--- rpl/lapack/lapack/zgelsx.f 2011/07/22 07:38:14 1.8 +++ rpl/lapack/lapack/zgelsx.f 2011/11/21 20:43:09 1.9 @@ -1,10 +1,193 @@ +*> \brief ZGELSX solves overdetermined or underdetermined systems for GE matrices +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZGELSX + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, +* WORK, RWORK, INFO ) +* +* .. Scalar Arguments .. +* INTEGER INFO, LDA, LDB, M, N, NRHS, RANK +* DOUBLE PRECISION RCOND +* .. +* .. Array Arguments .. +* INTEGER JPVT( * ) +* DOUBLE PRECISION RWORK( * ) +* COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> This routine is deprecated and has been replaced by routine ZGELSY. +*> +*> ZGELSX computes the minimum-norm solution to a complex 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 unitary 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**H [ inv(T11)*Q1**H*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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension +*> (min(M,N) + max( N, 2*min(M,N)+NRHS )), +*> \endverbatim +*> +*> \param[out] RWORK +*> \verbatim +*> RWORK is DOUBLE PRECISION array, dimension (2*N) +*> \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 November 2011 +* +*> \ingroup complex16GEsolve +* +* ===================================================================== SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, $ WORK, RWORK, INFO ) * -* -- LAPACK driver routine (version 3.3.1) -- +* -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* -- April 2011 -- +* November 2011 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, M, N, NRHS, RANK @@ -16,100 +199,6 @@ COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) * .. * -* Purpose -* ======= -* -* This routine is deprecated and has been replaced by routine ZGELSY. -* -* ZGELSX computes the minimum-norm solution to a complex 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 unitary 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**H [ inv(T11)*Q1**H*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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 array, dimension -* (min(M,N) + max( N, 2*min(M,N)+NRHS )), -* -* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) -* -* INFO (output) INTEGER -* = 0: successful exit -* < 0: if INFO = -i, the i-th argument had an illegal value -* * ===================================================================== * * .. Parameters ..