--- rpl/lapack/lapack/dgelsx.f 2011/07/22 07:38:04 1.8
+++ rpl/lapack/lapack/dgelsx.f 2023/08/07 08:38:48 1.18
@@ -1,10 +1,184 @@
+*> \brief DGELSX solves overdetermined or underdetermined systems for GE matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGELSX + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
+* WORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* INTEGER JPVT( * )
+* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> This routine is deprecated and has been replaced by routine DGELSY.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**T [ inv(T11)*Q1**T*B ]
+*> [ 0 ]
+*> where Q1 consists of the first RANK columns of Q.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the matrix A. M >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the matrix A. N >= 0.
+*> \endverbatim
+*>
+*> \param[in] NRHS
+*> \verbatim
+*> NRHS is INTEGER
+*> The number of right hand sides, i.e., the number of
+*> columns of matrices B and X. NRHS >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is DOUBLE PRECISION array, dimension (LDA,N)
+*> On entry, the M-by-N matrix A.
+*> On exit, A has been overwritten by details of its
+*> complete orthogonal factorization.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[in,out] B
+*> \verbatim
+*> B is DOUBLE PRECISION array, dimension (LDB,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.
+*> \endverbatim
+*>
+*> \param[in] LDB
+*> \verbatim
+*> LDB is INTEGER
+*> The leading dimension of the array B. LDB >= max(1,M,N).
+*> \endverbatim
+*>
+*> \param[in,out] JPVT
+*> \verbatim
+*> JPVT is 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.
+*> \endverbatim
+*>
+*> \param[in] RCOND
+*> \verbatim
+*> RCOND is 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.
+*> \endverbatim
+*>
+*> \param[out] RANK
+*> \verbatim
+*> RANK is 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.
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension
+*> (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit
+*> < 0: if INFO = -i, the i-th argument had an illegal value
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup doubleGEsolve
+*
+* =====================================================================
SUBROUTINE DGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$ WORK, INFO )
*
-* -- LAPACK driver routine (version 3.3.1) --
+* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* -- April 2011 --
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
@@ -15,98 +189,6 @@
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
-* 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**T [ inv(T11)*Q1**T*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 ..