dgelsx.c

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/* dgelsx.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__0 = 0;
static doublereal c_b13 = 0.;
static integer c__2 = 2;
static integer c__1 = 1;
static doublereal c_b36 = 1.;

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

    /* Local variables */
    integer i__, j, k;
    doublereal c1, c2, s1, s2, t1, t2;
    integer mn;
    doublereal anrm, bnrm, smin, smax;
    integer iascl, ibscl, 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 *), dorm2r_(
            char *, char *, integer *, integer *, integer *, doublereal *, 
            integer *, doublereal *, doublereal *, integer *, doublereal *, 
            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 *), dgeqpf_(integer *, integer *, 
            doublereal *, integer *, integer *, doublereal *, doublereal *, 
            integer *), dlaset_(char *, integer *, integer *, doublereal *, 
            doublereal *, doublereal *, integer *), xerbla_(char *, 
            integer *);
    doublereal bignum;
    extern /* Subroutine */ int dlatzm_(char *, integer *, integer *, 
            doublereal *, integer *, doublereal *, doublereal *, doublereal *, 
             integer *, doublereal *);
    doublereal sminpr, smaxpr, smlnum;
    extern /* Subroutine */ int dtzrqf_(integer *, integer *, doublereal *, 
            integer *, doublereal *, integer *);


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

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

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

/*  This routine is deprecated and has been replaced by routine DGELSY. */

/*  DGELSX 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. */

/*  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. */
/*          If m >= n and RANK = n, the residual sum-of-squares for */
/*          the solution in the i-th column is given by the sum of */
/*          squares of elements N+1:M in that column. */

/*  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 an */
/*          initial column, otherwise it is a free column.  Before */
/*          the QR factorization of A, all initial columns are */
/*          permuted to the leading positions; only the remaining */
/*          free columns are moved as a result of column pivoting */
/*          during the factorization. */
/*          On exit, if JPVT(i) = k, then the i-th column of A*P */
/*          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) DOUBLE PRECISION array, dimension */
/*                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )), */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */

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

/*     .. 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;
    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;
        }
    }

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

/*     Quick return if possible */

/* Computing MIN */
    i__1 = min(*m,*n);
    if (min(i__1,*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 elements 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_b13, &c_b13, &b[b_offset], ldb);
        *rank = 0;
        goto L100;
    }

    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 */

    dgeqpf_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], info);

/*     workspace 3*N. 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_b13, &c_b13, &b[b_offset], ldb);
        goto L100;
    } 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;
        }
    }

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

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

    if (*rank < *n) {
        dtzrqf_(rank, n, &a[a_offset], lda, &work[mn + 1], info);
    }

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

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

    dorm2r_("Left", "Transpose", m, nrhs, &mn, &a[a_offset], lda, &work[1], &
            b[b_offset], ldb, &work[(mn << 1) + 1], info);

/*     workspace NRHS */

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

    dtrsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b36, &
            a[a_offset], lda, &b[b_offset], ldb);

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

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

    if (*rank < *n) {
        i__1 = *rank;
        for (i__ = 1; i__ <= i__1; ++i__) {
            i__2 = *n - *rank + 1;
            dlatzm_("Left", &i__2, nrhs, &a[i__ + (*rank + 1) * a_dim1], lda, 
                    &work[mn + i__], &b[i__ + b_dim1], &b[*rank + 1 + b_dim1], 
                     ldb, &work[(mn << 1) + 1]);
/* L50: */
        }
    }

/*     workspace 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[(mn << 1) + i__] = 1.;
/* L60: */
        }
        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
            if (work[(mn << 1) + i__] == 1.) {
                if (jpvt[i__] != i__) {
                    k = i__;
                    t1 = b[k + j * b_dim1];
                    t2 = b[jpvt[k] + j * b_dim1];
L70:
                    b[jpvt[k] + j * b_dim1] = t1;
                    work[(mn << 1) + k] = 0.;
                    t1 = t2;
                    k = jpvt[k];
                    t2 = b[jpvt[k] + j * b_dim1];
                    if (jpvt[k] != i__) {
                        goto L70;
                    }
                    b[i__ + j * b_dim1] = t1;
                    work[(mn << 1) + k] = 0.;
                }
            }
/* L80: */
        }
/* L90: */
    }

/*     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);
    }

L100:

    return 0;

/*     End of DGELSX */

} /* dgelsx_ */