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NAME DGELS - solve overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A SYNOPSIS SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO ) CHARACTER TRANS INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( LWORK ) PURPOSE DGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank. The following options are provided: 1. If TRANS = 'N' and m >= n: find the least squares solu- tion of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||. 2. If TRANS = 'N' and m < n: find the minimum norm solution of an underdetermined system A * X = B. 3. If TRANS = 'T' and m >= n: find the minimum norm solu- tion of an undetermined system A**T * X = B. 4. If TRANS = 'T' and m < n: find the least squares solu- tion of an overdetermined system, i.e., solve the least squares problem minimize || B - A**T * X ||. 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. ARGUMENTS TRANS (input) CHARACTER = 'N': the linear system involves A; = 'T': the linear system involves A**T. 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 the 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, if M >= N, A is overwritten by details of its QR factorization as returned by DGEQRF; if M < N, A is overwritten by details of its LQ factorization as returned by DGELQF. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). (LDB,NRHS) B (input/output) DOUBLE PRECISION array, dimension On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'T'. On exit, B is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and m >= n, rows 1 to n of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and m < n, rows 1 to N of B contain the minimum norm solution vectors; if TRANS = 'T' and m >= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = 'T' and m < n, rows 1 to M of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column. LDB (input) INTEGER The leading dimension of the array B. LDB >= MAX(1,M,N). WORK (workspace) 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. LWORK >= min(M,N) + MAX(1,M,N,NRHS). For optimal performance, LWORK >= min(M,N) + MAX(1,M,N,NRHS) * NB where NB is the optimum block size. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value