sgelsd(3)

NAME

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 solu

tion 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 hexadeci

mal or decimal machines without guard digits, but we know of

none.

ARGUMENTS

M (input) INTEGER

The number of rows of A. M >= 0.

N (input) INTEGER

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 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 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 + (SMLSIZ+1)**2, if

M is less than N, the code will execute correctly. SMLSIZ is re

turned by ILAENV and is equal to the maximum size of the subprob

lems at the bottom of the computation tree (usually about 25),

and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

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.

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 interme

diate bidiagonal 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 version 3.0 15 June 2000

SGELSD(3)