sgelsy(3P) Sun Performance Library sgelsy(3P)NAMEsgelsy - compute the minimum-norm solution to a real linear least
squares problem
SYNOPSIS
SUBROUTINE SGELSY(M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
WORK, LWORK, INFO)
INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER JPVT(*)
REAL RCOND
REAL A(LDA,*), B(LDB,*), WORK(*)
SUBROUTINE SGELSY_64(M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
WORK, LWORK, INFO)
INTEGER*8 M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER*8 JPVT(*)
REAL RCOND
REAL A(LDA,*), B(LDB,*), WORK(*)
F95 INTERFACE
SUBROUTINE GELSY([M], [N], [NRHS], A, [LDA], B, [LDB], JPVT, RCOND,
RANK, [WORK], [LWORK], [INFO])
INTEGER :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER, DIMENSION(:) :: JPVT
REAL :: RCOND
REAL, DIMENSION(:) :: WORK
REAL, DIMENSION(:,:) :: A, B
SUBROUTINE GELSY_64([M], [N], [NRHS], A, [LDA], B, [LDB], JPVT,
RCOND, RANK, [WORK], [LWORK], [INFO])
INTEGER(8) :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
INTEGER(8), DIMENSION(:) :: JPVT
REAL :: RCOND
REAL, DIMENSION(:) :: WORK
REAL, DIMENSION(:,:) :: A, B
C INTERFACE
#include <sunperf.h>
void sgelsy(int m, int n, int nrhs, float *a, int lda, float *b, int
ldb, int *jpvt, float rcond, int *rank, int *info);
void sgelsy_64(long m, long n, long nrhs, float *a, long lda, float *b,
long ldb, long *jpvt, float rcond, long *rank, long *info);
PURPOSEsgelsy computes the minimum-norm solution to a real 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 con‐
dition 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
orthogonal 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' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first RANK columns of Q.
This routine is basically identical to the original xGELSX except three
differences:
o The call to the subroutine xGEQPF has been substituted by the
the call to the subroutine xGEQP3. This subroutine is a Blas-3
version of the QR factorization with column pivoting.
o Matrix B (the right hand side) is updated with Blas-3.
o The permutation of matrix B (the right hand side) is faster and
more simple.
ARGUMENTS
M (input) The number of rows of the matrix A. M >= 0.
N (input) The number of columns of the matrix A. N >= 0.
NRHS (input)
The number of right hand sides, i.e., the number of columns
of matrices B and X. NRHS >= 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, A has been overwrit‐
ten by details of its complete orthogonal factorization.
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,
the N-by-NRHS solution matrix X.
LDB (input)
The leading dimension of the array B. LDB >= max(1,M,N).
JPVT (input/output)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of AP, otherwise column i is a free column. On
exit, if JPVT(i) = k, then the i-th column of AP was the k-th
column of A.
RCOND (input)
RCOND is used to determine the effective rank of A, which is
defined as the order of the largest leading triangular subma‐
trix R11 in the QR factorization with pivoting of A, whose
estimated condition number < 1/RCOND.
RANK (output)
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)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. The unblocked strategy
requires that: LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN
= min( M, N ). The block algorithm requires that: LWORK >=
MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper
bound on the blocksize returned by ILAENV for the routines
SGEQP3, STZRZF, STZRQF, SORMQR, and SORMRZ.
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.
INFO (output)
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
Based on contributions by
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
6 Mar 2009 sgelsy(3P)
