/* ctgsna.f -- translated by f2c (version 20061008).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
#include "blaswrap.h"
/* Table of constant values */
static integer c__1 = 1;
static complex c_b19 = {1.f,0.f};
static complex c_b20 = {0.f,0.f};
static logical c_false = FALSE_;
static integer c__3 = 3;
/* Subroutine */ int ctgsna_(char *job, char *howmny, logical *select,
integer *n, complex *a, integer *lda, complex *b, integer *ldb,
complex *vl, integer *ldvl, complex *vr, integer *ldvr, real *s, real
*dif, integer *mm, integer *m, complex *work, integer *lwork, integer
*iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1,
vr_offset, i__1;
real r__1, r__2;
complex q__1;
/* Builtin functions */
double c_abs(complex *);
/* Local variables */
integer i__, k, n1, n2, ks;
real eps, cond;
integer ierr, ifst;
real lnrm;
complex yhax, yhbx;
integer ilst;
real rnrm, scale;
extern /* Complex */ VOID cdotc_(complex *, integer *, complex *, integer
*, complex *, integer *);
extern logical lsame_(char *, char *);
extern /* Subroutine */ int cgemv_(char *, integer *, integer *, complex *
, complex *, integer *, complex *, integer *, complex *, complex *
, integer *);
integer lwmin;
logical wants;
complex dummy[1];
extern doublereal scnrm2_(integer *, complex *, integer *), slapy2_(real *
, real *);
complex dummy1[1];
extern /* Subroutine */ int slabad_(real *, real *);
extern doublereal slamch_(char *);
extern /* Subroutine */ int clacpy_(char *, integer *, integer *, complex
*, integer *, complex *, integer *), ctgexc_(logical *,
logical *, integer *, complex *, integer *, complex *, integer *,
complex *, integer *, complex *, integer *, integer *, integer *,
integer *), xerbla_(char *, integer *);
real bignum;
logical wantbh, wantdf, somcon;
extern /* Subroutine */ int ctgsyl_(char *, integer *, integer *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, complex *, integer *, complex *, integer *, complex *, integer
*, real *, real *, complex *, integer *, integer *, integer *);
real smlnum;
logical lquery;
/* -- LAPACK routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* CTGSNA 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 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 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 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 CTGEVC. */
/* 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 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 CTGEVC. */
/* 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) REAL 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) REAL 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 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,N). */
/* If JOB = 'V' or 'B', LWORK >= max(1,2*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 CLATDF. */
/* 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. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic 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;
vl_dim1 = *ldvl;
vl_offset = 1 + vl_dim1;
vl -= vl_offset;
vr_dim1 = *ldvr;
vr_offset = 1 + vr_dim1;
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 (! 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 (*n == 0) {
lwmin = 1;
} else if (lsame_(job, "V") || lsame_(job,
"B")) {
lwmin = (*n << 1) * *n;
} else {
lwmin = *n;
}
work[1].r = (real) lwmin, work[1].i = 0.f;
if (*mm < *m) {
*info = -15;
} else if (*lwork < lwmin && ! lquery) {
*info = -18;
}
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("CTGSNA", &i__1);
return 0;
} else if (lquery) {
return 0;
}
/* Quick return if possible */
if (*n == 0) {
return 0;
}
/* Get machine constants */
eps = slamch_("P");
smlnum = slamch_("S") / eps;
bignum = 1.f / smlnum;
slabad_(&smlnum, &bignum);
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 = scnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1);
lnrm = scnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1);
cgemv_("N", n, n, &c_b19, &a[a_offset], lda, &vr[ks * vr_dim1 + 1]
, &c__1, &c_b20, &work[1], &c__1);
cdotc_(&q__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1);
yhax.r = q__1.r, yhax.i = q__1.i;
cgemv_("N", n, n, &c_b19, &b[b_offset], ldb, &vr[ks * vr_dim1 + 1]
, &c__1, &c_b20, &work[1], &c__1);
cdotc_(&q__1, n, &work[1], &c__1, &vl[ks * vl_dim1 + 1], &c__1);
yhbx.r = q__1.r, yhbx.i = q__1.i;
r__1 = c_abs(&yhax);
r__2 = c_abs(&yhbx);
cond = slapy2_(&r__1, &r__2);
if (cond == 0.f) {
s[ks] = -1.f;
} else {
s[ks] = cond / (rnrm * lnrm);
}
}
if (wantdf) {
if (*n == 1) {
r__1 = c_abs(&a[a_dim1 + 1]);
r__2 = c_abs(&b[b_dim1 + 1]);
dif[ks] = slapy2_(&r__1, &r__2);
} else {
/* 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. */
clacpy_("Full", n, n, &a[a_offset], lda, &work[1], n);
clacpy_("Full", n, n, &b[b_offset], ldb, &work[*n * *n + 1],
n);
ifst = k;
ilst = 1;
ctgexc_(&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.f;
} 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;
ctgsyl_("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], dummy, &c__1, &iwork[1], &ierr);
}
}
}
L20:
;
}
work[1].r = (real) lwmin, work[1].i = 0.f;
return 0;
/* End of CTGSNA */
} /* ctgsna_ */