634 lines
21 KiB
Fortran
634 lines
21 KiB
Fortran
*> \brief <b> SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SGELSD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgelsd.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgelsd.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgelsd.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
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* RANK, WORK, LWORK, IWORK, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
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* REAL RCOND
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* ..
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* .. Array Arguments ..
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* INTEGER IWORK( * )
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* REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SGELSD computes the minimum-norm solution to a real linear least
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*> squares problem:
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*> minimize 2-norm(| b - A*x |)
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*> using the singular value decomposition (SVD) of A. A is an M-by-N
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*> matrix which may be rank-deficient.
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*>
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*> Several right hand side vectors b and solution vectors x can be
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*> handled in a single call; they are stored as the columns of the
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*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
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*> matrix X.
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*>
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*> The problem is solved in three steps:
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*> (1) Reduce the coefficient matrix A to bidiagonal form with
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*> Householder transformations, reducing the original problem
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*> into a "bidiagonal least squares problem" (BLS)
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*> (2) Solve the BLS using a divide and conquer approach.
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*> (3) Apply back all the Householder tranformations to solve
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*> the original least squares problem.
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*>
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*> The effective rank of A is determined by treating as zero those
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*> singular values which are less than RCOND times the largest singular
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*> value.
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*>
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*> The divide and conquer algorithm makes very mild assumptions about
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*> floating point arithmetic. It will work on machines with a guard
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*> digit in add/subtract, or on those binary machines without guard
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*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
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*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
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*> without guard digits, but we know of none.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of A. M >= 0.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of columns
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*> of the matrices B and X. NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is REAL array, dimension (LDA,N)
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*> On entry, the M-by-N matrix A.
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*> On exit, A has been destroyed.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is REAL array, dimension (LDB,NRHS)
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*> On entry, the M-by-NRHS right hand side matrix B.
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*> On exit, B is overwritten by the N-by-NRHS solution
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*> matrix X. If m >= n and RANK = n, the residual
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*> sum-of-squares for the solution in the i-th column is given
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*> by the sum of squares of elements n+1:m in that column.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,max(M,N)).
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*> \endverbatim
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*>
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*> \param[out] S
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*> \verbatim
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*> S is REAL array, dimension (min(M,N))
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*> The singular values of A in decreasing order.
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*> The condition number of A in the 2-norm = S(1)/S(min(m,n)).
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*> \endverbatim
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*>
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*> \param[in] RCOND
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*> \verbatim
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*> RCOND is REAL
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*> RCOND is used to determine the effective rank of A.
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*> Singular values S(i) <= RCOND*S(1) are treated as zero.
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*> If RCOND < 0, machine precision is used instead.
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*> \endverbatim
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*>
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*> \param[out] RANK
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*> \verbatim
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*> RANK is INTEGER
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*> The effective rank of A, i.e., the number of singular values
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*> which are greater than RCOND*S(1).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (MAX(1,LWORK))
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*> \endverbatim
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*>
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*> \param[in] LWORK
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*> \verbatim
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*> LWORK is INTEGER
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*> The dimension of the array WORK. LWORK must be at least 1.
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*> The exact minimum amount of workspace needed depends on M,
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*> N and NRHS. As long as LWORK is at least
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*> 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
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*> if M is greater than or equal to N or
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*> 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
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*> if M is less than N, the code will execute correctly.
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*> SMLSIZ is returned by ILAENV and is equal to the maximum
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*> size of the subproblems at the bottom of the computation
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*> tree (usually about 25), and
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*> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
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*> For good performance, LWORK should generally be larger.
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*>
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*> If LWORK = -1, then a workspace query is assumed; the routine
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*> only calculates the optimal size of the array WORK and the
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*> minimum size of the array IWORK, and returns these values as
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*> the first entries of the WORK and IWORK arrays, and no error
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*> message related to LWORK is issued by XERBLA.
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*> \endverbatim
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*>
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*> \param[out] IWORK
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*> \verbatim
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*> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
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*> LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
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*> where MINMN = MIN( M,N ).
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*> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> > 0: the algorithm for computing the SVD failed to converge;
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*> if INFO = i, i off-diagonal elements of an intermediate
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*> bidiagonal form did not converge to zero.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \date November 2011
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*
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*> \ingroup realGEsolve
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*
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*> \par Contributors:
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* ==================
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*>
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*> Ming Gu and Ren-Cang Li, Computer Science Division, University of
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*> California at Berkeley, USA \n
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*> Osni Marques, LBNL/NERSC, USA \n
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*
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* =====================================================================
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SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
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$ RANK, WORK, LWORK, IWORK, INFO )
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*
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* -- LAPACK driver routine (version 3.4.0) --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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* November 2011
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
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REAL RCOND
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* ..
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* .. Array Arguments ..
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INTEGER IWORK( * )
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REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE, TWO
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY
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INTEGER IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
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$ LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,
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$ MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
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REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
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* ..
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* .. External Subroutines ..
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EXTERNAL SGEBRD, SGELQF, SGEQRF, SLABAD, SLACPY, SLALSD,
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$ SLASCL, SLASET, SORMBR, SORMLQ, SORMQR, XERBLA
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* ..
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* .. External Functions ..
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INTEGER ILAENV
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REAL SLAMCH, SLANGE
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EXTERNAL SLAMCH, SLANGE, ILAENV
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC INT, LOG, MAX, MIN, REAL
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* ..
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* .. Executable Statements ..
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*
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* Test the input arguments.
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*
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INFO = 0
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MINMN = MIN( M, N )
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MAXMN = MAX( M, N )
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LQUERY = ( LWORK.EQ.-1 )
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IF( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
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INFO = -7
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END IF
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*
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* Compute workspace.
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* (Note: Comments in the code beginning "Workspace:" describe the
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* minimal amount of workspace needed at that point in the code,
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* as well as the preferred amount for good performance.
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* NB refers to the optimal block size for the immediately
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* following subroutine, as returned by ILAENV.)
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*
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IF( INFO.EQ.0 ) THEN
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MINWRK = 1
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MAXWRK = 1
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LIWORK = 1
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IF( MINMN.GT.0 ) THEN
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SMLSIZ = ILAENV( 9, 'SGELSD', ' ', 0, 0, 0, 0 )
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MNTHR = ILAENV( 6, 'SGELSD', ' ', M, N, NRHS, -1 )
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NLVL = MAX( INT( LOG( REAL( MINMN ) / REAL( SMLSIZ + 1 ) ) /
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$ LOG( TWO ) ) + 1, 0 )
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LIWORK = 3*MINMN*NLVL + 11*MINMN
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MM = M
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IF( M.GE.N .AND. M.GE.MNTHR ) THEN
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*
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* Path 1a - overdetermined, with many more rows than
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* columns.
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*
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MM = N
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MAXWRK = MAX( MAXWRK, N + N*ILAENV( 1, 'SGEQRF', ' ', M,
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$ N, -1, -1 ) )
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MAXWRK = MAX( MAXWRK, N + NRHS*ILAENV( 1, 'SORMQR', 'LT',
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$ M, NRHS, N, -1 ) )
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END IF
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IF( M.GE.N ) THEN
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*
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* Path 1 - overdetermined or exactly determined.
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*
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MAXWRK = MAX( MAXWRK, 3*N + ( MM + N )*ILAENV( 1,
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$ 'SGEBRD', ' ', MM, N, -1, -1 ) )
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MAXWRK = MAX( MAXWRK, 3*N + NRHS*ILAENV( 1, 'SORMBR',
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$ 'QLT', MM, NRHS, N, -1 ) )
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MAXWRK = MAX( MAXWRK, 3*N + ( N - 1 )*ILAENV( 1,
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$ 'SORMBR', 'PLN', N, NRHS, N, -1 ) )
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WLALSD = 9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS +
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$ ( SMLSIZ + 1 )**2
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MAXWRK = MAX( MAXWRK, 3*N + WLALSD )
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MINWRK = MAX( 3*N + MM, 3*N + NRHS, 3*N + WLALSD )
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END IF
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IF( N.GT.M ) THEN
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WLALSD = 9*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS +
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$ ( SMLSIZ + 1 )**2
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IF( N.GE.MNTHR ) THEN
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*
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* Path 2a - underdetermined, with many more columns
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* than rows.
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*
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MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1,
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$ -1 )
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MAXWRK = MAX( MAXWRK, M*M + 4*M + 2*M*ILAENV( 1,
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$ 'SGEBRD', ' ', M, M, -1, -1 ) )
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MAXWRK = MAX( MAXWRK, M*M + 4*M + NRHS*ILAENV( 1,
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$ 'SORMBR', 'QLT', M, NRHS, M, -1 ) )
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MAXWRK = MAX( MAXWRK, M*M + 4*M + ( M - 1 )*ILAENV( 1,
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$ 'SORMBR', 'PLN', M, NRHS, M, -1 ) )
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IF( NRHS.GT.1 ) THEN
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MAXWRK = MAX( MAXWRK, M*M + M + M*NRHS )
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ELSE
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MAXWRK = MAX( MAXWRK, M*M + 2*M )
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END IF
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MAXWRK = MAX( MAXWRK, M + NRHS*ILAENV( 1, 'SORMLQ',
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$ 'LT', N, NRHS, M, -1 ) )
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MAXWRK = MAX( MAXWRK, M*M + 4*M + WLALSD )
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! XXX: Ensure the Path 2a case below is triggered. The workspace
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! calculation should use queries for all routines eventually.
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MAXWRK = MAX( MAXWRK,
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$ 4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )
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ELSE
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*
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* Path 2 - remaining underdetermined cases.
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*
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MAXWRK = 3*M + ( N + M )*ILAENV( 1, 'SGEBRD', ' ', M,
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$ N, -1, -1 )
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MAXWRK = MAX( MAXWRK, 3*M + NRHS*ILAENV( 1, 'SORMBR',
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$ 'QLT', M, NRHS, N, -1 ) )
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MAXWRK = MAX( MAXWRK, 3*M + M*ILAENV( 1, 'SORMBR',
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$ 'PLN', N, NRHS, M, -1 ) )
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MAXWRK = MAX( MAXWRK, 3*M + WLALSD )
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END IF
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MINWRK = MAX( 3*M + NRHS, 3*M + M, 3*M + WLALSD )
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END IF
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END IF
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MINWRK = MIN( MINWRK, MAXWRK )
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WORK( 1 ) = MAXWRK
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IWORK( 1 ) = LIWORK
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*
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IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
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INFO = -12
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGELSD', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible.
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*
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IF( M.EQ.0 .OR. N.EQ.0 ) THEN
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RANK = 0
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RETURN
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END IF
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*
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* Get machine parameters.
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*
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EPS = SLAMCH( 'P' )
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SFMIN = SLAMCH( 'S' )
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SMLNUM = SFMIN / EPS
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BIGNUM = ONE / SMLNUM
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CALL SLABAD( SMLNUM, BIGNUM )
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*
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* Scale A if max entry outside range [SMLNUM,BIGNUM].
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*
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ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
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IASCL = 0
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IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM.
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*
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CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
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IASCL = 1
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ELSE IF( ANRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM.
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*
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CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
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IASCL = 2
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ELSE IF( ANRM.EQ.ZERO ) THEN
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*
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* Matrix all zero. Return zero solution.
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*
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CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
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CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
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RANK = 0
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GO TO 10
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END IF
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*
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* Scale B if max entry outside range [SMLNUM,BIGNUM].
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*
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BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
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IBSCL = 0
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IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
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*
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* Scale matrix norm up to SMLNUM.
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*
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CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
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IBSCL = 1
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ELSE IF( BNRM.GT.BIGNUM ) THEN
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*
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* Scale matrix norm down to BIGNUM.
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*
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CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
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IBSCL = 2
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END IF
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*
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* If M < N make sure certain entries of B are zero.
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*
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IF( M.LT.N )
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$ CALL SLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
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*
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* Overdetermined case.
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*
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IF( M.GE.N ) THEN
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*
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* Path 1 - overdetermined or exactly determined.
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*
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MM = M
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IF( M.GE.MNTHR ) THEN
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*
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* Path 1a - overdetermined, with many more rows than columns.
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*
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MM = N
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ITAU = 1
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NWORK = ITAU + N
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*
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* Compute A=Q*R.
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* (Workspace: need 2*N, prefer N+N*NB)
|
|
*
|
|
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose(Q).
|
|
* (Workspace: need N+NRHS, prefer N+NRHS*NB)
|
|
*
|
|
CALL SORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Zero out below R.
|
|
*
|
|
IF( N.GT.1 ) THEN
|
|
CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
|
|
END IF
|
|
END IF
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + N
|
|
ITAUP = ITAUQ + N
|
|
NWORK = ITAUP + N
|
|
*
|
|
* Bidiagonalize R in A.
|
|
* (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
|
|
*
|
|
CALL SGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of R.
|
|
* (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL SLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of R.
|
|
*
|
|
CALL SORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
|
|
$ MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
|
|
*
|
|
* Path 2a - underdetermined, with many more columns than rows
|
|
* and sufficient workspace for an efficient algorithm.
|
|
*
|
|
LDWORK = M
|
|
IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
|
|
$ M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
|
|
ITAU = 1
|
|
NWORK = M + 1
|
|
*
|
|
* Compute A=L*Q.
|
|
* (Workspace: need 2*M, prefer M+M*NB)
|
|
*
|
|
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
IL = NWORK
|
|
*
|
|
* Copy L to WORK(IL), zeroing out above its diagonal.
|
|
*
|
|
CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
|
|
CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
|
|
$ LDWORK )
|
|
IE = IL + LDWORK*M
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
NWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize L in WORK(IL).
|
|
* (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
|
|
*
|
|
CALL SGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
|
|
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors of L.
|
|
* (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
|
|
$ WORK( ITAUQ ), B, LDB, WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL SLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of L.
|
|
*
|
|
CALL SORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
|
|
$ WORK( ITAUP ), B, LDB, WORK( NWORK ),
|
|
$ LWORK-NWORK+1, INFO )
|
|
*
|
|
* Zero out below first M rows of B.
|
|
*
|
|
CALL SLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
|
|
NWORK = ITAU + M
|
|
*
|
|
* Multiply transpose(Q) by B.
|
|
* (Workspace: need M+NRHS, prefer M+NRHS*NB)
|
|
*
|
|
CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
|
|
$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
ELSE
|
|
*
|
|
* Path 2 - remaining underdetermined cases.
|
|
*
|
|
IE = 1
|
|
ITAUQ = IE + M
|
|
ITAUP = ITAUQ + M
|
|
NWORK = ITAUP + M
|
|
*
|
|
* Bidiagonalize A.
|
|
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
|
|
*
|
|
CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
|
|
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
|
|
$ INFO )
|
|
*
|
|
* Multiply B by transpose of left bidiagonalizing vectors.
|
|
* (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
|
|
*
|
|
CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
* Solve the bidiagonal least squares problem.
|
|
*
|
|
CALL SLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
|
|
$ RCOND, RANK, WORK( NWORK ), IWORK, INFO )
|
|
IF( INFO.NE.0 ) THEN
|
|
GO TO 10
|
|
END IF
|
|
*
|
|
* Multiply B by right bidiagonalizing vectors of A.
|
|
*
|
|
CALL SORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
|
|
$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
|
|
*
|
|
END IF
|
|
*
|
|
* Undo scaling.
|
|
*
|
|
IF( IASCL.EQ.1 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
|
|
CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
ELSE IF( IASCL.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
|
|
CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
|
|
$ INFO )
|
|
END IF
|
|
IF( IBSCL.EQ.1 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
ELSE IF( IBSCL.EQ.2 ) THEN
|
|
CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
|
|
END IF
|
|
*
|
|
10 CONTINUE
|
|
WORK( 1 ) = MAXWRK
|
|
IWORK( 1 ) = LIWORK
|
|
RETURN
|
|
*
|
|
* End of SGELSD
|
|
*
|
|
END
|