sgelsd(3P) Sun Performance Library sgelsd(3P)NAMEsgelsd - compute the minimum-norm solution to a real linear least
squares problem
SYNOPSIS
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(*)
F95 INTERFACE
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
C INTERFACE
#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);
PURPOSEsgelsd 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 singu‐
lar 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 dig‐
its, but we know of none.
ARGUMENTS
M (input) The number of rows of A. M >= 0.
N (input) The number of columns of A. N >= 0.
NRHS (input)
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, A has been
destroyed.
LDA (input)
The leading dimension of the array A. LDA >= max(1,M).
B (input/output)
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 elements
n+1:m in that column.
LDB (input)
The leading dimension of the array B. LDB >= max(1,max(M,N)).
S (output)
The singular values of A in decreasing order. The condition
number of A in the 2-norm = S(1)/S(min(m,n)).
RCOND (input)
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.
RANK (output)
The effective rank of A, i.e., the number of singular values
which are greater than RCOND*S(1).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK >= 1. The exact mini‐
mum amount of workspace needed depends on M, N and NRHS. As
long as LWORK is at least N**2 + 13*N + 2*N*SMLSIZ + 8*N*NLVL
+ N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N
or M**2 + 13*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SML‐
SIZ+1)**2, if M is less than N, the code will execute cor‐
rectly. SMLSIZ is returned by ILAENV and is equal to the
maximum size of the subproblems at the bottom of the computa‐
tion tree (usually about 25), and NLVL = INT( LOG_2( MIN( M,N
)/(SMLSIZ+1) ) ) + 1 For good performance, LWORK should gen‐
erally be larger.
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.
IWORK (workspace)
LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN(
M,N ).
INFO (output)
= 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 bidi‐
agonal form did not converge to zero.
FURTHER DETAILS
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
6 Mar 2009 sgelsd(3P)
