dgelsy(3)

NAME

DGELSY - compute the minimum-norm solution to a real lin

ear least squares problem

SYNOPSIS

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

PURPOSE

DGELSY 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.

This routine is basically identical to the original xGELSX

except three differences:

o The call to the subroutine xGEQPF has been substituted

by the

the call to the subroutine xGEQP3. This subroutine is

a Blas-3
version of the QR factorization with column pivoting.

o Matrix B (the right hand side) is updated with Blas-3.
o The permutation of matrix B (the right hand side) is

faster and

more simple.

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.

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 permuted to the front of AP, otherwise column i is a free col

umn. On exit, if JPVT(i) = k, then the i-th column of AP 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/output) DOUBLE PRECISION array, dimen

sion (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. The unblocked

strategy requires that: LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),

where MN = min( M, N ). The block algorithm requires that: LWORK

>= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper

bound on the blocksize returned by ILAENV for the routines DGE

QP3, DTZRZF, STZRQF, DORMQR, and DORMRZ.

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.

INFO (output) INTEGER

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

gal value.

FURTHER DETAILS

Based on contributions by

A. Petitet, Computer Science Dept., Univ. of Tenn.,

Knoxville, USA
E. Quintana-Orti, Depto. de Informatica, Universidad

Jaime I, Spain
G. Quintana-Orti, Depto. de Informatica, Universidad

Jaime I, Spain

LAPACK version 3.0 15 June 2000

DGELSY(3)