f2c/c/dggsvd.c

/* ../netlib/dggsvd.f -- translated by f2c (version 20100827). 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 "FLA_f2c.h" /* Table of constant values */
static integer c__1 = 1;
/* > \brief <b> DGGSVD computes the singular value decomposition (SVD) for OTHER matrices</b> */
/* =========== DOCUMENTATION =========== */
/* Online html documentation available at */
/* http://www.netlib.org/lapack/explore-html/ */
/* > \htmlonly */
/* > Download DGGSVD + dependencies */
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/* > [TGZ]</a> */
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/* > [ZIP]</a> */
/* > <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggsvd. f"> */
/* > [TXT]</a> */
/* > \endhtmlonly */
/* Definition: */
/* =========== */
/* SUBROUTINE DGGSVD( JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, */
/* LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, */
/* IWORK, INFO ) */
/* .. Scalar Arguments .. */
/* CHARACTER JOBQ, JOBU, JOBV */
/* INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P */
/* .. */
/* .. Array Arguments .. */
/* INTEGER IWORK( * ) */
/* DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), */
/* $ BETA( * ), Q( LDQ, * ), U( LDU, * ), */
/* $ V( LDV, * ), WORK( * ) */
/* .. */
/* > \par Purpose: */
/* ============= */
/* > */
/* > \verbatim */
/* > */
/* > DGGSVD computes the generalized singular value decomposition (GSVD) */
/* > of an M-by-N real matrix A and P-by-N real matrix B: */
/* > */
/* > U**T*A*Q = D1*( 0 R ), V**T*B*Q = D2*( 0 R ) */
/* > */
/* > where U, V and Q are orthogonal matrices. */
/* > Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T, */
/* > 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**T. */
/* > If ( A**T,B**T)**T 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**T*A x = lambda* B**T*B x. */
/* > In some literature, the GSVD of A and B is presented in the form */
/* > U**T*A*X = ( 0 D1 ), V**T*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) ). */
/* > \endverbatim */
/* Arguments: */
/* ========== */
/* > \param[in] JOBU */
/* > \verbatim */
/* > JOBU is CHARACTER*1 */
/* > = 'U': Orthogonal matrix U is computed;
*/
/* > = 'N': U is not computed. */
/* > \endverbatim */
/* > */
/* > \param[in] JOBV */
/* > \verbatim */
/* > JOBV is CHARACTER*1 */
/* > = 'V': Orthogonal matrix V is computed;
*/
/* > = 'N': V is not computed. */
/* > \endverbatim */
/* > */
/* > \param[in] JOBQ */
/* > \verbatim */
/* > JOBQ is CHARACTER*1 */
/* > = 'Q': Orthogonal matrix Q is computed;
*/
/* > = 'N': Q is not computed. */
/* > \endverbatim */
/* > */
/* > \param[in] M */
/* > \verbatim */
/* > M is INTEGER */
/* > The number of rows of the matrix A. M >= 0. */
/* > \endverbatim */
/* > */
/* > \param[in] N */
/* > \verbatim */
/* > N is INTEGER */
/* > The number of columns of the matrices A and B. N >= 0. */
/* > \endverbatim */
/* > */
/* > \param[in] P */
/* > \verbatim */
/* > P is INTEGER */
/* > The number of rows of the matrix B. P >= 0. */
/* > \endverbatim */
/* > */
/* > \param[out] K */
/* > \verbatim */
/* > K is INTEGER */
/* > \endverbatim */
/* > */
/* > \param[out] L */
/* > \verbatim */
/* > L is INTEGER */
/* > */
/* > On exit, K and L specify the dimension of the subblocks */
/* > described in Purpose. */
/* > K + L = effective numerical rank of (A**T,B**T)**T. */
/* > \endverbatim */
/* > */
/* > \param[in,out] A */
/* > \verbatim */
/* > A is DOUBLE PRECISION 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. */
/* > \endverbatim */
/* > */
/* > \param[in] LDA */
/* > \verbatim */
/* > LDA is INTEGER */
/* > The leading dimension of the array A. LDA >= max(1,M). */
/* > \endverbatim */
/* > */
/* > \param[in,out] B */
/* > \verbatim */
/* > B is DOUBLE PRECISION 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. */
/* > \endverbatim */
/* > */
/* > \param[in] LDB */
/* > \verbatim */
/* > LDB is INTEGER */
/* > The leading dimension of the array B. LDB >= max(1,P). */
/* > \endverbatim */
/* > */
/* > \param[out] ALPHA */
/* > \verbatim */
/* > ALPHA is DOUBLE PRECISION array, dimension (N) */
/* > \endverbatim */
/* > */
/* > \param[out] BETA */
/* > \verbatim */
/* > BETA is DOUBLE PRECISION 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 */
/* > \endverbatim */
/* > */
/* > \param[out] U */
/* > \verbatim */
/* > U is DOUBLE PRECISION array, dimension (LDU,M) */
/* > If JOBU = 'U', U contains the M-by-M orthogonal matrix U. */
/* > If JOBU = 'N', U is not referenced. */
/* > \endverbatim */
/* > */
/* > \param[in] LDU */
/* > \verbatim */
/* > LDU is INTEGER */
/* > The leading dimension of the array U. LDU >= max(1,M) if */
/* > JOBU = 'U';
LDU >= 1 otherwise. */
/* > \endverbatim */
/* > */
/* > \param[out] V */
/* > \verbatim */
/* > V is DOUBLE PRECISION array, dimension (LDV,P) */
/* > If JOBV = 'V', V contains the P-by-P orthogonal matrix V. */
/* > If JOBV = 'N', V is not referenced. */
/* > \endverbatim */
/* > */
/* > \param[in] LDV */
/* > \verbatim */
/* > LDV is INTEGER */
/* > The leading dimension of the array V. LDV >= max(1,P) if */
/* > JOBV = 'V';
LDV >= 1 otherwise. */
/* > \endverbatim */
/* > */
/* > \param[out] Q */
/* > \verbatim */
/* > Q is DOUBLE PRECISION array, dimension (LDQ,N) */
/* > If JOBQ = 'Q', Q contains the N-by-N orthogonal matrix Q. */
/* > If JOBQ = 'N', Q is not referenced. */
/* > \endverbatim */
/* > */
/* > \param[in] LDQ */
/* > \verbatim */
/* > LDQ is INTEGER */
/* > The leading dimension of the array Q. LDQ >= max(1,N) if */
/* > JOBQ = 'Q';
LDQ >= 1 otherwise. */
/* > \endverbatim */
/* > */
/* > \param[out] WORK */
/* > \verbatim */
/* > WORK is DOUBLE PRECISION array, */
/* > dimension (max(3*N,M,P)+N) */
/* > \endverbatim */
/* > */
/* > \param[out] IWORK */
/* > \verbatim */
/* > IWORK is 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). */
/* > \endverbatim */
/* > */
/* > \param[out] INFO */
/* > \verbatim */
/* > INFO is 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 DTGSJA. */
/* > \endverbatim */
/* > \par Internal Parameters: */
/* ========================= */
/* > */
/* > \verbatim */
/* > TOLA DOUBLE PRECISION */
/* > TOLB DOUBLE PRECISION */
/* > TOLA and TOLB are the thresholds to determine the effective */
/* > rank of (A',B')**T. Generally, they are set to */
/* > TOLA = MAX(M,N)*norm(A)*MAZHEPS, */
/* > TOLB = MAX(P,N)*norm(B)*MAZHEPS. */
/* > The size of TOLA and TOLB may affect the size of backward */
/* > errors of the decomposition. */
/* > \endverbatim */
/* Authors: */
/* ======== */
/* > \author Univ. of Tennessee */
/* > \author Univ. of California Berkeley */
/* > \author Univ. of Colorado Denver */
/* > \author NAG Ltd. */
/* > \date November 2011 */
/* > \ingroup doubleOTHERsing */
/* > \par Contributors: */
/* ================== */
/* > */
/* > Ming Gu and Huan Ren, Computer Science Division, University of */
/* > California at Berkeley, USA */
/* > */
/* ===================================================================== */
/* Subroutine */
int dggsvd_(char *jobu, char *jobv, char *jobq, integer *m, integer *n, integer *p, integer *k, integer *l, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *alpha, doublereal *beta, doublereal *u, integer *ldu, doublereal *v, integer *ldv, doublereal *q, integer *ldq, doublereal *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;
    doublereal ulp;
    integer ibnd;
    doublereal tola;
    integer isub;
    doublereal tolb, unfl, temp, smax;
    extern logical lsame_(char *, char *);
    doublereal anorm, bnorm;
    extern /* Subroutine */
    int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *);
    logical wantq, wantu, wantv;
    extern doublereal dlamch_(char *), dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *);
    extern /* Subroutine */
    int dtgsja_(char *, char *, char *, integer *, integer *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *);
    integer ncycle;
    extern /* Subroutine */
    int xerbla_(char *, integer *), dggsvp_( char *, char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *);
    /* -- LAPACK driver routine (version 3.4.0) -- */
    /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
    /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
    /* November 2011 */
    /* .. Scalar Arguments .. */
    /* .. */
    /* .. Array Arguments .. */
    /* .. */
    /* ===================================================================== */
    /* .. 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_("DGGSVD", &i__1);
        return 0;
    }
    /* Compute the Frobenius norm of matrices A and B */
    anorm = dlange_("1", m, n, &a[a_offset], lda, &work[1]);
    bnorm = dlange_("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 = dlamch_("Precision");
    unfl = dlamch_("Safe Minimum");
    tola = max(*m,*n) * max(anorm,unfl) * ulp;
    tolb = max(*p,*n) * max(bnorm,unfl) * ulp;
    /* Preprocessing */
    dggsvp_(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 */
    dtgsja_(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 */
    dcopy_(n, &alpha[1], &c__1, &work[1], &c__1);
    /* Computing MIN */
    i__1 = *l;
    i__2 = *m - *k; // , expr subst
    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 DGGSVD */
}
/* dggsvd_ */