436 lines
13 KiB
Fortran
436 lines
13 KiB
Fortran
*> \brief <b> SGELSX solves overdetermined or underdetermined systems for GE matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SGELSX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgelsx.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgelsx.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelsx.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
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* WORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
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* REAL RCOND
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* ..
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* .. Array Arguments ..
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* INTEGER JPVT( * )
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* REAL A( LDA, * ), B( LDB, * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> This routine is deprecated and has been replaced by routine SGELSY.
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*>
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*> SGELSX 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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*>
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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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*>
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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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*>
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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**T [ inv(T11)*Q1**T*B ]
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*> [ 0 ]
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*> where Q1 consists of the first RANK columns of Q.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A. M >= 0.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is 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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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is REAL 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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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is REAL 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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*> If m >= n and RANK = n, the residual sum-of-squares for
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*> the solution in the i-th column is given by the sum of
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*> squares of elements N+1:M in that column.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,M,N).
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*> \endverbatim
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*>
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*> \param[in,out] JPVT
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*> \verbatim
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*> JPVT is INTEGER array, dimension (N)
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*> On entry, if JPVT(i) .ne. 0, the i-th column of A is an
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*> initial column, otherwise it is a free column. Before
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*> the QR factorization of A, all initial columns are
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*> permuted to the leading positions; only the remaining
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*> free columns are moved as a result of column pivoting
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*> during the factorization.
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*> On exit, if JPVT(i) = k, then the i-th column of A*P
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*> was the k-th column of A.
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*> \endverbatim
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*>
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*> \param[in] RCOND
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*> \verbatim
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*> RCOND is REAL
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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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*> \endverbatim
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*>
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*> \param[out] RANK
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*> \verbatim
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*> RANK is 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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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension
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*> (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is 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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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date December 2016
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*
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*> \ingroup realGEsolve
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*
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* =====================================================================
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SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
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$ WORK, INFO )
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*
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* -- LAPACK driver routine (version 3.7.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* December 2016
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
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REAL RCOND
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* ..
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* .. Array Arguments ..
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INTEGER JPVT( * )
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REAL A( LDA, * ), B( LDB, * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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INTEGER IMAX, IMIN
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PARAMETER ( IMAX = 1, IMIN = 2 )
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REAL ZERO, ONE, DONE, NTDONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, DONE = ZERO,
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$ NTDONE = ONE )
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* ..
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* .. Local Scalars ..
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INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
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REAL ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
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$ SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
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* ..
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* .. External Functions ..
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REAL SLAMCH, SLANGE
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EXTERNAL SLAMCH, SLANGE
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEQPF, SLABAD, SLAIC1, SLASCL, SLASET, SLATZM,
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$ SORM2R, STRSM, STZRQF, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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MN = MIN( M, N )
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ISMIN = MN + 1
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ISMAX = 2*MN + 1
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*
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* Test the input arguments.
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*
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INFO = 0
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IF( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
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INFO = -7
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGELSX', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( MIN( M, N, NRHS ).EQ.0 ) THEN
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RANK = 0
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RETURN
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END IF
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*
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* Get machine parameters
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*
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SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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*
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* Scale A, B if max elements outside range [SMLNUM,BIGNUM]
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*
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ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
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IASCL = 0
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM
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*
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CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
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IASCL = 1
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM
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*
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CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
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IASCL = 2
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ELSE IF( ANRM.EQ.ZERO ) THEN
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*
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* Matrix all zero. Return zero solution.
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*
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CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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RANK = 0
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GO TO 100
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END IF
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*
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BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
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IBSCL = 0
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IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM
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*
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CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
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IBSCL = 1
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ELSE IF( BNRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM
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*
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CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
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IBSCL = 2
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END IF
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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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*
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CALL SGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO )
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*
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* workspace 3*N. Details of Householder rotations stored
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* in WORK(1:MN).
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*
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* Determine RANK using incremental condition estimation
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*
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WORK( ISMIN ) = ONE
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WORK( ISMAX ) = ONE
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SMAX = ABS( A( 1, 1 ) )
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SMIN = SMAX
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IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
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RANK = 0
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CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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GO TO 100
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ELSE
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RANK = 1
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END IF
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*
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10 CONTINUE
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IF( RANK.LT.MN ) THEN
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I = RANK + 1
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CALL SLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
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$ A( I, I ), SMINPR, S1, C1 )
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CALL SLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
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$ A( I, I ), SMAXPR, S2, C2 )
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*
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IF( SMAXPR*RCOND.LE.SMINPR ) THEN
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DO 20 I = 1, RANK
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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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20 CONTINUE
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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 = RANK + 1
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GO TO 10
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END IF
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END IF
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*
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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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*
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* [R11,R12] = [ T11, 0 ] * Y
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*
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IF( RANK.LT.N )
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$ CALL STZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
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*
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* Details of Householder rotations stored in WORK(MN+1:2*MN)
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*
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* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
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*
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CALL SORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
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$ B, LDB, WORK( 2*MN+1 ), INFO )
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*
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* workspace NRHS
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*
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* B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)
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*
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CALL STRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,
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$ NRHS, ONE, A, LDA, B, LDB )
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*
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DO 40 I = RANK + 1, N
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DO 30 J = 1, NRHS
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B( I, J ) = ZERO
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30 CONTINUE
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40 CONTINUE
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*
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* B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
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*
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IF( RANK.LT.N ) THEN
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DO 50 I = 1, RANK
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CALL SLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
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$ WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
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$ WORK( 2*MN+1 ) )
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50 CONTINUE
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END IF
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*
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* workspace NRHS
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*
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* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
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*
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DO 90 J = 1, NRHS
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DO 60 I = 1, N
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WORK( 2*MN+I ) = NTDONE
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60 CONTINUE
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DO 80 I = 1, N
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IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
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IF( JPVT( I ).NE.I ) THEN
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K = I
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T1 = B( K, J )
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T2 = B( JPVT( K ), J )
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70 CONTINUE
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B( JPVT( K ), J ) = T1
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WORK( 2*MN+K ) = DONE
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T1 = T2
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K = JPVT( K )
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T2 = B( JPVT( K ), J )
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IF( JPVT( K ).NE.I )
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$ GO TO 70
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B( I, J ) = T1
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WORK( 2*MN+K ) = DONE
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END IF
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END IF
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80 CONTINUE
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90 CONTINUE
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*
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* Undo scaling
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*
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IF( IASCL.EQ.1 ) THEN
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CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
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CALL SLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
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$ INFO )
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ELSE IF( IASCL.EQ.2 ) THEN
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CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
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CALL SLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
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$ INFO )
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END IF
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IF( IBSCL.EQ.1 ) THEN
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CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
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ELSE IF( IBSCL.EQ.2 ) THEN
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CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
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END IF
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*
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100 CONTINUE
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*
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RETURN
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*
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* End of SGELSX
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*
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END
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