zgelsd()

subroutine zgelsd	(	integer	m,
		integer	n,
		integer	nrhs,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		double precision, dimension( * )	s,
		double precision	rcond,
		integer	rank,
		complex*16, dimension( * )	work,
		integer	lwork,
		double precision, dimension( * )	rwork,
		integer, dimension( * )	iwork,
		integer	info )

ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices

Download ZGELSD + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> ZGELSD 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 problem is solved in three steps:
!> (1) Reduce the coefficient matrix A to bidiagonal form with
!>     Householder transformations, reducing the original problem
!>     into a  (BLS)
!> (2) Solve the BLS using a divide and conquer approach.
!> (3) Apply back all the Householder transformations to solve
!>     the original least squares problem.
!>
!> The effective rank of A is determined by treating as zero those
!> singular values which are less than RCOND times the largest singular
!> value.
!>
!> 

Parameters

[in]	M	!> M is INTEGER !> The number of rows of the matrix A. M >= 0. !>
[in]	N	!> N is INTEGER !> The number of columns of the matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrices B and X. NRHS >= 0. !>
[in,out]	A	!> A is COMPLEX*16 array, dimension (LDA,N) !> On entry, the M-by-N matrix A. !> On exit, A has been destroyed. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,M). !>
[in,out]	B	!> B is COMPLEX*16 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 the modulus of elements n+1:m in that column. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,M,N). !>
[out]	S	!> 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)). !>
[in]	RCOND	!> 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. !>
[out]	RANK	!> RANK is INTEGER !> The effective rank of A, i.e., the number of singular values !> which are greater than RCOND*S(1). !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. LWORK must be at least 1. !> The exact minimum amount of workspace needed depends on M, !> N and NRHS. As long as LWORK is at least !> 2*N + N*NRHS !> if M is greater than or equal to N or !> 2*M + M*NRHS !> if M is less than N, the code will execute correctly. !> 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 array WORK and the !> minimum sizes of the arrays RWORK and IWORK, and returns !> these values as the first entries of the WORK, RWORK and !> IWORK arrays, and no error message related to LWORK is issued !> by XERBLA. !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) !> LRWORK >= !> 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + !> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) !> if M is greater than or equal to N or !> 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS + !> MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) !> if M is less than N, the code will execute correctly. !> SMLSIZ is returned by ILAENV and is equal to the maximum !> size of the subproblems at the bottom of the computation !> tree (usually about 25), and !> NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) !> On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK. !>
[out]	IWORK	!> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) !> LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), !> where MINMN = MIN( M,N ). !> On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. !>
[out]	INFO	!> 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. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA