dgelsx(3)

NAME

DGELSX - routine is deprecated and has been replaced by

routine DGELSY

SYNOPSIS

SUBROUTINE  DGELSX(  M,  N,  NRHS,  A,  LDA, B, LDB, JPVT,
RCOND, RANK, WORK, INFO )
    INTEGER        INFO, LDA, LDB, M, N, NRHS, RANK
    DOUBLE         PRECISION RCOND
    INTEGER        JPVT( * )
    DOUBLE         PRECISION A( LDA, * ),  B(  LDB,  *  ),
WORK( * )

PURPOSE

This routine is deprecated and has been replaced by rou

tine DGELSY. DGELSX computes the minimum-norm solution to a real

linear least squares problem:

minimize || A * X - B

using a complete orthogonal factorization 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 routine first computes a QR factorization with column

pivoting:

A * P = Q * [ R11 R12 ]

[ 0 R22 ]

with R11 defined as the largest leading submatrix whose

estimated condition number is less than 1/RCOND. The order of

R11, RANK, is the effective rank of A.

Then, R22 is considered to be negligible, and R12 is anni

hilated by orthogonal transformations from the right, arriving at

the complete orthogonal factorization:

A * P = Q * [ T11 0 ] * Z

[ 0 0 ]

The minimum-norm solution is then

X = P * Z' [ inv(T11)*Q1'*B ]

[ 0 ]

where Q1 consists of the first RANK columns of Q.

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 matrices B and X. NRHS >= 0.

A (input/output) DOUBLE PRECISION array, dimension

(LDA,N)

On entry, the M-by-N matrix A. On exit, A has

been overwritten by details of its complete orthogonal factoriza

tion.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,M).

B (input/output) DOUBLE PRECISION array, dimension

(LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B.

On exit, 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 col

umn is given by the sum of squares of elements N+1:M in that col

umn.

LDB (input) INTEGER

The leading dimension of the array B. LDB >=

max(1,M,N).

JPVT (input/output) INTEGER array, dimension (N)

On entry, if JPVT(i) .ne. 0, the i-th column of A

is an initial column, otherwise it is a free column. Before the

QR factorization of A, all initial columns are permuted to the

leading positions; only the remaining free columns are moved as a

result of column pivoting during the factorization. On exit, if

JPVT(i) = k, then the i-th column of A*P was the k-th column of

A.

RCOND (input) DOUBLE PRECISION

RCOND is used to determine the effective rank of

A, which is defined as the order of the largest leading triangu

lar submatrix R11 in the QR factorization with pivoting of A,

whose estimated condition number < 1/RCOND.

RANK (output) INTEGER

The effective rank of A, i.e., the order of the

submatrix R11. This is the same as the order of the submatrix

T11 in the complete orthogonal factorization of A.

WORK (workspace) DOUBLE PRECISION array, dimension

(max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value

LAPACK version 3.0 15 June 2000

DGELSX(3)