}
} else {
*m = *n;
}
if (*mm < *m) {
*info = -15;
} else if (*lwork < lwmin && ! lquery) {
*info = -18;
}
}
if (*info == 0) {
work[1].r = (doublereal) lwmin, work[1].i = 0.;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTGSNA", &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") / eps;
bignum = 1. / smlnum;
dlabad_(&smlnum, &bignum);
llwrk = *lwork - (*n << 1) * *n;
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
/* Determine whether condition numbers are required for the k-th
eigenpair. */
if (somcon) {
if (! select[k]) {
goto L20;
}
}
++ks;
if (wants) {
/* Compute the reciprocal condition number of the k-th
eigenvalue. */
rnrm = dznrm2_(n, &vr_ref(1, ks), &c__1);
lnrm = dznrm2_(n, &vl_ref(1, ks), &c__1);
zgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr_ref(1, ks), &
c__1, &c_b20, &work[1], &c__1);
zdotc_(&z__1, n, &work[1], &c__1, &vl_ref(1, ks), &c__1);
yhax.r = z__1.r, yhax.i = z__1.i;
zgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr_ref(1, ks), &
c__1, &c_b20, &work[1], &c__1);
zdotc_(&z__1, n, &work[1], &c__1, &vl_ref(1, ks), &c__1);
yhbx.r = z__1.r, yhbx.i = z__1.i;
d__1 = z_abs(&yhax);
d__2 = z_abs(&yhbx);
cond = dlapy2_(&d__1, &d__2);
if (cond == 0.) {
s[ks] = -1.;
} else {
s[ks] = cond / (rnrm * lnrm);
}
}
if (wantdf) {
if (*n == 1) {
d__1 = z_abs(&a_ref(1, 1));
d__2 = z_abs(&b_ref(1, 1));
dif[ks] = dlapy2_(&d__1, &d__2);
goto L20;
}
/* Estimate the reciprocal condition number of the k-th
eigenvectors.
Copy the matrix (A, B) to the array WORK and move the
(k,k)th pair to the (1,1) position. */
zlacpy_("Full", n, n, &a[a_offset], lda, &work[1], n);
zlacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1], n);
ifst = k;
ilst = 1;
ztgexc_(&c_false, &c_false, n, &work[1], n, &work[*n * *n + 1], n,
dummy, &c__1, dummy1, &c__1, &ifst, &ilst, &ierr);
if (ierr > 0) {
/* Ill-conditioned problem - swap rejected. */
dif[ks] = 0.;
} else {
/* Reordering successful, solve generalized Sylvester
equation for R and L,
A22 * R - L * A11 = A12
B22 * R - L * B11 = B12,
and compute estimate of Difl[(A11,B11), (A22, B22)]. */
n1 = 1;
n2 = *n - n1;
i__ = *n * *n + 1;
ztgsyl_("N", &c__3, &n2, &n1, &work[*n * n1 + n1 + 1], n, &
work[1], n, &work[n1 + 1], n, &work[*n * n1 + n1 +
i__], n, &work[i__], n, &work[n1 + i__], n, &scale, &
dif[ks], &work[(*n * *n << 1) + 1], &llwrk, &iwork[1],
&ierr);
}
}
L20:
;
}
work[1].r = (doublereal) lwmin, work[1].i = 0.;
return 0;
/* End of ZTGSNA */
} /* ztgsna_ */
int ztgsen_(int *ijob, int *wantq, int *wantz,
int *select, int *n, doublecomplex *a, int *lda,
doublecomplex *b, int *ldb, doublecomplex *alpha, doublecomplex *
beta, doublecomplex *q, int *ldq, doublecomplex *z__, int *
ldz, int *m, double *pl, double *pr, double *dif,
doublecomplex *work, int *lwork, int *iwork, int *liwork,
int *info)
{
/* System generated locals */
int a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, z_dim1,
z_offset, i__1, i__2, i__3;
doublecomplex z__1, z__2;
/* Builtin functions */
double sqrt(double), z_abs(doublecomplex *);
void d_cnjg(doublecomplex *, doublecomplex *);
/* Local variables */
int i__, k, n1, n2, ks, mn2, ijb, kase, ierr;
double dsum;
int swap;
doublecomplex temp1, temp2;
int isave[3];
extern int zscal_(int *, doublecomplex *,
doublecomplex *, int *);
int wantd;
int lwmin;
int wantp;
extern int zlacn2_(int *, doublecomplex *,
doublecomplex *, double *, int *, int *);
int wantd1, wantd2;
extern double dlamch_(char *);
double dscale, rdscal, safmin;
extern int xerbla_(char *, int *);
int liwmin;
extern int zlacpy_(char *, int *, int *,
doublecomplex *, int *, doublecomplex *, int *),
ztgexc_(int *, int *, int *, doublecomplex *, int
*, doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, int *, int *, int *),
zlassq_(int *, doublecomplex *, int *, double *,
double *);
int lquery;
extern int ztgsyl_(char *, int *, int *, int
*, doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, doublecomplex *, int *,
doublecomplex *, int *, doublecomplex *, int *,
double *, double *, doublecomplex *, int *, int *,
int *);
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* January 2007 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* ZTGSEN reorders the generalized Schur decomposition of a complex */
/* matrix pair (A, B) (in terms of an unitary equivalence trans- */
/* formation Q' * (A, B) * Z), so that a selected cluster of eigenvalues */
/* appears in the leading diagonal blocks of the pair (A,B). The leading */
/* columns of Q and Z form unitary bases of the corresponding left and */
/* right eigenspaces (deflating subspaces). (A, B) must be in */
/* generalized Schur canonical form, that is, A and B are both upper */
/* triangular. */
/* ZTGSEN also computes the generalized eigenvalues */
/* w(j)= ALPHA(j) / BETA(j) */
/* of the reordered matrix pair (A, B). */
/* Optionally, the routine computes estimates of reciprocal condition */
/* numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), */
/* (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) */
/* between the matrix pairs (A11, B11) and (A22,B22) that correspond to */
/* the selected cluster and the eigenvalues outside the cluster, resp., */
/* and norms of "projections" onto left and right eigenspaces w.r.t. */
/* the selected cluster in the (1,1)-block. */
/* Arguments */
/* ========= */
/* IJOB (input) int */
/* Specifies whether condition numbers are required for the */
/* cluster of eigenvalues (PL and PR) or the deflating subspaces */
/* (Difu and Difl): */
/* =0: Only reorder w.r.t. SELECT. No extras. */
/* =1: Reciprocal of norms of "projections" onto left and right */
/* eigenspaces w.r.t. the selected cluster (PL and PR). */
/* =2: Upper bounds on Difu and Difl. F-norm-based estimate */
/* (DIF(1:2)). */
/* =3: Estimate of Difu and Difl. 1-norm-based estimate */
/* (DIF(1:2)). */
/* About 5 times as expensive as IJOB = 2. */
/* =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic */
/* version to get it all. */
/* =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) */
/* WANTQ (input) LOGICAL */
/* .TRUE. : update the left transformation matrix Q; */
/* .FALSE.: do not update Q. */
/* WANTZ (input) LOGICAL */
/* .TRUE. : update the right transformation matrix Z; */
/* .FALSE.: do not update Z. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* SELECT specifies the eigenvalues in the selected cluster. To */
/* select an eigenvalue w(j), SELECT(j) must be set to */
/* .TRUE.. */
/* N (input) INTEGER */
/* The order of the matrices A and B. N >= 0. */
/* A (input/output) COMPLEX*16 array, dimension(LDA,N) */
/* On entry, the upper triangular matrix A, in generalized */
/* Schur canonical form. */
/* On exit, A is overwritten by the reordered matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= MAX(1,N). */
/* B (input/output) COMPLEX*16 array, dimension(LDB,N) */
/* On entry, the upper triangular matrix B, in generalized */
/* Schur canonical form. */
/* On exit, B is overwritten by the reordered matrix B. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= MAX(1,N). */
/* ALPHA (output) COMPLEX*16 array, dimension (N) */
/* BETA (output) COMPLEX*16 array, dimension (N) */
/* The diagonal elements of A and B, respectively, */
/* when the pair (A,B) has been reduced to generalized Schur */
/* form. ALPHA(i)/BETA(i) i=1,...,N are the generalized */
/* eigenvalues. */
/* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) */
/* On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. */
/* On exit, Q has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Q form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTQ = .FALSE., Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= 1. */
/* If WANTQ = .TRUE., LDQ >= N. */
/* Z (input/output) COMPLEX*16 array, dimension (LDZ,N) */
/* On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. */
/* On exit, Z has been postmultiplied by the left unitary */
/* transformation matrix which reorder (A, B); The leading M */
/* columns of Z form orthonormal bases for the specified pair of */
/* left eigenspaces (deflating subspaces). */
/* If WANTZ = .FALSE., Z is not referenced. */
/* LDZ (input) INTEGER */
/* The leading dimension of the array Z. LDZ >= 1. */
/* If WANTZ = .TRUE., LDZ >= N. */
/* M (output) INTEGER */
/* The dimension of the specified pair of left and right */
/* eigenspaces, (deflating subspaces) 0 <= M <= N. */
/* PL (output) DOUBLE PRECISION */
/* PR (output) DOUBLE PRECISION */
/* If IJOB = 1, 4 or 5, PL, PR are lower bounds on the */
/* reciprocal of the norm of "projections" onto left and right */
/* eigenspace with respect to the selected cluster. */
/* 0 < PL, PR <= 1. */
/* If M = 0 or M = N, PL = PR = 1. */
/* If IJOB = 0, 2 or 3 PL, PR are not referenced. */
/* DIF (output) DOUBLE PRECISION array, dimension (2). */
/* If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. */
/* If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on */
/* Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based */
/* estimates of Difu and Difl, computed using reversed */
/* communication with ZLACN2. */
/* If M = 0 or N, DIF(1:2) = F-norm([A, B]). */
/* If IJOB = 0 or 1, DIF is not referenced. */
/* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) */
/* IF IJOB = 0, WORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, WORK(1) returns the optimal LWORK. */
/* LWORK (input) INTEGER */
/* The dimension of the array WORK. LWORK >= 1 */
/* If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) */
/* If IJOB = 3 or 5, LWORK >= 4*M*(N-M) */
/* If LWORK = -1, then a workspace query is assumed; the routine */
/* only calculates the optimal size of the WORK array, returns */
/* this value as the first entry of the WORK array, and no error */
/* message related to LWORK is issued by XERBLA. */
/* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) */
/* IF IJOB = 0, IWORK is not referenced. Otherwise, */
/* on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. */
/* LIWORK (input) INTEGER */
/* The dimension of the array IWORK. LIWORK >= 1. */
/* If IJOB = 1, 2 or 4, LIWORK >= N+2; */
/* If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); */
/* If LIWORK = -1, then a workspace query is assumed; the */
/* routine only calculates the optimal size of the IWORK array, */
/* returns this value as the first entry of the IWORK array, and */
/* no error message related to LIWORK is issued by XERBLA. */
/* INFO (output) INTEGER */
/* =0: Successful exit. */
/* <0: If INFO = -i, the i-th argument had an illegal value. */
/* =1: Reordering of (A, B) failed because the transformed */
/* matrix pair (A, B) would be too far from generalized */
/* Schur form; the problem is very ill-conditioned. */
/* (A, B) may have been partially reordered. */
/* If requested, 0 is returned in DIF(*), PL and PR. */
/* Further Details */
/* =============== */
/* ZTGSEN first collects the selected eigenvalues by computing unitary */
/* U and W that move them to the top left corner of (A, B). In other */
/* words, the selected eigenvalues are the eigenvalues of (A11, B11) in */
/* U'*(A, B)*W = (A11 A12) (B11 B12) n1 */
/* ( 0 A22),( 0 B22) n2 */
/* n1 n2 n1 n2 */
/* where N = n1+n2 and U' means the conjugate transpose of U. The first */
/* n1 columns of U and W span the specified pair of left and right */
/* eigenspaces (deflating subspaces) of (A, B). */
/* If (A, B) has been obtained from the generalized float Schur */
/* decomposition of a matrix pair (C, D) = Q*(A, B)*Z', then the */
/* reordered generalized Schur form of (C, D) is given by */
/* (C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)', */
/* and the first n1 columns of Q*U and Z*W span the corresponding */
/* deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). */
/* Note that if the selected eigenvalue is sufficiently ill-conditioned, */
/* then its value may differ significantly from its value before */
/* reordering. */
/* The reciprocal condition numbers of the left and right eigenspaces */
/* spanned by the first n1 columns of U and W (or Q*U and Z*W) may */
/* be returned in DIF(1:2), corresponding to Difu and Difl, resp. */
/* The Difu and Difl are defined as: */
/* Difu[(A11, B11), (A22, B22)] = sigma-MIN( Zu ) */
/* and */
/* Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], */
/* where sigma-MIN(Zu) is the smallest singular value of the */
/* (2*n1*n2)-by-(2*n1*n2) matrix */
/* Zu = [ kron(In2, A11) -kron(A22', In1) ] */
/* [ kron(In2, B11) -kron(B22', In1) ]. */
/* Here, Inx is the identity matrix of size nx and A22' is the */
/* transpose of A22. kron(X, Y) is the Kronecker product between */
/* the matrices X and Y. */
/* When DIF(2) is small, small changes in (A, B) can cause large changes */
/* in the deflating subspace. An approximate (asymptotic) bound on the */
/* maximum angular error in the computed deflating subspaces is */
/* EPS * norm((A, B)) / DIF(2), */
/* where EPS is the machine precision. */
/* The reciprocal norm of the projectors on the left and right */
/* eigenspaces associated with (A11, B11) may be returned in PL and PR. */
/* They are computed as follows. First we compute L and R so that */
/* P*(A, B)*Q is block diagonal, where */
/* P = ( I -L ) n1 Q = ( I R ) n1 */
/* ( 0 I ) n2 and ( 0 I ) n2 */
/* n1 n2 n1 n2 */
/* and (L, R) is the solution to the generalized Sylvester equation */
/* A11*R - L*A22 = -A12 */
/* B11*R - L*B22 = -B12 */
/* Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). */
/* An approximate (asymptotic) bound on the average absolute error of */
/* the selected eigenvalues is */
/* EPS * norm((A, B)) / PL. */
/* There are also global error bounds which valid for perturbations up */
/* to a certain restriction: A lower bound (x) on the smallest */
/* F-norm(E,F) for which an eigenvalue of (A11, B11) may move and */
/* coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), */
/* (i.e. (A + E, B + F), is */
/* x = MIN(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*MAX(1/PL,1/PR)). */
/* An approximate bound on x can be computed from DIF(1:2), PL and PR. */
/* If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed */
/* (L', R') and unperturbed (L, R) left and right deflating subspaces */
/* associated with the selected cluster in the (1,1)-blocks can be */
/* bounded as */
/* max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) */
/* max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) */
/* See LAPACK User's Guide section 4.11 or the following references */
/* for more information. */
/* Note that if the default method for computing the Frobenius-norm- */
/* based estimate DIF is not wanted (see ZLATDF), then the parameter */
/* IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF */
/* (IJOB = 2 will be used)). See ZTGSYL for more details. */
/* Based on contributions by */
/* Bo Kagstrom and Peter Poromaa, Department of Computing Science, */
/* Umea University, S-901 87 Umea, Sweden. */
/* References */
/* ========== */
/* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the */
/* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in */
/* M.S. Moonen et al (eds), Linear Algebra for Large Scale and */
/* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. */
/* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified */
/* Eigenvalues of a Regular Matrix Pair (A, B) and Condition */
/* Estimation: Theory, Algorithms and Software, Report */
/* UMINF - 94.04, Department of Computing Science, Umea University, */
/* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. */
/* To appear in Numerical Algorithms, 1996. */
/* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software */
/* for Solving the Generalized Sylvester Equation and Estimating the */
/* Separation between Regular Matrix Pairs, Report UMINF - 93.23, */
/* Department of Computing Science, Umea University, S-901 87 Umea, */
/* Sweden, December 1993, Revised April 1994, Also as LAPACK working */
/* Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, */
/* 1996. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters */
/* Parameter adjustments */
--select;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
z_dim1 = *ldz;
z_offset = 1 + z_dim1;
z__ -= z_offset;
--dif;
--work;
--iwork;
/* Function Body */
*info = 0;
lquery = *lwork == -1 || *liwork == -1;
if (*ijob < 0 || *ijob > 5) {
*info = -1;
} else if (*n < 0) {
*info = -5;
} else if (*lda < MAX(1,*n)) {
*info = -7;
} else if (*ldb < MAX(1,*n)) {
*info = -9;
} else if (*ldq < 1 || *wantq && *ldq < *n) {
*info = -13;
} else if (*ldz < 1 || *wantz && *ldz < *n) {
*info = -15;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTGSEN", &i__1);
return 0;
}
ierr = 0;
wantp = *ijob == 1 || *ijob >= 4;
wantd1 = *ijob == 2 || *ijob == 4;
wantd2 = *ijob == 3 || *ijob == 5;
wantd = wantd1 || wantd2;
/* Set M to the dimension of the specified pair of deflating */
/* subspaces. */
*m = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
if (k < *n) {
if (select[k]) {
++(*m);
}
} else {
if (select[*n]) {
++(*m);
}
}
/* L10: */
}
if (*ijob == 1 || *ijob == 2 || *ijob == 4) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m);
lwmin = MAX(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = *n + 2;
liwmin = MAX(i__1,i__2);
} else if (*ijob == 3 || *ijob == 5) {
/* Computing MAX */
i__1 = 1, i__2 = (*m << 2) * (*n - *m);
lwmin = MAX(i__1,i__2);
/* Computing MAX */
i__1 = 1, i__2 = (*m << 1) * (*n - *m), i__1 = MAX(i__1,i__2), i__2 =
*n + 2;
liwmin = MAX(i__1,i__2);
} else {
lwmin = 1;
liwmin = 1;
}
work[1].r = (double) lwmin, work[1].i = 0.;
iwork[1] = liwmin;
if (*lwork < lwmin && ! lquery) {
*info = -21;
} else if (*liwork < liwmin && ! lquery) {
*info = -23;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("ZTGSEN", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible. */
if (*m == *n || *m == 0) {
if (wantp) {
*pl = 1.;
*pr = 1.;
}
if (wantd) {
dscale = 0.;
dsum = 1.;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
zlassq_(n, &a[i__ * a_dim1 + 1], &c__1, &dscale, &dsum);
zlassq_(n, &b[i__ * b_dim1 + 1], &c__1, &dscale, &dsum);
/* L20: */
}
dif[1] = dscale * sqrt(dsum);
dif[2] = dif[1];
}
goto L70;
}
/* Get machine constant */
safmin = dlamch_("S");
/* Collect the selected blocks at the top-left corner of (A, B). */
ks = 0;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
swap = select[k];
if (swap) {
++ks;
/* Swap the K-th block to position KS. Compute unitary Q */
/* and Z that will swap adjacent diagonal blocks in (A, B). */
if (k != ks) {
ztgexc_(wantq, wantz, n, &a[a_offset], lda, &b[b_offset], ldb,
&q[q_offset], ldq, &z__[z_offset], ldz, &k, &ks, &
ierr);
}
if (ierr > 0) {
/* Swap is rejected: exit. */
*info = 1;
if (wantp) {
*pl = 0.;
*pr = 0.;
}
if (wantd) {
dif[1] = 0.;
dif[2] = 0.;
}
goto L70;
}
}
/* L30: */
}
if (wantp) {
/* Solve generalized Sylvester equation for R and L: */
/* A11 * R - L * A22 = A12 */
/* B11 * R - L * B22 = B12 */
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
zlacpy_("Full", &n1, &n2, &a[i__ * a_dim1 + 1], lda, &work[1], &n1);
zlacpy_("Full", &n1, &n2, &b[i__ * b_dim1 + 1], ldb, &work[n1 * n2 +
1], &n1);
ijb = 0;
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ * a_dim1]
, lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ + i__ *
b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &dif[1], &
work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &ierr);
/* Estimate the reciprocal of norms of "projections" onto */
/* left and right eigenspaces */
rdscal = 0.;
dsum = 1.;
i__1 = n1 * n2;
zlassq_(&i__1, &work[1], &c__1, &rdscal, &dsum);
*pl = rdscal * sqrt(dsum);
if (*pl == 0.) {
*pl = 1.;
} else {
*pl = dscale / (sqrt(dscale * dscale / *pl + *pl) * sqrt(*pl));
}
rdscal = 0.;
dsum = 1.;
i__1 = n1 * n2;
zlassq_(&i__1, &work[n1 * n2 + 1], &c__1, &rdscal, &dsum);
*pr = rdscal * sqrt(dsum);
if (*pr == 0.) {
*pr = 1.;
} else {
*pr = dscale / (sqrt(dscale * dscale / *pr + *pr) * sqrt(*pr));
}
}
if (wantd) {
/* Compute estimates Difu and Difl. */
if (wantd1) {
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 3;
/* Frobenius norm-based Difu estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ + i__ *
a_dim1], lda, &work[1], &n1, &b[b_offset], ldb, &b[i__ +
i__ * b_dim1], ldb, &work[n1 * n2 + 1], &n1, &dscale, &
dif[1], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
/* Frobenius norm-based Difl estimate. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda, &a[
a_offset], lda, &work[1], &n2, &b[i__ + i__ * b_dim1],
ldb, &b[b_offset], ldb, &work[n1 * n2 + 1], &n2, &dscale,
&dif[2], &work[(n1 * n2 << 1) + 1], &i__1, &iwork[1], &
ierr);
} else {
/* Compute 1-norm-based estimates of Difu and Difl using */
/* reversed communication with ZLACN2. In each step a */
/* generalized Sylvester equation or a transposed variant */
/* is solved. */
kase = 0;
n1 = *m;
n2 = *n - *m;
i__ = n1 + 1;
ijb = 0;
mn2 = (n1 << 1) * n2;
/* 1-norm-based estimate of Difu. */
L40:
zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[1], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("C", &ijb, &n1, &n2, &a[a_offset], lda, &a[i__ +
i__ * a_dim1], lda, &work[1], &n1, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n1, &dscale, &dif[1], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L40;
}
dif[1] = dscale / dif[1];
/* 1-norm-based estimate of Difl. */
L50:
zlacn2_(&mn2, &work[mn2 + 1], &work[1], &dif[2], &kase, isave);
if (kase != 0) {
if (kase == 1) {
/* Solve generalized Sylvester equation */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("N", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[i__ + i__ *
b_dim1], ldb, &b[b_offset], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
} else {
/* Solve the transposed variant. */
i__1 = *lwork - (n1 << 1) * n2;
ztgsyl_("C", &ijb, &n2, &n1, &a[i__ + i__ * a_dim1], lda,
&a[a_offset], lda, &work[1], &n2, &b[b_offset],
ldb, &b[i__ + i__ * b_dim1], ldb, &work[n1 * n2 +
1], &n2, &dscale, &dif[2], &work[(n1 * n2 << 1) +
1], &i__1, &iwork[1], &ierr);
}
goto L50;
}
dif[2] = dscale / dif[2];
}
}
/* If B(K,K) is complex, make it float and positive (normalization */
/* of the generalized Schur form) and Store the generalized */
/* eigenvalues of reordered pair (A, B) */
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
dscale = z_abs(&b[k + k * b_dim1]);
if (dscale > safmin) {
i__2 = k + k * b_dim1;
z__2.r = b[i__2].r / dscale, z__2.i = b[i__2].i / dscale;
d_cnjg(&z__1, &z__2);
temp1.r = z__1.r, temp1.i = z__1.i;
i__2 = k + k * b_dim1;
z__1.r = b[i__2].r / dscale, z__1.i = b[i__2].i / dscale;
temp2.r = z__1.r, temp2.i = z__1.i;
i__2 = k + k * b_dim1;
b[i__2].r = dscale, b[i__2].i = 0.;
i__2 = *n - k;
zscal_(&i__2, &temp1, &b[k + (k + 1) * b_dim1], ldb);
i__2 = *n - k + 1;
zscal_(&i__2, &temp1, &a[k + k * a_dim1], lda);
if (*wantq) {
zscal_(n, &temp2, &q[k * q_dim1 + 1], &c__1);
}
} else {
i__2 = k + k * b_dim1;
b[i__2].r = 0., b[i__2].i = 0.;
}
i__2 = k;
i__3 = k + k * a_dim1;
alpha[i__2].r = a[i__3].r, alpha[i__2].i = a[i__3].i;
i__2 = k;
i__3 = k + k * b_dim1;
beta[i__2].r = b[i__3].r, beta[i__2].i = b[i__3].i;
/* L60: */
}
L70:
work[1].r = (double) lwmin, work[1].i = 0.;
iwork[1] = liwmin;
return 0;
/* End of ZTGSEN */
} /* ztgsen_ */