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/* sggsvd.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;
/* Subroutine */ int sggsvd_(char *jobu, char *jobv, char *jobq, integer *m,
integer *n, integer *p, integer *k, integer *l, real *a, integer *lda,
real *b, integer *ldb, real *alpha, real *beta, real *u, integer *
ldu, real *v, integer *ldv, real *q, integer *ldq, real *work,
integer *iwork, integer *info)
{
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, q_dim1, q_offset, u_dim1,
u_offset, v_dim1, v_offset, i__1, i__2;
/* Local variables */
integer i__, j;
real ulp;
integer ibnd;
real tola;
integer isub;
real tolb, unfl, temp, smax;
extern logical lsame_(char *, char *);
real anorm, bnorm;
logical wantq;
extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *,
integer *);
logical wantu, wantv;
extern doublereal slamch_(char *), slange_(char *, integer *,
integer *, real *, integer *, real *);
integer ncycle;
extern /* Subroutine */ int xerbla_(char *, integer *), stgsja_(
char *, char *, char *, integer *, integer *, integer *, integer *
, integer *, real *, integer *, real *, integer *, real *, real *,
real *, real *, real *, integer *, real *, integer *, real *,
integer *, real *, integer *, integer *),
sggsvp_(char *, char *, char *, integer *, integer *, integer *,
real *, integer *, real *, integer *, real *, real *, integer *,
integer *, real *, integer *, real *, integer *, real *, integer *
, integer *, real *, real *, integer *);
/* -- LAPACK driver routine (version 3.2) -- */
/* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/* November 2006 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SGGSVD computes the generalized singular value decomposition (GSVD) */
/* of an M-by-N real matrix A and P-by-N real matrix B: */
/* U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R ) */
/* where U, V and Q are orthogonal matrices, and Z' is the transpose */
/* of Z. Let K+L = the effective numerical rank of the matrix (A',B')', */
/* then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and */
/* D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the */
/* following structures, respectively: */
/* If M-K-L >= 0, */
/* K L */
/* D1 = K ( I 0 ) */
/* L ( 0 C ) */
/* M-K-L ( 0 0 ) */
/* K L */
/* D2 = L ( 0 S ) */
/* P-L ( 0 0 ) */
/* N-K-L K L */
/* ( 0 R ) = K ( 0 R11 R12 ) */
/* L ( 0 0 R22 ) */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(K+L) ), */
/* S = diag( BETA(K+1), ... , BETA(K+L) ), */
/* C**2 + S**2 = I. */
/* R is stored in A(1:K+L,N-K-L+1:N) on exit. */
/* If M-K-L < 0, */
/* K M-K K+L-M */
/* D1 = K ( I 0 0 ) */
/* M-K ( 0 C 0 ) */
/* K M-K K+L-M */
/* D2 = M-K ( 0 S 0 ) */
/* K+L-M ( 0 0 I ) */
/* P-L ( 0 0 0 ) */
/* N-K-L K M-K K+L-M */
/* ( 0 R ) = K ( 0 R11 R12 R13 ) */
/* M-K ( 0 0 R22 R23 ) */
/* K+L-M ( 0 0 0 R33 ) */
/* where */
/* C = diag( ALPHA(K+1), ... , ALPHA(M) ), */
/* S = diag( BETA(K+1), ... , BETA(M) ), */
/* C**2 + S**2 = I. */
/* (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored */
/* ( 0 R22 R23 ) */
/* in B(M-K+1:L,N+M-K-L+1:N) on exit. */
/* The routine computes C, S, R, and optionally the orthogonal */
/* transformation matrices U, V and Q. */
/* In particular, if B is an N-by-N nonsingular matrix, then the GSVD of */
/* A and B implicitly gives the SVD of A*inv(B): */
/* A*inv(B) = U*(D1*inv(D2))*V'. */
/* If ( A',B')' has orthonormal columns, then the GSVD of A and B is */
/* also equal to the CS decomposition of A and B. Furthermore, the GSVD */
/* can be used to derive the solution of the eigenvalue problem: */
/* A'*A x = lambda* B'*B x. */
/* In some literature, the GSVD of A and B is presented in the form */
/* U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 ) */
/* where U and V are orthogonal and X is nonsingular, D1 and D2 are */
/* ``diagonal''. The former GSVD form can be converted to the latter */
/* form by taking the nonsingular matrix X as */
/* X = Q*( I 0 ) */
/* ( 0 inv(R) ). */
/* Arguments */
/* ========= */
/* JOBU (input) CHARACTER*1 */
/* = 'U': Orthogonal matrix U is computed; */
/* = 'N': U is not computed. */
/* JOBV (input) CHARACTER*1 */
/* = 'V': Orthogonal matrix V is computed; */
/* = 'N': V is not computed. */
/* JOBQ (input) CHARACTER*1 */
/* = 'Q': Orthogonal matrix Q is computed; */
/* = 'N': Q is not computed. */
/* M (input) INTEGER */
/* The number of rows of the matrix A. M >= 0. */
/* N (input) INTEGER */
/* The number of columns of the matrices A and B. N >= 0. */
/* P (input) INTEGER */
/* The number of rows of the matrix B. P >= 0. */
/* K (output) INTEGER */
/* L (output) INTEGER */
/* On exit, K and L specify the dimension of the subblocks */
/* described in the Purpose section. */
/* K + L = effective numerical rank of (A',B')'. */
/* A (input/output) REAL array, dimension (LDA,N) */
/* On entry, the M-by-N matrix A. */
/* On exit, A contains the triangular matrix R, or part of R. */
/* See Purpose for details. */
/* LDA (input) INTEGER */
/* The leading dimension of the array A. LDA >= max(1,M). */
/* B (input/output) REAL array, dimension (LDB,N) */
/* On entry, the P-by-N matrix B. */
/* On exit, B contains the triangular matrix R if M-K-L < 0. */
/* See Purpose for details. */
/* LDB (input) INTEGER */
/* The leading dimension of the array B. LDB >= max(1,P). */
/* ALPHA (output) REAL array, dimension (N) */
/* BETA (output) REAL array, dimension (N) */
/* On exit, ALPHA and BETA contain the generalized singular */
/* value pairs of A and B; */
/* ALPHA(1:K) = 1, */
/* BETA(1:K) = 0, */
/* and if M-K-L >= 0, */
/* ALPHA(K+1:K+L) = C, */
/* BETA(K+1:K+L) = S, */
/* or if M-K-L < 0, */
/* ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0 */
/* BETA(K+1:M) =S, BETA(M+1:K+L) =1 */
/* and */
/* ALPHA(K+L+1:N) = 0 */
/* BETA(K+L+1:N) = 0 */
/* U (output) REAL array, dimension (LDU,M) */
/* If JOBU = 'U', U contains the M-by-M orthogonal matrix U. */
/* If JOBU = 'N', U is not referenced. */
/* LDU (input) INTEGER */
/* The leading dimension of the array U. LDU >= max(1,M) if */
/* JOBU = 'U'; LDU >= 1 otherwise. */
/* V (output) REAL array, dimension (LDV,P) */
/* If JOBV = 'V', V contains the P-by-P orthogonal matrix V. */
/* If JOBV = 'N', V is not referenced. */
/* LDV (input) INTEGER */
/* The leading dimension of the array V. LDV >= max(1,P) if */
/* JOBV = 'V'; LDV >= 1 otherwise. */
/* Q (output) REAL array, dimension (LDQ,N) */
/* If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. */
/* If JOBQ = 'N', Q is not referenced. */
/* LDQ (input) INTEGER */
/* The leading dimension of the array Q. LDQ >= max(1,N) if */
/* JOBQ = 'Q'; LDQ >= 1 otherwise. */
/* WORK (workspace) REAL array, */
/* dimension (max(3*N,M,P)+N) */
/* IWORK (workspace/output) INTEGER array, dimension (N) */
/* On exit, IWORK stores the sorting information. More */
/* precisely, the following loop will sort ALPHA */
/* for I = K+1, min(M,K+L) */
/* swap ALPHA(I) and ALPHA(IWORK(I)) */
/* endfor */
/* such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N). */
/* INFO (output) INTEGER */
/* = 0: successful exit */
/* < 0: if INFO = -i, the i-th argument had an illegal value. */
/* > 0: if INFO = 1, the Jacobi-type procedure failed to */
/* converge. For further details, see subroutine STGSJA. */
/* Internal Parameters */
/* =================== */
/* TOLA REAL */
/* TOLB REAL */
/* TOLA and TOLB are the thresholds to determine the effective */
/* rank of (A',B')'. Generally, they are set to */
/* TOLA = MAX(M,N)*norm(A)*MACHEPS, */
/* TOLB = MAX(P,N)*norm(B)*MACHEPS. */
/* The size of TOLA and TOLB may affect the size of backward */
/* errors of the decomposition. */
/* Further Details */
/* =============== */
/* 2-96 Based on modifications by */
/* Ming Gu and Huan Ren, Computer Science Division, University of */
/* California at Berkeley, USA */
/* ===================================================================== */
/* .. Local Scalars .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Executable Statements .. */
/* Test the input parameters */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
--alpha;
--beta;
u_dim1 = *ldu;
u_offset = 1 + u_dim1;
u -= u_offset;
v_dim1 = *ldv;
v_offset = 1 + v_dim1;
v -= v_offset;
q_dim1 = *ldq;
q_offset = 1 + q_dim1;
q -= q_offset;
--work;
--iwork;
/* Function Body */
wantu = lsame_(jobu, "U");
wantv = lsame_(jobv, "V");
wantq = lsame_(jobq, "Q");
*info = 0;
if (! (wantu || lsame_(jobu, "N"))) {
*info = -1;
} else if (! (wantv || lsame_(jobv, "N"))) {
*info = -2;
} else if (! (wantq || lsame_(jobq, "N"))) {
*info = -3;
} else if (*m < 0) {
*info = -4;
} else if (*n < 0) {
*info = -5;
} else if (*p < 0) {
*info = -6;
} else if (*lda < max(1,*m)) {
*info = -10;
} else if (*ldb < max(1,*p)) {
*info = -12;
} else if (*ldu < 1 || wantu && *ldu < *m) {
*info = -16;
} else if (*ldv < 1 || wantv && *ldv < *p) {
*info = -18;
} else if (*ldq < 1 || wantq && *ldq < *n) {
*info = -20;
}
if (*info != 0) {
i__1 = -(*info);
xerbla_("SGGSVD", &i__1);
return 0;
}
/* Compute the Frobenius norm of matrices A and B */
anorm = slange_("1", m, n, &a[a_offset], lda, &work[1]);
bnorm = slange_("1", p, n, &b[b_offset], ldb, &work[1]);
/* Get machine precision and set up threshold for determining */
/* the effective numerical rank of the matrices A and B. */
ulp = slamch_("Precision");
unfl = slamch_("Safe Minimum");
tola = max(*m,*n) * dmax(anorm,unfl) * ulp;
tolb = max(*p,*n) * dmax(bnorm,unfl) * ulp;
/* Preprocessing */
sggsvp_(jobu, jobv, jobq, m, p, n, &a[a_offset], lda, &b[b_offset], ldb, &
tola, &tolb, k, l, &u[u_offset], ldu, &v[v_offset], ldv, &q[
q_offset], ldq, &iwork[1], &work[1], &work[*n + 1], info);
/* Compute the GSVD of two upper "triangular" matrices */
stgsja_(jobu, jobv, jobq, m, p, n, k, l, &a[a_offset], lda, &b[b_offset],
ldb, &tola, &tolb, &alpha[1], &beta[1], &u[u_offset], ldu, &v[
v_offset], ldv, &q[q_offset], ldq, &work[1], &ncycle, info);
/* Sort the singular values and store the pivot indices in IWORK */
/* Copy ALPHA to WORK, then sort ALPHA in WORK */
scopy_(n, &alpha[1], &c__1, &work[1], &c__1);
/* Computing MIN */
i__1 = *l, i__2 = *m - *k;
ibnd = min(i__1,i__2);
i__1 = ibnd;
for (i__ = 1; i__ <= i__1; ++i__) {
/* Scan for largest ALPHA(K+I) */
isub = i__;
smax = work[*k + i__];
i__2 = ibnd;
for (j = i__ + 1; j <= i__2; ++j) {
temp = work[*k + j];
if (temp > smax) {
isub = j;
smax = temp;
}
/* L10: */
}
if (isub != i__) {
work[*k + isub] = work[*k + i__];
work[*k + i__] = smax;
iwork[*k + i__] = *k + isub;
} else {
iwork[*k + i__] = *k + i__;
}
/* L20: */
}
return 0;
/* End of SGGSVD */
} /* sggsvd_ */
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