dgelsy.c

Go to the documentation of this file.

/* dgelsy.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 integer c_n1 = -1;
static integer c__0 = 0;
static doublereal c_b31 = 0.;
static integer c__2 = 2;
static doublereal c_b54 = 1.;

/* Subroutine */ int dgelsy_(integer *m, integer *n, integer *nrhs, 
        doublereal *a, integer *lda, doublereal *b, integer *ldb, integer *
        jpvt, doublereal *rcond, integer *rank, doublereal *work, integer *
        lwork, integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
    doublereal d__1, d__2;

    /* Local variables */
    integer i__, j;
    doublereal c1, c2, s1, s2;
    integer nb, mn, nb1, nb2, nb3, nb4;
    doublereal anrm, bnrm, smin, smax;
    integer iascl, ibscl;
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
            doublereal *, integer *);
    integer ismin, ismax;
    extern /* Subroutine */ int dtrsm_(char *, char *, char *, char *, 
            integer *, integer *, doublereal *, doublereal *, integer *, 
            doublereal *, integer *), dlaic1_(
            integer *, integer *, doublereal *, doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *, doublereal *);
    doublereal wsize;
    extern /* Subroutine */ int dgeqp3_(integer *, integer *, doublereal *, 
            integer *, integer *, doublereal *, doublereal *, integer *, 
            integer *), dlabad_(doublereal *, doublereal *);
    extern doublereal dlamch_(char *), dlange_(char *, integer *, 
            integer *, doublereal *, integer *, doublereal *);
    extern /* Subroutine */ int dlascl_(char *, integer *, integer *, 
            doublereal *, doublereal *, integer *, integer *, doublereal *, 
            integer *, integer *), dlaset_(char *, integer *, integer 
            *, doublereal *, doublereal *, doublereal *, integer *), 
            xerbla_(char *, integer *);
    extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 
            integer *, integer *);
    doublereal bignum;
    integer lwkmin;
    extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, 
            integer *, doublereal *, integer *, doublereal *, doublereal *, 
            integer *, doublereal *, integer *, integer *);
    doublereal sminpr, smaxpr, smlnum;
    extern /* Subroutine */ int dormrz_(char *, char *, integer *, integer *, 
            integer *, integer *, doublereal *, integer *, doublereal *, 
            doublereal *, integer *, doublereal *, integer *, integer *);
    integer lwkopt;
    logical lquery;
    extern /* Subroutine */ int dtzrzf_(integer *, integer *, doublereal *, 
            integer *, doublereal *, doublereal *, integer *, integer *);


/*  -- LAPACK driver routine (version 3.2) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

/*     .. Scalar Arguments .. */
/*     .. */
/*     .. Array Arguments .. */
/*     .. */

/*  Purpose */
/*  ======= */

/*  DGELSY computes the minimum-norm solution to a real linear least */
/*  squares problem: */
/*      minimize || A * X - B || */
/*  using a complete orthogonal factorization of A.  A is an M-by-N */
/*  matrix which may be rank-deficient. */

/*  Several right hand side vectors b and solution vectors x can be */
/*  handled in a single call; they are stored as the columns of the */
/*  M-by-NRHS right hand side matrix B and the N-by-NRHS solution */
/*  matrix X. */

/*  The routine first computes a QR factorization with column pivoting: */
/*      A * P = Q * [ R11 R12 ] */
/*                  [  0  R22 ] */
/*  with R11 defined as the largest leading submatrix whose estimated */
/*  condition number is less than 1/RCOND.  The order of R11, RANK, */
/*  is the effective rank of A. */

/*  Then, R22 is considered to be negligible, and R12 is annihilated */
/*  by orthogonal transformations from the right, arriving at the */
/*  complete orthogonal factorization: */
/*     A * P = Q * [ T11 0 ] * Z */
/*                 [  0  0 ] */
/*  The minimum-norm solution is then */
/*     X = P * Z' [ inv(T11)*Q1'*B ] */
/*                [        0       ] */
/*  where Q1 consists of the first RANK columns of Q. */

/*  This routine is basically identical to the original xGELSX except */
/*  three differences: */
/*    o The call to the subroutine xGEQPF has been substituted by the */
/*      the call to the subroutine xGEQP3. This subroutine is a Blas-3 */
/*      version of the QR factorization with column pivoting. */
/*    o Matrix B (the right hand side) is updated with Blas-3. */
/*    o The permutation of matrix B (the right hand side) is faster and */
/*      more simple. */

/*  Arguments */
/*  ========= */

/*  M       (input) INTEGER */
/*          The number of rows of the matrix A.  M >= 0. */

/*  N       (input) INTEGER */
/*          The number of columns of the matrix A.  N >= 0. */

/*  NRHS    (input) INTEGER */
/*          The number of right hand sides, i.e., the number of */
/*          columns of matrices B and X. NRHS >= 0. */

/*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N) */
/*          On entry, the M-by-N matrix A. */
/*          On exit, A has been overwritten by details of its */
/*          complete orthogonal factorization. */

/*  LDA     (input) INTEGER */
/*          The leading dimension of the array A.  LDA >= max(1,M). */

/*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */
/*          On entry, the M-by-NRHS right hand side matrix B. */
/*          On exit, the N-by-NRHS solution matrix X. */

/*  LDB     (input) INTEGER */
/*          The leading dimension of the array B. LDB >= max(1,M,N). */

/*  JPVT    (input/output) INTEGER array, dimension (N) */
/*          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted */
/*          to the front of AP, otherwise column i is a free column. */
/*          On exit, if JPVT(i) = k, then the i-th column of AP */
/*          was the k-th column of A. */

/*  RCOND   (input) DOUBLE PRECISION */
/*          RCOND is used to determine the effective rank of A, which */
/*          is defined as the order of the largest leading triangular */
/*          submatrix R11 in the QR factorization with pivoting of A, */
/*          whose estimated condition number < 1/RCOND. */

/*  RANK    (output) INTEGER */
/*          The effective rank of A, i.e., the order of the submatrix */
/*          R11.  This is the same as the order of the submatrix T11 */
/*          in the complete orthogonal factorization of A. */

/*  WORK    (workspace/output) DOUBLE PRECISION 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. */
/*          The unblocked strategy requires that: */
/*             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), */
/*          where MN = min( M, N ). */
/*          The block algorithm requires that: */
/*             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), */
/*          where NB is an upper bound on the blocksize returned */
/*          by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR, */
/*          and DORMRZ. */

/*          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 */
/*  =============== */

/*  Based on contributions by */
/*    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA */
/*    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */
/*    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain */

/*  ===================================================================== */

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. Executable Statements .. */

    /* Parameter adjustments */
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1;
    b -= b_offset;
    --jpvt;
    --work;

    /* Function Body */
    mn = min(*m,*n);
    ismin = mn + 1;
    ismax = (mn << 1) + 1;

/*     Test the input arguments. */

    *info = 0;
    lquery = *lwork == -1;
    if (*m < 0) {
        *info = -1;
    } else if (*n < 0) {
        *info = -2;
    } else if (*nrhs < 0) {
        *info = -3;
    } else if (*lda < max(1,*m)) {
        *info = -5;
    } else /* if(complicated condition) */ {
/* Computing MAX */
        i__1 = max(1,*m);
        if (*ldb < max(i__1,*n)) {
            *info = -7;
        }
    }

/*     Figure out optimal block size */

    if (*info == 0) {
        if (mn == 0 || *nrhs == 0) {
            lwkmin = 1;
            lwkopt = 1;
        } else {
            nb1 = ilaenv_(&c__1, "DGEQRF", " ", m, n, &c_n1, &c_n1);
            nb2 = ilaenv_(&c__1, "DGERQF", " ", m, n, &c_n1, &c_n1);
            nb3 = ilaenv_(&c__1, "DORMQR", " ", m, n, nrhs, &c_n1);
            nb4 = ilaenv_(&c__1, "DORMRQ", " ", m, n, nrhs, &c_n1);
/* Computing MAX */
            i__1 = max(nb1,nb2), i__1 = max(i__1,nb3);
            nb = max(i__1,nb4);
/* Computing MAX */
            i__1 = mn << 1, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = mn + 
                    *nrhs;
            lwkmin = mn + max(i__1,i__2);
/* Computing MAX */
            i__1 = lwkmin, i__2 = mn + (*n << 1) + nb * (*n + 1), i__1 = max(
                    i__1,i__2), i__2 = (mn << 1) + nb * *nrhs;
            lwkopt = max(i__1,i__2);
        }
        work[1] = (doublereal) lwkopt;

        if (*lwork < lwkmin && ! lquery) {
            *info = -12;
        }
    }

    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("DGELSY", &i__1);
        return 0;
    } else if (lquery) {
        return 0;
    }

/*     Quick return if possible */

    if (mn == 0 || *nrhs == 0) {
        *rank = 0;
        return 0;
    }

/*     Get machine parameters */

    smlnum = dlamch_("S") / dlamch_("P");
    bignum = 1. / smlnum;
    dlabad_(&smlnum, &bignum);

/*     Scale A, B if max entries outside range [SMLNUM,BIGNUM] */

    anrm = dlange_("M", m, n, &a[a_offset], lda, &work[1]);
    iascl = 0;
    if (anrm > 0. && anrm < smlnum) {

/*        Scale matrix norm up to SMLNUM */

        dlascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda, 
                info);
        iascl = 1;
    } else if (anrm > bignum) {

/*        Scale matrix norm down to BIGNUM */

        dlascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda, 
                info);
        iascl = 2;
    } else if (anrm == 0.) {

/*        Matrix all zero. Return zero solution. */

        i__1 = max(*m,*n);
        dlaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
        *rank = 0;
        goto L70;
    }

    bnrm = dlange_("M", m, nrhs, &b[b_offset], ldb, &work[1]);
    ibscl = 0;
    if (bnrm > 0. && bnrm < smlnum) {

/*        Scale matrix norm up to SMLNUM */

        dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, nrhs, &b[b_offset], ldb, 
                 info);
        ibscl = 1;
    } else if (bnrm > bignum) {

/*        Scale matrix norm down to BIGNUM */

        dlascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb, 
                 info);
        ibscl = 2;
    }

/*     Compute QR factorization with column pivoting of A: */
/*        A * P = Q * R */

    i__1 = *lwork - mn;
    dgeqp3_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], &i__1, 
             info);
    wsize = mn + work[mn + 1];

/*     workspace: MN+2*N+NB*(N+1). */
/*     Details of Householder rotations stored in WORK(1:MN). */

/*     Determine RANK using incremental condition estimation */

    work[ismin] = 1.;
    work[ismax] = 1.;
    smax = (d__1 = a[a_dim1 + 1], abs(d__1));
    smin = smax;
    if ((d__1 = a[a_dim1 + 1], abs(d__1)) == 0.) {
        *rank = 0;
        i__1 = max(*m,*n);
        dlaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
        goto L70;
    } else {
        *rank = 1;
    }

L10:
    if (*rank < mn) {
        i__ = *rank + 1;
        dlaic1_(&c__2, rank, &work[ismin], &smin, &a[i__ * a_dim1 + 1], &a[
                i__ + i__ * a_dim1], &sminpr, &s1, &c1);
        dlaic1_(&c__1, rank, &work[ismax], &smax, &a[i__ * a_dim1 + 1], &a[
                i__ + i__ * a_dim1], &smaxpr, &s2, &c2);

        if (smaxpr * *rcond <= sminpr) {
            i__1 = *rank;
            for (i__ = 1; i__ <= i__1; ++i__) {
                work[ismin + i__ - 1] = s1 * work[ismin + i__ - 1];
                work[ismax + i__ - 1] = s2 * work[ismax + i__ - 1];
/* L20: */
            }
            work[ismin + *rank] = c1;
            work[ismax + *rank] = c2;
            smin = sminpr;
            smax = smaxpr;
            ++(*rank);
            goto L10;
        }
    }

/*     workspace: 3*MN. */

/*     Logically partition R = [ R11 R12 ] */
/*                             [  0  R22 ] */
/*     where R11 = R(1:RANK,1:RANK) */

/*     [R11,R12] = [ T11, 0 ] * Y */

    if (*rank < *n) {
        i__1 = *lwork - (mn << 1);
        dtzrzf_(rank, n, &a[a_offset], lda, &work[mn + 1], &work[(mn << 1) + 
                1], &i__1, info);
    }

/*     workspace: 2*MN. */
/*     Details of Householder rotations stored in WORK(MN+1:2*MN) */

/*     B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) */

    i__1 = *lwork - (mn << 1);
    dormqr_("Left", "Transpose", m, nrhs, &mn, &a[a_offset], lda, &work[1], &
            b[b_offset], ldb, &work[(mn << 1) + 1], &i__1, info);
/* Computing MAX */
    d__1 = wsize, d__2 = (mn << 1) + work[(mn << 1) + 1];
    wsize = max(d__1,d__2);

/*     workspace: 2*MN+NB*NRHS. */

/*     B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) */

    dtrsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b54, &
            a[a_offset], lda, &b[b_offset], ldb);

    i__1 = *nrhs;
    for (j = 1; j <= i__1; ++j) {
        i__2 = *n;
        for (i__ = *rank + 1; i__ <= i__2; ++i__) {
            b[i__ + j * b_dim1] = 0.;
/* L30: */
        }
/* L40: */
    }

/*     B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) */

    if (*rank < *n) {
        i__1 = *n - *rank;
        i__2 = *lwork - (mn << 1);
        dormrz_("Left", "Transpose", n, nrhs, rank, &i__1, &a[a_offset], lda, 
                &work[mn + 1], &b[b_offset], ldb, &work[(mn << 1) + 1], &i__2, 
                 info);
    }

/*     workspace: 2*MN+NRHS. */

/*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS) */

    i__1 = *nrhs;
    for (j = 1; j <= i__1; ++j) {
        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
            work[jpvt[i__]] = b[i__ + j * b_dim1];
/* L50: */
        }
        dcopy_(n, &work[1], &c__1, &b[j * b_dim1 + 1], &c__1);
/* L60: */
    }

/*     workspace: N. */

/*     Undo scaling */

    if (iascl == 1) {
        dlascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb, 
                 info);
        dlascl_("U", &c__0, &c__0, &smlnum, &anrm, rank, rank, &a[a_offset], 
                lda, info);
    } else if (iascl == 2) {
        dlascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb, 
                 info);
        dlascl_("U", &c__0, &c__0, &bignum, &anrm, rank, rank, &a[a_offset], 
                lda, info);
    }
    if (ibscl == 1) {
        dlascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb, 
                 info);
    } else if (ibscl == 2) {
        dlascl_("G", &c__0, &c__0, &bignum, &bnrm, n, nrhs, &b[b_offset], ldb, 
                 info);
    }

L70:
    work[1] = (doublereal) lwkopt;

    return 0;

/*     End of DGELSY */

} /* dgelsy_ */