--- rpl/lapack/lapack/dgelss.f 2010/01/26 15:22:46 1.1
+++ rpl/lapack/lapack/dgelss.f 2023/08/07 08:38:48 1.18
@@ -1,10 +1,178 @@
+*> \brief DGELSS solves overdetermined or underdetermined systems for GE matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGELSS + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
+* WORK, LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
+* DOUBLE PRECISION RCOND
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGELSS computes the minimum norm solution to a real linear least
+*> squares problem:
+*>
+*> Minimize 2-norm(| b - A*x |).
+*>
+*> using the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those
+*> singular values which are less than RCOND times the largest singular
+*> value.
+*> \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 the 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, the first min(m,n) rows of A are overwritten with
+*> its right singular vectors, stored rowwise.
+*> \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, B is overwritten by 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,max(M,N)).
+*> \endverbatim
+*>
+*> \param[out] S
+*> \verbatim
+*> S is DOUBLE PRECISION array, dimension (min(M,N))
+*> The singular values of A in decreasing order.
+*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
+*> \endverbatim
+*>
+*> \param[in] RCOND
+*> \verbatim
+*> RCOND is DOUBLE PRECISION
+*> RCOND is used to determine the effective rank of A.
+*> Singular values S(i) <= RCOND*S(1) are treated as zero.
+*> If RCOND < 0, machine precision is used instead.
+*> \endverbatim
+*>
+*> \param[out] RANK
+*> \verbatim
+*> RANK is INTEGER
+*> The effective rank of A, i.e., the number of singular values
+*> which are greater than RCOND*S(1).
+*> \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. LWORK >= 1, and also:
+*> LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
+*> For good performance, LWORK should generally be larger.
+*>
+*> 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.
+*> > 0: the algorithm for computing the SVD failed to converge;
+*> if INFO = i, i off-diagonal elements of an intermediate
+*> bidiagonal form did not converge to zero.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup doubleGEsolve
+*
+* =====================================================================
SUBROUTINE DGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
$ WORK, LWORK, INFO )
*
-* -- LAPACK driver routine (version 3.2) --
+* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
@@ -14,90 +182,6 @@
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* DGELSS computes the minimum norm solution to a real linear least
-* squares problem:
-*
-* Minimize 2-norm(| b - A*x |).
-*
-* using the singular value decomposition (SVD) 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 effective rank of A is determined by treating as zero those
-* singular values which are less than RCOND times the largest singular
-* value.
-*
-* 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 the 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, the first min(m,n) rows of A are overwritten with
-* its right singular vectors, stored rowwise.
-*
-* 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, B is overwritten by 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,max(M,N)).
-*
-* S (output) DOUBLE PRECISION array, dimension (min(M,N))
-* The singular values of A in decreasing order.
-* The condition number of A in the 2-norm = S(1)/S(min(m,n)).
-*
-* RCOND (input) DOUBLE PRECISION
-* RCOND is used to determine the effective rank of A.
-* Singular values S(i) <= RCOND*S(1) are treated as zero.
-* If RCOND < 0, machine precision is used instead.
-*
-* RANK (output) INTEGER
-* The effective rank of A, i.e., the number of singular values
-* which are greater than RCOND*S(1).
-*
-* 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. LWORK >= 1, and also:
-* LWORK >= 3*min(M,N) + max( 2*min(M,N), max(M,N), NRHS )
-* For good performance, LWORK should generally be larger.
-*
-* 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.
-* > 0: the algorithm for computing the SVD failed to converge;
-* if INFO = i, i off-diagonal elements of an intermediate
-* bidiagonal form did not converge to zero.
-*
* =====================================================================
*
* .. Parameters ..
@@ -109,10 +193,13 @@
INTEGER BDSPAC, BL, CHUNK, I, IASCL, IBSCL, IE, IL,
$ ITAU, ITAUP, ITAUQ, IWORK, LDWORK, MAXMN,
$ MAXWRK, MINMN, MINWRK, MM, MNTHR
+ INTEGER LWORK_DGEQRF, LWORK_DORMQR, LWORK_DGEBRD,
+ $ LWORK_DORMBR, LWORK_DORGBR, LWORK_DORMLQ,
+ $ LWORK_DGELQF
DOUBLE PRECISION ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, THR
* ..
* .. Local Arrays ..
- DOUBLE PRECISION VDUM( 1 )
+ DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DCOPY, DGEBRD, DGELQF, DGEMM, DGEMV,
@@ -165,11 +252,16 @@
* Path 1a - overdetermined, with many more rows than
* columns
*
+* Compute space needed for DGEQRF
+ CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, INFO )
+ LWORK_DGEQRF = INT( DUM(1) )
+* Compute space needed for DORMQR
+ CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, DUM(1), B,
+ $ LDB, DUM(1), -1, INFO )
+ LWORK_DORMQR = INT( DUM(1) )
MM = N
- MAXWRK = MAX( MAXWRK, N + N*ILAENV( 1, 'DGEQRF', ' ', M,
- $ N, -1, -1 ) )
- MAXWRK = MAX( MAXWRK, N + NRHS*ILAENV( 1, 'DORMQR', 'LT',
- $ M, NRHS, N, -1 ) )
+ MAXWRK = MAX( MAXWRK, N + LWORK_DGEQRF )
+ MAXWRK = MAX( MAXWRK, N + LWORK_DORMQR )
END IF
IF( M.GE.N ) THEN
*
@@ -178,12 +270,22 @@
* Compute workspace needed for DBDSQR
*
BDSPAC = MAX( 1, 5*N )
- MAXWRK = MAX( MAXWRK, 3*N + ( MM + N )*ILAENV( 1,
- $ 'DGEBRD', ' ', MM, N, -1, -1 ) )
- MAXWRK = MAX( MAXWRK, 3*N + NRHS*ILAENV( 1, 'DORMBR',
- $ 'QLT', MM, NRHS, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 3*N + ( N - 1 )*ILAENV( 1,
- $ 'DORGBR', 'P', N, N, N, -1 ) )
+* Compute space needed for DGEBRD
+ CALL DGEBRD( MM, N, A, LDA, S, DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, INFO )
+ LWORK_DGEBRD = INT( DUM(1) )
+* Compute space needed for DORMBR
+ CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, DUM(1),
+ $ B, LDB, DUM(1), -1, INFO )
+ LWORK_DORMBR = INT( DUM(1) )
+* Compute space needed for DORGBR
+ CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
+ $ DUM(1), -1, INFO )
+ LWORK_DORGBR = INT( DUM(1) )
+* Compute total workspace needed
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORMBR )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
MINWRK = MAX( 3*N + MM, 3*N + NRHS, BDSPAC )
@@ -200,33 +302,57 @@
* Path 2a - underdetermined, with many more columns
* than rows
*
- MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
- $ 'DGEBRD', ' ', M, M, -1, -1 ) )
- MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
- $ 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
- MAXWRK = MAX( MAXWRK, M*M + 4*M +
- $ ( M - 1 )*ILAENV( 1, 'DORGBR', 'P', M,
- $ M, M, -1 ) )
+* Compute space needed for DGELQF
+ CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1),
+ $ -1, INFO )
+ LWORK_DGELQF = INT( DUM(1) )
+* Compute space needed for DGEBRD
+ CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, INFO )
+ LWORK_DGEBRD = INT( DUM(1) )
+* Compute space needed for DORMBR
+ CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA,
+ $ DUM(1), B, LDB, DUM(1), -1, INFO )
+ LWORK_DORMBR = INT( DUM(1) )
+* Compute space needed for DORGBR
+ CALL DORGBR( 'P', M, M, M, A, LDA, DUM(1),
+ $ DUM(1), -1, INFO )
+ LWORK_DORGBR = INT( DUM(1) )
+* Compute space needed for DORMLQ
+ CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, DUM(1),
+ $ B, LDB, DUM(1), -1, INFO )
+ LWORK_DORMLQ = INT( DUM(1) )
+* Compute total workspace needed
+ MAXWRK = M + LWORK_DGELQF
+ MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DGEBRD )
+ MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORMBR )
+ MAXWRK = MAX( MAXWRK, M*M + 4*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, M*M + M + BDSPAC )
IF( NRHS.GT.1 ) THEN
MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
ELSE
MAXWRK = MAX( MAXWRK, M*M + 2*M )
END IF
- MAXWRK = MAX( MAXWRK, M + NRHS*ILAENV( 1, 'DORMLQ',
- $ 'LT', N, NRHS, M, -1 ) )
+ MAXWRK = MAX( MAXWRK, M + LWORK_DORMLQ )
ELSE
*
* Path 2 - underdetermined
*
- MAXWRK = 3*M + ( N + M )*ILAENV( 1, 'DGEBRD', ' ', M,
- $ N, -1, -1 )
- MAXWRK = MAX( MAXWRK, 3*M + NRHS*ILAENV( 1, 'DORMBR',
- $ 'QLT', M, NRHS, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 3*M + M*ILAENV( 1, 'DORGBR',
- $ 'P', M, N, M, -1 ) )
+* Compute space needed for DGEBRD
+ CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, INFO )
+ LWORK_DGEBRD = INT( DUM(1) )
+* Compute space needed for DORMBR
+ CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, A, LDA,
+ $ DUM(1), B, LDB, DUM(1), -1, INFO )
+ LWORK_DORMBR = INT( DUM(1) )
+* Compute space needed for DORGBR
+ CALL DORGBR( 'P', M, N, M, A, LDA, DUM(1),
+ $ DUM(1), -1, INFO )
+ LWORK_DORGBR = INT( DUM(1) )
+ MAXWRK = 3*M + LWORK_DGEBRD
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORMBR )
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR )
MAXWRK = MAX( MAXWRK, BDSPAC )
MAXWRK = MAX( MAXWRK, N*NRHS )
END IF
@@ -282,7 +408,7 @@
* Matrix all zero. Return zero solution.
*
CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
- CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
+ CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, MINMN )
RANK = 0
GO TO 70
END IF
@@ -368,7 +494,7 @@
* compute right singular vectors in A
* (Workspace: need BDSPAC)
*
- CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM,
+ CALL DBDSQR( 'U', N, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70
@@ -551,7 +677,7 @@
* multiplying B by transpose of left singular vectors
* (Workspace: need BDSPAC)
*
- CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, VDUM,
+ CALL DBDSQR( 'L', M, N, 0, NRHS, S, WORK( IE ), A, LDA, DUM,
$ 1, B, LDB, WORK( IWORK ), INFO )
IF( INFO.NE.0 )
$ GO TO 70