Contents
sgelsx - routine is deprecated and has been replaced by rou-
tine SGELSY
SUBROUTINE SGELSX(M, N, NRHS, A, LDA, B, LDB, JPIVOT, RCOND, IRANK,
WORK, INFO)
INTEGER M, N, NRHS, LDA, LDB, IRANK, INFO
INTEGER JPIVOT(*)
REAL RCOND
REAL A(LDA,*), B(LDB,*), WORK(*)
SUBROUTINE SGELSX_64(M, N, NRHS, A, LDA, B, LDB, JPIVOT, RCOND,
IRANK, WORK, INFO)
INTEGER*8 M, N, NRHS, LDA, LDB, IRANK, INFO
INTEGER*8 JPIVOT(*)
REAL RCOND
REAL A(LDA,*), B(LDB,*), WORK(*)
F95 INTERFACE
SUBROUTINE GELSX([M], [N], [NRHS], A, [LDA], B, [LDB], JPIVOT, RCOND,
IRANK, [WORK], [INFO])
INTEGER :: M, N, NRHS, LDA, LDB, IRANK, INFO
INTEGER, DIMENSION(:) :: JPIVOT
REAL :: RCOND
REAL, DIMENSION(:) :: WORK
REAL, DIMENSION(:,:) :: A, B
SUBROUTINE GELSX_64([M], [N], [NRHS], A, [LDA], B, [LDB], JPIVOT,
RCOND, IRANK, [WORK], [INFO])
INTEGER(8) :: M, N, NRHS, LDA, LDB, IRANK, INFO
INTEGER(8), DIMENSION(:) :: JPIVOT
REAL :: RCOND
REAL, DIMENSION(:) :: WORK
REAL, DIMENSION(:,:) :: A, B
C INTERFACE
#include <sunperf.h>
void sgelsx(int m, int n, int nrhs, float *a, int lda, float
*b, int ldb, int *jpivot, float rcond, int *irank,
int *info);
void sgelsx_64(long m, long n, long nrhs, float *a, long
lda, float *b, long ldb, long *jpivot, float
rcond, long *irank, long *info);
sgelsx routine is deprecated and has been replaced by rou-
tine SGELSY.
SGELSX 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 annihi-
lated 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.
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 overwritten 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. If m >=
n and IRANK = 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,M,N).
JPIVOT (input/output)
On entry, if JPIVOT(i) .ne. 0, the i-th column of
A is an initial column, otherwise it is a free
column. Before the QR factorization of A, all
initial columns are permuted to the leading posi-
tions; only the remaining free columns are moved
as a result of column pivoting during the factori-
zation. On exit, if JPIVOT(i) = k, then the i-th
column of A*P 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 submatrix R11 in the QR factor-
ization with pivoting of A, whose estimated condi-
tion number < 1/RCOND.
IRANK (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 fac-
torization of A.
WORK (workspace)
(max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal value