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(*)
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
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 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.