sgelsd - compute the minimum-norm solution to a real linear least squares problem
SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, * LWORK, IWORK, INFO) INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, INFO INTEGER IWORK(*) REAL RCOND REAL A(LDA,*), B(LDB,*), S(*), WORK(*)
SUBROUTINE SGELSD_64( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, * WORK, LWORK, IWORK, INFO) INTEGER*8 M, N, NRHS, LDA, LDB, RANK, LWORK, INFO INTEGER*8 IWORK(*) REAL RCOND REAL A(LDA,*), B(LDB,*), S(*), WORK(*)
SUBROUTINE GELSD( [M], [N], [NRHS], A, [LDA], B, [LDB], S, RCOND, * RANK, [WORK], [LWORK], [IWORK], [INFO]) INTEGER :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK REAL :: RCOND REAL, DIMENSION(:) :: S, WORK REAL, DIMENSION(:,:) :: A, B
SUBROUTINE GELSD_64( [M], [N], [NRHS], A, [LDA], B, [LDB], S, RCOND, * RANK, [WORK], [LWORK], [IWORK], [INFO]) INTEGER(8) :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK REAL :: RCOND REAL, DIMENSION(:) :: S, WORK REAL, DIMENSION(:,:) :: A, B
#include <sunperf.h>
void sgelsd(int m, int n, int nrhs, float *a, int lda, float *b, int ldb, float *s, float rcond, int *rank, int *info);
void sgelsd_64(long m, long n, long nrhs, float *a, long lda, float *b, long ldb, float *s, float rcond, long *rank, long *info);
sgelsd 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 ``bidiagonal least squares problem'' (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder tranformations 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.
The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
S(i)
< = RCOND*S(1) are treated as zero.
If RCOND < 0, machine precision is used instead.
WORK(1)
returns the optimal LWORK.
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.
= 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.
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA