--- rpl/lapack/lapack/dgelsy.f 2010/12/21 13:53:25 1.7
+++ rpl/lapack/lapack/dgelsy.f 2017/06/17 11:06:16 1.16
@@ -1,10 +1,213 @@
+*> \brief DGELSY solves overdetermined or underdetermined systems for GE matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGELSY + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
+* WORK, LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* INTEGER JPVT( * )
+* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> 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**T [ inv(T11)*Q1**T*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.
+*> \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.
+*> \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 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.
+*> \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(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is 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.
+*> \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.
+*
+*> \date December 2016
+*
+*> \ingroup doubleGEsolve
+*
+*> \par Contributors:
+* ==================
+*>
+*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n
+*> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
+*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
+*>
+* =====================================================================
SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$ WORK, LWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
+* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
@@ -15,122 +218,6 @@
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
-* 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 ..
@@ -325,7 +412,7 @@
* 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)
+* B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
$ B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
@@ -344,7 +431,7 @@
30 CONTINUE
40 CONTINUE
*
-* B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS)
+* B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
IF( RANK.LT.N ) THEN
CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,