
ZGELSD(l) ) ZGELSD(l)
NAME
ZGELSD  compute the minimumnorm 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 minimumnorm solution to a real linear least squares problem: min
imize 2norm( b  A*x )
using the singular value decomposition (SVD) of A. A is an MbyN matrix which may be
rankdeficient.
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 MbyNRHS right hand side matrix B and the NbyNRHS
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 assumptions about floating point arith
metic. 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 XMP, Cray YMP, Cray C90, or
Cray2. It could conceivably fail on hexadecimal or decimal machines without guard digits,
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 MbyN 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 MbyNRHS right hand side matrix B. On exit, B is overwritten by
the NbyNRHS solution matrix X. If m >= n and RANK = n, the residual sumof
squares for the solution in the ith 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,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
2norm = 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 (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 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 subproblems at the bottom of the computation
tree (usually 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 ith 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.
FURTHER DETAILS
Based on contributions by
Ming Gu and RenCang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA
LAPACK version 3.0 15 June 2000 ZGELSD(l) 
