SGELSS(l) ) SGELSS(l)
SGELSS - compute the minimum norm solution to a real linear least squares problem
SUBROUTINE SGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
SGELSS 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
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 effective rank of A is determined by treating as zero those singular values which are
less than RCOND times the largest singular value.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrices B and
X. NRHS >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, the first min(m,n) rows of A are over-
written with its right singular vectors, stored rowwise.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) REAL array, dimension (LDB,NRHS)
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 ele-
ments n+1:m in that column.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,max(M,N)).
S (output) REAL array, dimension (min(M,N))
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) REAL
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) INTEGER
The effective rank of A, i.e., the number of singular values which are greater
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1, and also: LWORK >= 3*min(M,N) + max(
2*min(M,N), max(M,N), NRHS ) For good performance, LWORK should generally be
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) INTEGER
= 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 bidiagonal form did not converge to zero.
LAPACK version 3.0 15 June 2000 SGELSS(l)