CGELSS - compute the minimum norm solution to a complex
linear least squares problem
SYNOPSIS
SUBROUTINE CGELSS( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
RANK, WORK, LWORK, RWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
REAL RCOND
REAL RWORK( * ), S( * )
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
CGELSS computes the minimum norm solution to a complex
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 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 effective rank of A is determined by treating as zero
those singular values which are less than RCOND times the
largest singular value.
ARGUMENTS
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) COMPLEX array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, the first
min(m,n) rows of A are overwritten with its right
singular vectors, stored rowwise.
LDA (input) INTEGER
The leading dimension of the array A. LDA >=
max(1,M).
On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solu
tion matrix X. If m >= n and RANK = n, the resid
ual 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) INTEGER
The leading dimension of the array B. LDB >=
max(1,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 sin
gular values which are greater than RCOND*S(1).
WORK (workspace/output) COMPLEX 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 >= 2*min(M,N) + max(M,N,NRHS) For
good performance, LWORK should generally be
larger.
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.
RWORK (workspace) REAL array, dimension (5*min(M,N))
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal 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 con
verge to zero.
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