ZGELSD - compute the minimum-norm solution to a real lin
ear least squares problem
SYNOPSIS
SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND,
RANK, WORK, LWORK, RWORK, IWORK, INFO )
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
DOUBLE PRECISION RCOND
INTEGER IWORK( * )
DOUBLE PRECISION RWORK( * ), S( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
ZGELSD 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 tranformations, 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
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) COMPLEX*16 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) COMPLEX*16 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,M,N).
S (output) DOUBLE PRECISION 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) DOUBLE PRECISION
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*16 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
needed depends on M, N and NRHS. As long as LWORK
is at least 2 * N + N * NRHS if M is greater than
or equal to N or 2 * M + M * NRHS if M is less
than N, the code will execute correctly. 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) DOUBLE PRECISION array, dimension at
least
10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
(SMLSIZ+1)**2 if M is greater than or equal to N
or 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
(SMLSIZ+1)**2 if M is less than N, the code will
execute correctly. SMLSIZ is returned by ILAENV
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 )
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
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