SYNOPSIS

SUBROUTINE DGGLSE(

M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO )

    

INTEGER INFO, LDA, LDB, LWORK, M, N, P

    

DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( * ), D( * ), WORK( * ), X( * )

PURPOSE

DGGLSE solves the linear equality-constrained least squares (LSE) problem:
        minimize || c - A*x ||_2   subject to   B*x = d
where A is an M-by-N matrix, B is a P-by-N matrix, c is a given M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and

         rank(B) = P and  rank( (A) ) = N.

                              ( (B) )
These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized RQ factorization of the matrices (B, A) given by

   B = (0 R)*Q,   A = Z*T*Q.

ARGUMENTS

M (input) INTEGER

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

N (input) INTEGER

The number of columns of the matrices A and B. N >= 0.

P (input) INTEGER

The number of rows of the matrix B. 0 <= P <= N <= M+P.

A (input/output) DOUBLE PRECISION array, dimension (LDA,N)

On entry, the M-by-N matrix A. On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix T.

LDA (input) INTEGER

The leading dimension of the array A. LDA >= max(1,M).

B (input/output) DOUBLE PRECISION array, dimension (LDB,N)

On entry, the P-by-N matrix B. On exit, the upper triangle of the subarray B(1:P,N-P+1:N) contains the P-by-P upper triangular matrix R.

LDB (input) INTEGER

The leading dimension of the array B. LDB >= max(1,P).

C (input/output) DOUBLE PRECISION array, dimension (M)

On entry, C contains the right hand side vector for the least squares part of the LSE problem. On exit, the residual sum of squares for the solution is given by the sum of squares of elements N-P+1 to M of vector C.

D (input/output) DOUBLE PRECISION array, dimension (P)

On entry, D contains the right hand side vector for the constrained equation. On exit, D is destroyed.

X (output) DOUBLE PRECISION array, dimension (N)

On exit, X is the solution of the LSE problem.

WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

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

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= max(1,M+N+P). For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB, where NB is an upper bound for the optimal blocksizes for DGEQRF, SGERQF, DORMQR and SORMRQ. 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.
= 1: the upper triangular factor R associated with B in the generalized RQ factorization of the pair (B, A) is singular, so that rank(B) < P; the least squares solution could not be computed. = 2: the (N-P) by (N-P) part of the upper trapezoidal factor T associated with A in the generalized RQ factorization of the pair (B, A) is singular, so that rank( (A) ) < N; the least squares solution could not ( (B) ) be computed.