SGELSD - compute the minimum-norm solution to a real lin
ear least squares problem
SYNOPSIS
SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
RANK, WORK, LWORK, IWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
REAL RCOND
INTEGER IWORK( * )
REAL A( LDA, * ), B( LDB, * ), S( * ), WORK(
* )
PURPOSE
SGELSD 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 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 problem is solved in three steps:
(1) Reduce the coefficient matrix A to bidiagonal form
with
Householder transformations, reducing the original
problem
into a "bidiagonal least squares problem" (BLS)
(2) Solve the BLS using a divide and conquer approach.
(3) Apply back all the Householder tranformations to solve
the original least squares problem.
The effective rank of A is determined by treating as zero
those singular values which are less than RCOND times the
largest singular value.
The divide and conquer algorithm makes very mild assump
tions about floating point arithmetic. It will work on
machines with a guard digit in add/subtract, or on those
binary machines without guard digits which subtract like
the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
conceivably fail on hexadecimal or decimal machines with
out guard digits, but we know of none.
ARGUMENTS
M (input) INTEGER
The number of rows of A. M >= 0.
The number of columns of 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) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, A has
been destroyed.
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 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,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 sin
gular values which are greater than RCOND*S(1).
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 must be at
least 1. The exact minimum amount of workspace
needed depends on M, N and NRHS. As long as LWORK
is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS
+ (SMLSIZ+1)**2, if M is greater than or equal to
N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SML
SIZ+1)**2, if M is less than N, the code will
and is equal to the maximum size of the subprob
lems at the bottom of the computation tree (usu
ally about 25), and NLVL = MAX( 0, INT( LOG_2(
MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good perfor
mance, 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.
IWORK (workspace) INTEGER array, dimension (LIWORK)
LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where
MINMN = 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.
FURTHER DETAILS
Based on contributions by
Ming Gu and Ren-Cang Li, Computer Science Division,
University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
Man(1) output converted with
man2html