SYNOPSIS
 SUBROUTINE DGELSD(
 M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO )
 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
 DOUBLE PRECISION RCOND
 INTEGER IWORK( * )
 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
PURPOSE
DGELSD computes the minimumnorm solution to a real linear least squares problem:minimize 2norm( b  A*x )
using the singular value decomposition (SVD) of A. A is an MbyN matrix which may be rankdeficient.
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 MbyNRHS right hand side matrix B and the NbyNRHS 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 XMP, Cray YMP, Cray C90, or Cray2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.
ARGUMENTS
 M (input) INTEGER
 The number of rows of A. M >= 0.
 N (input) INTEGER
 The number of columns of 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) DOUBLE PRECISION array, dimension (LDA,N)
 On entry, the MbyN matrix A. On exit, A has been destroyed.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,M).
 B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
 On entry, the MbyNRHS right hand side matrix B. On exit, B is overwritten by the NbyNRHS solution matrix X. If m >= n and RANK = n, the residual sumofsquares for the solution in the ith column is given by the sum of squares of elements n+1:m in that column.
 LDB (input) INTEGER
 The leading dimension of the array B. LDB >= max(1,max(M,N)).
 S (output) DOUBLE PRECISION array, dimension (min(M,N))
 The singular values of A in decreasing order. The condition number of A in the 2norm = S(1)/S(min(m,n)).
 RCOND (input) 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.
 RANK (output) INTEGER
 The effective rank of A, i.e., the number of singular values which are greater than RCOND*S(1).
 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 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 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly. SMLSIZ is returned by 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 XERBLA.
 IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
 LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N ).
 INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge; if INFO = i, i offdiagonal elements of an intermediate bidiagonal form did not converge to zero.
FURTHER DETAILS
Based on contributions byMing Gu and RenCang Li, Computer Science Division, University of
California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA