ESMF_DGELSY Subroutine

subroutine ESMF_DGELSY(M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO)

\brief ESMF_DGELSY solves overdetermined or underdetermined systems for GE matrices \htmlonly Download ESMF_DGELSY + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_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 * ZT [ inv(T11)*Q1T*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. \endverbatim \param[in] M \verbatim M is INTEGER The number of rows of the matrix A. M >= 0. \endverbatim

\param[in] N \verbatim N is INTEGER The number of columns of the matrix A. N >= 0. \endverbatim

\param[in] NRHS \verbatim NRHS is INTEGER The number of right hand sides, i.e., the number of columns of matrices B and X. NRHS >= 0. \endverbatim

\param[in,out] A \verbatim A is 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. \endverbatim

\param[in] LDA \verbatim LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). \endverbatim

\param[in,out] B \verbatim B is 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. \endverbatim

\param[in] LDB \verbatim LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M,N). \endverbatim

\param[in,out] JPVT \verbatim JPVT is 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. \endverbatim

\param[in] RCOND \verbatim RCOND is 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. \endverbatim

\param[out] RANK \verbatim RANK is 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. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The dimension of the array WORK. The unblocked strategy requires that: LWORK >= MAX( MN+3N+1, 2MN+NRHS ), where MN = min( M, N ). The block algorithm requires that: LWORK >= MAX( MN+2N+NB(N+1), 2MN+NBNRHS ), where NB is an upper bound on the blocksize returned by ESMF_ILAENV for the routines ESMF_DGEQP3, ESMF_DTZRZF, STZRQF, ESMF_DORMQR, and ESMF_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 ESMF_XERBLA.

\endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit < 0: If INFO = -i, the i-th argument had an illegal value. \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date November 2011 \ingroup doubleGEsolve \par Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n

Arguments