dgelsy - compute the minimum-norm solution to a real linear least squares problem
SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, * WORK, LWORK, INFO) INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, INFO INTEGER JPVT(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(LDA,*), B(LDB,*), WORK(*)
SUBROUTINE DGELSY_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(*) DOUBLE PRECISION RCOND DOUBLE PRECISION A(LDA,*), B(LDB,*), WORK(*)
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(8) :: RCOND REAL(8), DIMENSION(:) :: WORK REAL(8), 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(8) :: RCOND REAL(8), DIMENSION(:) :: WORK REAL(8), DIMENSION(:,:) :: A, B
#include <sunperf.h>
void dgelsy(int m, int n, int nrhs, double *a, int lda, double *b, int ldb, int *jpvt, double rcond, int *rank, int *info);
void dgelsy_64(long m, long n, long nrhs, double *a, long lda, double *b, long ldb, long *jpvt, double rcond, long *rank, long *info);
dgelsy 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 condition 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.
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.
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.
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