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|
/* lapack/complex16/ztrsen.f -- translated by f2c (version 20090411).
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
*/
#ifdef __cplusplus
extern "C" {
#endif
#include "v3p_netlib.h"
/* Table of constant values */
static integer c_n1 = -1;
/*< >*/
/* Subroutine */ int ztrsen_(char *job, char *compq, logical *select, integer
*n, doublecomplex *t, integer *ldt, doublecomplex *q, integer *ldq,
doublecomplex *w, integer *m, doublereal *s, doublereal *sep,
doublecomplex *work, integer *lwork, integer *info, ftnlen job_len,
ftnlen compq_len)
{
/* System generated locals */
integer q_dim1, q_offset, t_dim1, t_offset, i__1, i__2, i__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer k, n1, n2, nn, ks;
doublereal est;
integer kase, ierr;
doublereal scale;
extern logical lsame_(const char *, const char *, ftnlen, ftnlen);
integer isave[3], lwmin;
logical wantq, wants;
doublereal rnorm, rwork[1];
extern /* Subroutine */ int zlacn2_(integer *, doublecomplex *,
doublecomplex *, doublereal *, integer *, integer *), xerbla_(
char *, integer *, ftnlen);
extern doublereal zlange_(char *, integer *, integer *, doublecomplex *,
integer *, doublereal *, ftnlen);
logical wantbh;
extern /* Subroutine */ int zlacpy_(char *, integer *, integer *,
doublecomplex *, integer *, doublecomplex *, integer *, ftnlen);
logical wantsp;
extern /* Subroutine */ int ztrexc_(char *, integer *, doublecomplex *,
integer *, doublecomplex *, integer *, integer *, integer *,
integer *, ftnlen);
logical lquery;
extern /* Subroutine */ int ztrsyl_(char *, char *, integer *, integer *,
integer *, doublecomplex *, integer *, doublecomplex *, integer *,
doublecomplex *, integer *, doublereal *, integer *, ftnlen,
ftnlen);
(void)job_len;
(void)compq_len;
/* -- LAPACK routine (version 3.2) -- */
/* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
/* November 2006 */
/* Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH. */
/* .. Scalar Arguments .. */
/*< CHARACTER COMPQ, JOB >*/
/*< INTEGER INFO, LDQ, LDT, LWORK, M, N >*/
/*< DOUBLE PRECISION S, SEP >*/
/* .. */
/* .. Array Arguments .. */
/*< LOGICAL SELECT( * ) >*/
/*< COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * ) >*/
/* .. */
/* Purpose */
/* ======= */
/* ZTRSEN reorders the Schur factorization of a complex matrix */
/* A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in */
/* the leading positions on the diagonal of the upper triangular matrix */
/* T, and the leading columns of Q form an orthonormal basis of the */
/* corresponding right invariant subspace. */
/* Optionally the routine computes the reciprocal condition numbers of */
/* the cluster of eigenvalues and/or the invariant subspace. */
/* Arguments */
/* ========= */
/* JOB (input) CHARACTER*1 */
/* Specifies whether condition numbers are required for the */
/* cluster of eigenvalues (S) or the invariant subspace (SEP): */
/* = 'N': none; */
/* = 'E': for eigenvalues only (S); */
/* = 'V': for invariant subspace only (SEP); */
/* = 'B': for both eigenvalues and invariant subspace (S and */
/* SEP). */
/* COMPQ (input) CHARACTER*1 */
/* = 'V': update the matrix Q of Schur vectors; */
/* = 'N': do not update Q. */
/* SELECT (input) LOGICAL array, dimension (N) */
/* SELECT specifies the eigenvalues in the selected cluster. To */
/* select the j-th eigenvalue, SELECT(j) must be set to .TRUE.. */
/* N (input) INTEGER */
/* The order of the matrix T. N >= 0. */
/* T (input/output) COMPLEX*16 array, dimension (LDT,N) */
/* On entry, the upper triangular matrix T. */
/* On exit, T is overwritten by the reordered matrix T, with the */
/* selected eigenvalues as the leading diagonal elements. */
/* LDT (input) INTEGER */
/* The leading dimension of the array T. LDT >= max(1,N). */
/* Q (input/output) COMPLEX*16 array, dimension (LDQ,N) */
/* On entry, if COMPQ = 'V', the matrix Q of Schur vectors. */
/* On exit, if COMPQ = 'V', Q has been postmultiplied by the */
/* unitary transformation matrix which reorders T; the leading M */
/* columns of Q form an orthonormal basis for the specified */
/* invariant subspace. */
/* If COMPQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. */
/* LDQ >= 1; and if COMPQ = 'V', LDQ >= N. */
/* W (output) COMPLEX*16 array, dimension (N) */
/* The reordered eigenvalues of T, in the same order as they */
/* appear on the diagonal of T. */
/* M (output) INTEGER */
/* The dimension of the specified invariant subspace. */
/* 0 <= M <= N. */
/* S (output) DOUBLE PRECISION */
/* If JOB = 'E' or 'B', S is a lower bound on the reciprocal */
/* condition number for the selected cluster of eigenvalues. */
/* S cannot underestimate the true reciprocal condition number */
/* by more than a factor of sqrt(N). If M = 0 or N, S = 1. */
/* If JOB = 'N' or 'V', S is not referenced. */
/* SEP (output) DOUBLE PRECISION */
/* If JOB = 'V' or 'B', SEP is the estimated reciprocal */
/* condition number of the specified invariant subspace. If */
/* M = 0 or N, SEP = norm(T). */
/* If JOB = 'N' or 'E', SEP is not referenced. */
/* WORK (workspace/output) COMPLEX*16 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 JOB = 'N', LWORK >= 1; */
/* if JOB = 'E', LWORK = max(1,M*(N-M)); */
/* if JOB = 'V' or 'B', LWORK >= max(1,2*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. */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value */
/* Further Details */
/* =============== */
/* ZTRSEN first collects the selected eigenvalues by computing a unitary */
/* transformation Z to move them to the top left corner of T. In other */
/* words, the selected eigenvalues are the eigenvalues of T11 in: */
/* Z'*T*Z = ( T11 T12 ) n1 */
/* ( 0 T22 ) n2 */
/* n1 n2 */
/* where N = n1+n2 and Z' means the conjugate transpose of Z. The first */
/* n1 columns of Z span the specified invariant subspace of T. */
/* If T has been obtained from the Schur factorization of a matrix */
/* A = Q*T*Q', then the reordered Schur factorization of A is given by */
/* A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the */
/* corresponding invariant subspace of A. */
/* The reciprocal condition number of the average of the eigenvalues of */
/* T11 may be returned in S. S lies between 0 (very badly conditioned) */
/* and 1 (very well conditioned). It is computed as follows. First we */
/* compute R so that */
/* P = ( I R ) n1 */
/* ( 0 0 ) n2 */
/* n1 n2 */
/* is the projector on the invariant subspace associated with T11. */
/* R is the solution of the Sylvester equation: */
/* T11*R - R*T22 = T12. */
/* Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote */
/* the two-norm of M. Then S is computed as the lower bound */
/* (1 + F-norm(R)**2)**(-1/2) */
/* on the reciprocal of 2-norm(P), the true reciprocal condition number. */
/* S cannot underestimate 1 / 2-norm(P) by more than a factor of */
/* sqrt(N). */
/* An approximate error bound for the computed average of the */
/* eigenvalues of T11 is */
/* EPS * norm(T) / S */
/* where EPS is the machine precision. */
/* The reciprocal condition number of the right invariant subspace */
/* spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. */
/* SEP is defined as the separation of T11 and T22: */
/* sep( T11, T22 ) = sigma-min( C ) */
/* where sigma-min(C) is the smallest singular value of the */
/* n1*n2-by-n1*n2 matrix */
/* C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) */
/* I(m) is an m by m identity matrix, and kprod denotes the Kronecker */
/* product. We estimate sigma-min(C) by the reciprocal of an estimate of */
/* the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) */
/* cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). */
/* When SEP is small, small changes in T can cause large changes in */
/* the invariant subspace. An approximate bound on the maximum angular */
/* error in the computed right invariant subspace is */
/* EPS * norm(T) / SEP */
/* ===================================================================== */
/* .. Parameters .. */
/*< DOUBLE PRECISION ZERO, ONE >*/
/*< PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) >*/
/* .. */
/* .. Local Scalars .. */
/*< LOGICAL LQUERY, WANTBH, WANTQ, WANTS, WANTSP >*/
/*< INTEGER IERR, K, KASE, KS, LWMIN, N1, N2, NN >*/
/*< DOUBLE PRECISION EST, RNORM, SCALE >*/
/* .. */
/* .. Local Arrays .. */
/*< INTEGER ISAVE( 3 ) >*/
/*< DOUBLE PRECISION RWORK( 1 ) >*/
/* .. */
/* .. External Functions .. */
/*< LOGICAL LSAME >*/
/*< DOUBLE PRECISION ZLANGE >*/
/*< EXTERNAL LSAME, ZLANGE >*/
/* .. */
/* .. External Subroutines .. */
/*< EXTERNAL XERBLA, ZLACN2, ZLACPY, ZTREXC, ZTRSYL >*/
/* .. */
/* .. Intrinsic Functions .. */
/*< INTRINSIC MAX, SQRT >*/
/* .. */
/* .. Executable Statements .. */
/* Decode and test the input parameters. */
/*< WANTBH = LSAME( JOB, 'B' ) >*/
/* Parameter adjustments */
--select;
t_dim1 = *ldt;
t_offset = 1 + t_dim1;
t -= t_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--w;
--work;
/* Function Body */
wantbh = lsame_(job, "B", (ftnlen)1, (ftnlen)1);
/*< WANTS = LSAME( JOB, 'E' ) .OR. WANTBH >*/
wants = lsame_(job, "E", (ftnlen)1, (ftnlen)1) || wantbh;
/*< WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH >*/
wantsp = lsame_(job, "V", (ftnlen)1, (ftnlen)1) || wantbh;
/*< WANTQ = LSAME( COMPQ, 'V' ) >*/
wantq = lsame_(compq, "V", (ftnlen)1, (ftnlen)1);
/* Set M to the number of selected eigenvalues. */
/*< M = 0 >*/
*m = 0;
/*< DO 10 K = 1, N >*/
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
/*< >*/
if (select[k]) {
++(*m);
}
/*< 10 CONTINUE >*/
/* L10: */
}
/*< N1 = M >*/
n1 = *m;
/*< N2 = N - M >*/
n2 = *n - *m;
/*< NN = N1*N2 >*/
nn = n1 * n2;
/*< INFO = 0 >*/
*info = 0;
/*< LQUERY = ( LWORK.EQ.-1 ) >*/
lquery = *lwork == -1;
/*< IF( WANTSP ) THEN >*/
if (wantsp) {
/*< LWMIN = MAX( 1, 2*NN ) >*/
/* Computing MAX */
i__1 = 1, i__2 = nn << 1;
lwmin = max(i__1,i__2);
/*< ELSE IF( LSAME( JOB, 'N' ) ) THEN >*/
} else if (lsame_(job, "N", (ftnlen)1, (ftnlen)1)) {
/*< LWMIN = 1 >*/
lwmin = 1;
/*< ELSE IF( LSAME( JOB, 'E' ) ) THEN >*/
} else if (lsame_(job, "E", (ftnlen)1, (ftnlen)1)) {
/*< LWMIN = MAX( 1, NN ) >*/
lwmin = max(1,nn);
/*< END IF >*/
}
/*< >*/
if (! lsame_(job, "N", (ftnlen)1, (ftnlen)1) && ! wants && ! wantsp) {
/*< INFO = -1 >*/
*info = -1;
/*< ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN >*/
} else if (! lsame_(compq, "N", (ftnlen)1, (ftnlen)1) && ! wantq) {
/*< INFO = -2 >*/
*info = -2;
/*< ELSE IF( N.LT.0 ) THEN >*/
} else if (*n < 0) {
/*< INFO = -4 >*/
*info = -4;
/*< ELSE IF( LDT.LT.MAX( 1, N ) ) THEN >*/
} else if (*ldt < max(1,*n)) {
/*< INFO = -6 >*/
*info = -6;
/*< ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN >*/
} else if (*ldq < 1 || (wantq && *ldq) < *n) {
/*< INFO = -8 >*/
*info = -8;
/*< ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN >*/
} else if (*lwork < lwmin && ! lquery) {
/*< INFO = -14 >*/
*info = -14;
/*< END IF >*/
}
/*< IF( INFO.EQ.0 ) THEN >*/
if (*info == 0) {
/*< WORK( 1 ) = LWMIN >*/
work[1].r = (doublereal) lwmin, work[1].i = 0.;
/*< END IF >*/
}
/*< IF( INFO.NE.0 ) THEN >*/
if (*info != 0) {
/*< CALL XERBLA( 'ZTRSEN', -INFO ) >*/
i__1 = -(*info);
xerbla_("ZTRSEN", &i__1, (ftnlen)6);
/*< RETURN >*/
return 0;
/*< ELSE IF( LQUERY ) THEN >*/
} else if (lquery) {
/*< RETURN >*/
return 0;
/*< END IF >*/
}
/* Quick return if possible */
/*< IF( M.EQ.N .OR. M.EQ.0 ) THEN >*/
if (*m == *n || *m == 0) {
/*< >*/
if (wants) {
*s = 1.;
}
/*< >*/
if (wantsp) {
*sep = zlange_("1", n, n, &t[t_offset], ldt, rwork, (ftnlen)1);
}
/*< GO TO 40 >*/
goto L40;
/*< END IF >*/
}
/* Collect the selected eigenvalues at the top left corner of T. */
/*< KS = 0 >*/
ks = 0;
/*< DO 20 K = 1, N >*/
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
/*< IF( SELECT( K ) ) THEN >*/
if (select[k]) {
/*< KS = KS + 1 >*/
++ks;
/* Swap the K-th eigenvalue to position KS. */
/*< >*/
if (k != ks) {
ztrexc_(compq, n, &t[t_offset], ldt, &q[q_offset], ldq, &k, &
ks, &ierr, (ftnlen)1);
}
/*< END IF >*/
}
/*< 20 CONTINUE >*/
/* L20: */
}
/*< IF( WANTS ) THEN >*/
if (wants) {
/* Solve the Sylvester equation for R: */
/* T11*R - R*T22 = scale*T12 */
/*< CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) >*/
zlacpy_("F", &n1, &n2, &t[(n1 + 1) * t_dim1 + 1], ldt, &work[1], &n1,
(ftnlen)1);
/*< >*/
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 + 1 + (n1
+ 1) * t_dim1], ldt, &work[1], &n1, &scale, &ierr, (ftnlen)1,
(ftnlen)1);
/* Estimate the reciprocal of the condition number of the cluster */
/* of eigenvalues. */
/*< RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK ) >*/
rnorm = zlange_("F", &n1, &n2, &work[1], &n1, rwork, (ftnlen)1);
/*< IF( RNORM.EQ.ZERO ) THEN >*/
if (rnorm == 0.) {
/*< S = ONE >*/
*s = 1.;
/*< ELSE >*/
} else {
/*< >*/
*s = scale / (sqrt(scale * scale / rnorm + rnorm) * sqrt(rnorm));
/*< END IF >*/
}
/*< END IF >*/
}
/*< IF( WANTSP ) THEN >*/
if (wantsp) {
/* Estimate sep(T11,T22). */
/*< EST = ZERO >*/
est = 0.;
/*< KASE = 0 >*/
kase = 0;
/*< 30 CONTINUE >*/
L30:
/*< CALL ZLACN2( NN, WORK( NN+1 ), WORK, EST, KASE, ISAVE ) >*/
zlacn2_(&nn, &work[nn + 1], &work[1], &est, &kase, isave);
/*< IF( KASE.NE.0 ) THEN >*/
if (kase != 0) {
/*< IF( KASE.EQ.1 ) THEN >*/
if (kase == 1) {
/* Solve T11*R - R*T22 = scale*X. */
/*< >*/
ztrsyl_("N", "N", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
ierr, (ftnlen)1, (ftnlen)1);
/*< ELSE >*/
} else {
/* Solve T11'*R - R*T22' = scale*X. */
/*< >*/
ztrsyl_("C", "C", &c_n1, &n1, &n2, &t[t_offset], ldt, &t[n1 +
1 + (n1 + 1) * t_dim1], ldt, &work[1], &n1, &scale, &
ierr, (ftnlen)1, (ftnlen)1);
/*< END IF >*/
}
/*< GO TO 30 >*/
goto L30;
/*< END IF >*/
}
/*< SEP = SCALE / EST >*/
*sep = scale / est;
/*< END IF >*/
}
/*< 40 CONTINUE >*/
L40:
/* Copy reordered eigenvalues to W. */
/*< DO 50 K = 1, N >*/
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
/*< W( K ) = T( K, K ) >*/
i__2 = k;
i__3 = k + k * t_dim1;
w[i__2].r = t[i__3].r, w[i__2].i = t[i__3].i;
/*< 50 CONTINUE >*/
/* L50: */
}
/*< WORK( 1 ) = LWMIN >*/
work[1].r = (doublereal) lwmin, work[1].i = 0.;
/*< RETURN >*/
return 0;
/* End of ZTRSEN */
/*< END >*/
} /* ztrsen_ */
#ifdef __cplusplus
}
#endif
|
