DGELSY - compute the minimum-norm solution to a real linear least squares problem

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( * )

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 solution 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 annihilated 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.

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 factorization.

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 column. 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 triangular 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, dimension (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 DGEQP3, 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 illegal value.

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