ESMF_DGELSD Subroutine

subroutine ESMF_DGELSD(M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO)

\brief ESMF_DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices \htmlonly Download ESMF_DGELSD + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) 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 problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a “bidiagonal least squares problem” (BLS) (2) Solve the BLS using a divide and conquer approach. (3) Apply back all the Householder tranformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none. \endverbatim \param[in] M \verbatim M is INTEGER The number of rows of A. M >= 0. \endverbatim

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

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

\param[in] A \verbatim A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, A has been destroyed. \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, B is overwritten by the N-by-NRHS solution matrix X. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column. \endverbatim

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

\param[out] S \verbatim S is DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order. The condition number of A in the 2-norm = S(1)/S(min(m,n)). \endverbatim

\param[in] RCOND \verbatim RCOND is DOUBLE PRECISION RCOND is used to determine the effective rank of A. Singular values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0, machine precision is used instead. \endverbatim

\param[out] RANK \verbatim RANK is INTEGER The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1). \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. LWORK must be at least 1. The exact minimum amount of workspace needed depends on M, N and NRHS. As long as LWORK is at least 12N + 2NSMLSIZ + 8NNLVL + NNRHS + (SMLSIZ+1)2, if M is greater than or equal to N or 12M + 2MSMLSIZ + 8MNLVL + MNRHS + (SMLSIZ+1)2, if M is less than N, the code will execute correctly. SMLSIZ is returned by ESMF_ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger.

     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] IWORK \verbatim IWORK is INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN), where MINMN = MIN( M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK. \endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero. \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:

Ming Gu and Ren-Cang Li, Computer Science Division, University of
  California at Berkeley, USA \n
Osni Marques, LBNL/NERSC, USA \n

Arguments