475 lines
15 KiB
C
475 lines
15 KiB
C
#include "blaswrap.h"
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#ifdef _cpluscplus
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extern "C" {
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#endif
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#include "f2c.h"
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/* Subroutine */ int dgelsy_(integer *m, integer *n, integer *nrhs,
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doublereal *a, integer *lda, doublereal *b, integer *ldb, integer *
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jpvt, doublereal *rcond, integer *rank, doublereal *work, integer *
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lwork, integer *info)
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{
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/* -- LAPACK driver routine (version 3.0) --
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Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
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Courant Institute, Argonne National Lab, and Rice University
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June 30, 1999
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Purpose
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=======
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DGELSY computes the minimum-norm solution to a real linear least
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squares problem:
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minimize || A * X - B ||
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using a complete orthogonal factorization of A. A is an M-by-N
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matrix which may be rank-deficient.
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Several right hand side vectors b and solution vectors x can be
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handled in a single call; they are stored as the columns of the
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M-by-NRHS right hand side matrix B and the N-by-NRHS solution
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matrix X.
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The routine first computes a QR factorization with column pivoting:
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A * P = Q * [ R11 R12 ]
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[ 0 R22 ]
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with R11 defined as the largest leading submatrix whose estimated
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condition number is less than 1/RCOND. The order of R11, RANK,
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is the effective rank of A.
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Then, R22 is considered to be negligible, and R12 is annihilated
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by orthogonal transformations from the right, arriving at the
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complete orthogonal factorization:
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A * P = Q * [ T11 0 ] * Z
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[ 0 0 ]
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The minimum-norm solution is then
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X = P * Z' [ inv(T11)*Q1'*B ]
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[ 0 ]
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where Q1 consists of the first RANK columns of Q.
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This routine is basically identical to the original xGELSX except
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three differences:
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o The call to the subroutine xGEQPF has been substituted by the
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the call to the subroutine xGEQP3. This subroutine is a Blas-3
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version of the QR factorization with column pivoting.
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o Matrix B (the right hand side) is updated with Blas-3.
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o The permutation of matrix B (the right hand side) is faster and
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more simple.
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Arguments
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=========
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M (input) INTEGER
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The number of rows of the matrix A. M >= 0.
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N (input) INTEGER
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The number of columns of the matrix A. N >= 0.
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NRHS (input) INTEGER
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The number of right hand sides, i.e., the number of
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columns of matrices B and X. NRHS >= 0.
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A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
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On entry, the M-by-N matrix A.
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On exit, A has been overwritten by details of its
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complete orthogonal factorization.
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LDA (input) INTEGER
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The leading dimension of the array A. LDA >= max(1,M).
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B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
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On entry, the M-by-NRHS right hand side matrix B.
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On exit, the N-by-NRHS solution matrix X.
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LDB (input) INTEGER
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The leading dimension of the array B. LDB >= max(1,M,N).
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JPVT (input/output) INTEGER array, dimension (N)
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On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
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to the front of AP, otherwise column i is a free column.
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On exit, if JPVT(i) = k, then the i-th column of AP
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was the k-th column of A.
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RCOND (input) DOUBLE PRECISION
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RCOND is used to determine the effective rank of A, which
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is defined as the order of the largest leading triangular
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submatrix R11 in the QR factorization with pivoting of A,
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whose estimated condition number < 1/RCOND.
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RANK (output) INTEGER
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The effective rank of A, i.e., the order of the submatrix
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R11. This is the same as the order of the submatrix T11
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in the complete orthogonal factorization of A.
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WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
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On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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LWORK (input) INTEGER
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The dimension of the array WORK.
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The unblocked strategy requires that:
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LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
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where MN = min( M, N ).
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The block algorithm requires that:
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LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
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where NB is an upper bound on the blocksize returned
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by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
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and DORMRZ.
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If LWORK = -1, then a workspace query is assumed; the routine
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only calculates the optimal size of the WORK array, returns
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this value as the first entry of the WORK array, and no error
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message related to LWORK is issued by XERBLA.
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INFO (output) INTEGER
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= 0: successful exit
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< 0: If INFO = -i, the i-th argument had an illegal value.
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Further Details
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===============
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Based on contributions by
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A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
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G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
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=====================================================================
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Parameter adjustments */
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/* Table of constant values */
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static integer c__1 = 1;
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static integer c_n1 = -1;
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static integer c__0 = 0;
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static doublereal c_b31 = 0.;
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static integer c__2 = 2;
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static doublereal c_b54 = 1.;
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/* System generated locals */
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integer a_dim1, a_offset, b_dim1, b_offset, i__1, i__2;
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doublereal d__1, d__2;
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/* Local variables */
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static doublereal anrm, bnrm, smin, smax;
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static integer i__, j, iascl, ibscl;
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extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *,
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doublereal *, integer *);
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static integer ismin, ismax;
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static doublereal c1, c2;
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extern /* Subroutine */ int dtrsm_(char *, char *, char *, char *,
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integer *, integer *, doublereal *, doublereal *, integer *,
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doublereal *, integer *), dlaic1_(
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integer *, integer *, doublereal *, doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *);
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static doublereal wsize, s1, s2;
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extern /* Subroutine */ int dgeqp3_(integer *, integer *, doublereal *,
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integer *, integer *, doublereal *, doublereal *, integer *,
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integer *), dlabad_(doublereal *, doublereal *);
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static integer nb;
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extern doublereal dlamch_(char *), dlange_(char *, integer *,
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integer *, doublereal *, integer *, doublereal *);
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static integer mn;
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extern /* Subroutine */ int dlascl_(char *, integer *, integer *,
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doublereal *, doublereal *, integer *, integer *, doublereal *,
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integer *, integer *), dlaset_(char *, integer *, integer
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*, doublereal *, doublereal *, doublereal *, integer *),
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xerbla_(char *, integer *);
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extern integer ilaenv_(integer *, char *, char *, integer *, integer *,
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integer *, integer *, ftnlen, ftnlen);
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static doublereal bignum;
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static integer nb1, nb2, nb3, nb4;
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extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *,
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integer *, doublereal *, integer *, doublereal *, doublereal *,
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integer *, doublereal *, integer *, integer *);
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static doublereal sminpr, smaxpr, smlnum;
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extern /* Subroutine */ int dormrz_(char *, char *, integer *, integer *,
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integer *, integer *, doublereal *, integer *, doublereal *,
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doublereal *, integer *, doublereal *, integer *, integer *);
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static integer lwkopt;
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static logical lquery;
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extern /* Subroutine */ int dtzrzf_(integer *, integer *, doublereal *,
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integer *, doublereal *, doublereal *, integer *, integer *);
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#define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1]
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#define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1]
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a_dim1 = *lda;
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a_offset = 1 + a_dim1 * 1;
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a -= a_offset;
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b_dim1 = *ldb;
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b_offset = 1 + b_dim1 * 1;
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b -= b_offset;
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--jpvt;
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--work;
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/* Function Body */
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mn = min(*m,*n);
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ismin = mn + 1;
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ismax = (mn << 1) + 1;
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/* Test the input arguments. */
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*info = 0;
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nb1 = ilaenv_(&c__1, "DGEQRF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (
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ftnlen)1);
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nb2 = ilaenv_(&c__1, "DGERQF", " ", m, n, &c_n1, &c_n1, (ftnlen)6, (
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ftnlen)1);
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nb3 = ilaenv_(&c__1, "DORMQR", " ", m, n, nrhs, &c_n1, (ftnlen)6, (ftnlen)
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1);
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nb4 = ilaenv_(&c__1, "DORMRQ", " ", m, n, nrhs, &c_n1, (ftnlen)6, (ftnlen)
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1);
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/* Computing MAX */
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i__1 = max(nb1,nb2), i__1 = max(i__1,nb3);
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nb = max(i__1,nb4);
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/* Computing MAX */
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i__1 = 1, i__2 = mn + (*n << 1) + nb * (*n + 1), i__1 = max(i__1,i__2),
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i__2 = (mn << 1) + nb * *nrhs;
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lwkopt = max(i__1,i__2);
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work[1] = (doublereal) lwkopt;
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lquery = *lwork == -1;
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if (*m < 0) {
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*info = -1;
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} else if (*n < 0) {
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*info = -2;
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} else if (*nrhs < 0) {
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*info = -3;
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} else if (*lda < max(1,*m)) {
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*info = -5;
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} else /* if(complicated condition) */ {
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/* Computing MAX */
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i__1 = max(1,*m);
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if (*ldb < max(i__1,*n)) {
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*info = -7;
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} else /* if(complicated condition) */ {
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/* Computing MAX */
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i__1 = 1, i__2 = mn + *n * 3 + 1, i__1 = max(i__1,i__2), i__2 = (
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mn << 1) + *nrhs;
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if (*lwork < max(i__1,i__2) && ! lquery) {
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*info = -12;
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}
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}
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}
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if (*info != 0) {
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i__1 = -(*info);
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xerbla_("DGELSY", &i__1);
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return 0;
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} else if (lquery) {
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return 0;
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}
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/* Quick return if possible
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Computing MIN */
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i__1 = min(*m,*n);
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if (min(i__1,*nrhs) == 0) {
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*rank = 0;
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return 0;
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}
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/* Get machine parameters */
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smlnum = dlamch_("S") / dlamch_("P");
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bignum = 1. / smlnum;
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dlabad_(&smlnum, &bignum);
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/* Scale A, B if max entries outside range [SMLNUM,BIGNUM] */
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anrm = dlange_("M", m, n, &a[a_offset], lda, &work[1]);
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iascl = 0;
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if (anrm > 0. && anrm < smlnum) {
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/* Scale matrix norm up to SMLNUM */
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dlascl_("G", &c__0, &c__0, &anrm, &smlnum, m, n, &a[a_offset], lda,
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info);
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iascl = 1;
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} else if (anrm > bignum) {
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/* Scale matrix norm down to BIGNUM */
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dlascl_("G", &c__0, &c__0, &anrm, &bignum, m, n, &a[a_offset], lda,
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info);
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iascl = 2;
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} else if (anrm == 0.) {
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/* Matrix all zero. Return zero solution. */
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i__1 = max(*m,*n);
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dlaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
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*rank = 0;
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goto L70;
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}
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bnrm = dlange_("M", m, nrhs, &b[b_offset], ldb, &work[1]);
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ibscl = 0;
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if (bnrm > 0. && bnrm < smlnum) {
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/* Scale matrix norm up to SMLNUM */
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dlascl_("G", &c__0, &c__0, &bnrm, &smlnum, m, nrhs, &b[b_offset], ldb,
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info);
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ibscl = 1;
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} else if (bnrm > bignum) {
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/* Scale matrix norm down to BIGNUM */
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dlascl_("G", &c__0, &c__0, &bnrm, &bignum, m, nrhs, &b[b_offset], ldb,
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info);
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ibscl = 2;
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}
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/* Compute QR factorization with column pivoting of A:
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A * P = Q * R */
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i__1 = *lwork - mn;
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dgeqp3_(m, n, &a[a_offset], lda, &jpvt[1], &work[1], &work[mn + 1], &i__1,
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info);
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wsize = mn + work[mn + 1];
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/* workspace: MN+2*N+NB*(N+1).
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Details of Householder rotations stored in WORK(1:MN).
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Determine RANK using incremental condition estimation */
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work[ismin] = 1.;
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work[ismax] = 1.;
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smax = (d__1 = a_ref(1, 1), abs(d__1));
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smin = smax;
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if ((d__1 = a_ref(1, 1), abs(d__1)) == 0.) {
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*rank = 0;
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i__1 = max(*m,*n);
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dlaset_("F", &i__1, nrhs, &c_b31, &c_b31, &b[b_offset], ldb);
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goto L70;
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} else {
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*rank = 1;
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}
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L10:
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if (*rank < mn) {
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i__ = *rank + 1;
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dlaic1_(&c__2, rank, &work[ismin], &smin, &a_ref(1, i__), &a_ref(i__,
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i__), &sminpr, &s1, &c1);
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dlaic1_(&c__1, rank, &work[ismax], &smax, &a_ref(1, i__), &a_ref(i__,
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i__), &smaxpr, &s2, &c2);
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if (smaxpr * *rcond <= sminpr) {
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i__1 = *rank;
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for (i__ = 1; i__ <= i__1; ++i__) {
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work[ismin + i__ - 1] = s1 * work[ismin + i__ - 1];
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work[ismax + i__ - 1] = s2 * work[ismax + i__ - 1];
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/* L20: */
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}
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work[ismin + *rank] = c1;
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work[ismax + *rank] = c2;
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smin = sminpr;
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smax = smaxpr;
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++(*rank);
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goto L10;
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}
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}
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/* workspace: 3*MN.
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Logically partition R = [ R11 R12 ]
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[ 0 R22 ]
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where R11 = R(1:RANK,1:RANK)
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[R11,R12] = [ T11, 0 ] * Y */
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if (*rank < *n) {
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i__1 = *lwork - (mn << 1);
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dtzrzf_(rank, n, &a[a_offset], lda, &work[mn + 1], &work[(mn << 1) +
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1], &i__1, info);
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}
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/* workspace: 2*MN.
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Details of Householder rotations stored in WORK(MN+1:2*MN)
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B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS) */
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i__1 = *lwork - (mn << 1);
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dormqr_("Left", "Transpose", m, nrhs, &mn, &a[a_offset], lda, &work[1], &
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b[b_offset], ldb, &work[(mn << 1) + 1], &i__1, info);
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/* Computing MAX */
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d__1 = wsize, d__2 = (mn << 1) + work[(mn << 1) + 1];
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wsize = max(d__1,d__2);
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/* workspace: 2*MN+NB*NRHS.
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B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS) */
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dtrsm_("Left", "Upper", "No transpose", "Non-unit", rank, nrhs, &c_b54, &
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a[a_offset], lda, &b[b_offset], ldb);
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i__1 = *nrhs;
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for (j = 1; j <= i__1; ++j) {
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i__2 = *n;
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for (i__ = *rank + 1; i__ <= i__2; ++i__) {
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b_ref(i__, j) = 0.;
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/* L30: */
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}
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/* L40: */
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}
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/* B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS) */
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if (*rank < *n) {
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i__1 = *n - *rank;
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i__2 = *lwork - (mn << 1);
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dormrz_("Left", "Transpose", n, nrhs, rank, &i__1, &a[a_offset], lda,
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&work[mn + 1], &b[b_offset], ldb, &work[(mn << 1) + 1], &i__2,
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info);
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}
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/* workspace: 2*MN+NRHS.
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B(1:N,1:NRHS) := P * B(1:N,1:NRHS) */
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i__1 = *nrhs;
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for (j = 1; j <= i__1; ++j) {
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i__2 = *n;
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for (i__ = 1; i__ <= i__2; ++i__) {
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work[jpvt[i__]] = b_ref(i__, j);
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/* L50: */
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}
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dcopy_(n, &work[1], &c__1, &b_ref(1, j), &c__1);
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/* L60: */
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}
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/* workspace: N.
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Undo scaling */
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if (iascl == 1) {
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dlascl_("G", &c__0, &c__0, &anrm, &smlnum, n, nrhs, &b[b_offset], ldb,
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info);
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dlascl_("U", &c__0, &c__0, &smlnum, &anrm, rank, rank, &a[a_offset],
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lda, info);
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} else if (iascl == 2) {
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dlascl_("G", &c__0, &c__0, &anrm, &bignum, n, nrhs, &b[b_offset], ldb,
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info);
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dlascl_("U", &c__0, &c__0, &bignum, &anrm, rank, rank, &a[a_offset],
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lda, info);
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}
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if (ibscl == 1) {
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dlascl_("G", &c__0, &c__0, &smlnum, &bnrm, n, nrhs, &b[b_offset], ldb,
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info);
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} else if (ibscl == 2) {
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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_ */
|
|
|
|
#undef b_ref
|
|
#undef a_ref
|
|
|
|
|
|
#ifdef _cpluscplus
|
|
}
|
|
#endif
|