NAME

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


SYNOPSIS

  SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, 
 *      LWORK, IWORK, INFO)
  INTEGER M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
  INTEGER IWORK(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), B(LDB,*), S(*), WORK(*)
  SUBROUTINE DGELSD_64( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, 
 *      WORK, LWORK, IWORK, INFO)
  INTEGER*8 M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
  INTEGER*8 IWORK(*)
  DOUBLE PRECISION RCOND
  DOUBLE PRECISION A(LDA,*), B(LDB,*), S(*), WORK(*)

F95 INTERFACE

  SUBROUTINE GELSD( [M], [N], [NRHS], A, [LDA], B, [LDB], S, RCOND, 
 *       RANK, [WORK], [LWORK], [IWORK], [INFO])
  INTEGER :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: S, WORK
  REAL(8), DIMENSION(:,:) :: A, B
  SUBROUTINE GELSD_64( [M], [N], [NRHS], A, [LDA], B, [LDB], S, RCOND, 
 *       RANK, [WORK], [LWORK], [IWORK], [INFO])
  INTEGER(8) :: M, N, NRHS, LDA, LDB, RANK, LWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  REAL(8) :: RCOND
  REAL(8), DIMENSION(:) :: S, WORK
  REAL(8), DIMENSION(:,:) :: A, B

C INTERFACE

#include <sunperf.h>

void dgelsd(int m, int n, int nrhs, double *a, int lda, double *b, int ldb, double *s, double rcond, int *rank, int *info);

void dgelsd_64(long m, long n, long nrhs, double *a, long lda, double *b, long ldb, double *s, double rcond, long *rank, long *info);


PURPOSE

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.


ARGUMENTS


FURTHER DETAILS

Based on contributions by

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