sggsvd.c

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/* lapack/single/sggsvd.f -- translated by f2c (version 20050501).
   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__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, ftnlen jobu_len, ftnlen jobv_len, 
        ftnlen jobq_len)
{
    /* 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 *, ftnlen, ftnlen);
    real anorm, bnorm;
    logical wantq;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
            integer *);
    logical wantu, wantv;
    extern doublereal slamch_(char *, ftnlen), slange_(char *, integer *, 
            integer *, real *, integer *, real *, ftnlen);
    integer ncycle;
    extern /* Subroutine */ int xerbla_(char *, integer *, ftnlen), 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 *, ftnlen, ftnlen, ftnlen), 
            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 *, ftnlen, ftnlen, ftnlen);
    (void)jobu_len;
    (void)jobv_len;
    (void)jobq_len;

/*  -- 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 .. */
/*<       CHARACTER          JOBQ, JOBU, JOBV >*/
/*<       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P >*/
/*     .. */
/*     .. Array Arguments .. */
/*<       INTEGER            IWORK( * ) >*/
/*<    >*/
/*     .. */

/*  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. LDA >= 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, */