ZGELSD(1) LAPACK driver routine (version 3.2) ZGELSD(1)NAME
ZGELSD - computes the minimum-norm solution to a real linear 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 singu‐
lar values which are less than RCOND times the largest singular value.
The divide and conquer algorithm makes very mild assumptions 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 without guard dig‐
its, but we know of none.
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) 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 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 the modulus
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 singular values
which are greater than RCOND*S(1).
WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,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 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 array WORK and the minimum sizes of the arrays RWORK and
IWORK, and returns these values as the first entries of the
WORK, RWORK and IWORK arrays, and no error message related to
LWORK is issued by XERBLA.
RWORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
LRWORK >= 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SML‐
SIZ+1)**2 if M is greater than or equal to N or 10*M + 2*M*SML‐
SIZ + 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 subproblems at
the bottom of the computation tree (usually about 25), and NLVL
= MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) On exit,
if INFO = 0, RWORK(1) returns the minimum LRWORK.
IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN(
M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum
LIWORK.
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 bidiag‐
onal form did not converge 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
LAPACK driver routine (version 3November 2008 ZGELSD(1)
