--- rpl/lapack/lapack/dgels.f 2010/08/07 13:22:13 1.5
+++ rpl/lapack/lapack/dgels.f 2023/08/07 08:38:48 1.18
@@ -1,10 +1,189 @@
+*> \brief DGELS solves overdetermined or underdetermined systems for GE matrices
+*
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download DGELS + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
+* INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER TRANS
+* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGELS solves overdetermined or underdetermined real linear systems
+*> involving an M-by-N matrix A, or its transpose, using a QR or LQ
+*> factorization of A. It is assumed that A has full rank.
+*>
+*> The following options are provided:
+*>
+*> 1. If TRANS = 'N' and m >= n: find the least squares solution of
+*> an overdetermined system, i.e., solve the least squares problem
+*> minimize || B - A*X ||.
+*>
+*> 2. If TRANS = 'N' and m < n: find the minimum norm solution of
+*> an underdetermined system A * X = B.
+*>
+*> 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
+*> an underdetermined system A**T * X = B.
+*>
+*> 4. If TRANS = 'T' and m < n: find the least squares solution of
+*> an overdetermined system, i.e., solve the least squares problem
+*> minimize || B - A**T * X ||.
+*>
+*> 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] TRANS
+*> \verbatim
+*> TRANS is CHARACTER*1
+*> = 'N': the linear system involves A;
+*> = 'T': the linear system involves A**T.
+*> \endverbatim
+*>
+*> \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,
+*> if M >= N, A is overwritten by details of its QR
+*> factorization as returned by DGEQRF;
+*> if M < N, A is overwritten by details of its LQ
+*> factorization as returned by DGELQF.
+*> \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 matrix B of right hand side vectors, stored
+*> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
+*> if TRANS = 'T'.
+*> On exit, if INFO = 0, B is overwritten by the solution
+*> vectors, stored columnwise:
+*> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
+*> squares solution vectors; the residual sum of squares for the
+*> solution in each column is given by the sum of squares of
+*> elements N+1 to M in that column;
+*> if TRANS = 'N' and m < n, rows 1 to N of B contain the
+*> minimum norm solution vectors;
+*> if TRANS = 'T' and m >= n, rows 1 to M of B contain the
+*> minimum norm solution vectors;
+*> if TRANS = 'T' and m < n, rows 1 to M of B contain the
+*> least squares solution vectors; the residual sum of squares
+*> for the solution in each column is given by the sum of
+*> squares of elements M+1 to N 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[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 >= max( 1, MN + max( MN, NRHS ) ).
+*> For optimal performance,
+*> LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
+*> where MN = min(M,N) and NB is the optimum block size.
+*>
+*> 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: if INFO = i, the i-th diagonal element of the
+*> triangular factor of A is zero, so that A does not have
+*> full rank; the least squares solution could not be
+*> computed.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \ingroup doubleGEsolve
+*
+* =====================================================================
SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, 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 ..
CHARACTER TRANS
@@ -14,107 +193,6 @@
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* DGELS solves overdetermined or underdetermined real linear systems
-* involving an M-by-N matrix A, or its transpose, using a QR or LQ
-* factorization of A. It is assumed that A has full rank.
-*
-* The following options are provided:
-*
-* 1. If TRANS = 'N' and m >= n: find the least squares solution of
-* an overdetermined system, i.e., solve the least squares problem
-* minimize || B - A*X ||.
-*
-* 2. If TRANS = 'N' and m < n: find the minimum norm solution of
-* an underdetermined system A * X = B.
-*
-* 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
-* an undetermined system A**T * X = B.
-*
-* 4. If TRANS = 'T' and m < n: find the least squares solution of
-* an overdetermined system, i.e., solve the least squares problem
-* minimize || B - A**T * X ||.
-*
-* 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.
-*
-* Arguments
-* =========
-*
-* TRANS (input) CHARACTER*1
-* = 'N': the linear system involves A;
-* = 'T': the linear system involves A**T.
-*
-* 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,
-* if M >= N, A is overwritten by details of its QR
-* factorization as returned by DGEQRF;
-* if M < N, A is overwritten by details of its LQ
-* factorization as returned by DGELQF.
-*
-* 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 matrix B of right hand side vectors, stored
-* columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
-* if TRANS = 'T'.
-* On exit, if INFO = 0, B is overwritten by the solution
-* vectors, stored columnwise:
-* if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
-* squares solution vectors; the residual sum of squares for the
-* solution in each column is given by the sum of squares of
-* elements N+1 to M in that column;
-* if TRANS = 'N' and m < n, rows 1 to N of B contain the
-* minimum norm solution vectors;
-* if TRANS = 'T' and m >= n, rows 1 to M of B contain the
-* minimum norm solution vectors;
-* if TRANS = 'T' and m < n, rows 1 to M of B contain the
-* least squares solution vectors; the residual sum of squares
-* for the solution in each column is given by the sum of
-* squares of elements M+1 to N in that column.
-*
-* LDB (input) INTEGER
-* The leading dimension of the array B. LDB >= MAX(1,M,N).
-*
-* 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 >= max( 1, MN + max( MN, NRHS ) ).
-* For optimal performance,
-* LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
-* where MN = min(M,N) and NB is the optimum block size.
-*
-* 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: if INFO = i, the i-th diagonal element of the
-* triangular factor of A is zero, so that A does not have
-* full rank; the least squares solution could not be
-* computed.
-*
* =====================================================================
*
* .. Parameters ..
@@ -277,7 +355,7 @@
*
* Least-Squares Problem min || A * X - B ||
*
-* 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, N, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
@@ -298,9 +376,9 @@
*
ELSE
*
-* Overdetermined system of equations A' * X = B
+* Underdetermined system of equations A**T * X = B
*
-* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS)
+* B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS)
*
CALL DTRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS,
$ A, LDA, B, LDB, INFO )
@@ -359,7 +437,7 @@
30 CONTINUE
40 CONTINUE
*
-* B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS)
+* B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS)
*
CALL DORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
@@ -371,7 +449,7 @@
*
ELSE
*
-* overdetermined system min || A' * X - B ||
+* overdetermined system min || A**T * X - B ||
*
* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
@@ -381,7 +459,7 @@
*
* workspace at least NRHS, optimally NRHS*NB
*
-* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS)
+* B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS)
*
CALL DTRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS,
$ A, LDA, B, LDB, INFO )