NAME

       sgelsd.f -

SYNOPSIS

   Functions/Subroutines
       subroutine sgelsd (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, IWORK, INFO)
            SGELSD computes the minimum-norm solution to a linear least squares problem for GE
           matrices

Function/Subroutine Documentation

   subroutine sgelsd (integerM, integerN, integerNRHS, real, dimension( lda, * )A, integerLDA,
       real, dimension( ldb, * )B, integerLDB, real, dimension( * )S, realRCOND, integerRANK,
       real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
        SGELSD computes the minimum-norm solution to a linear least squares problem for GE
       matrices

       Purpose:

            SGELSD 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.

       Parameters:
           M

                     M is INTEGER
                     The number of rows of A. M >= 0.

           N

                     N is INTEGER
                     The number of columns of A. N >= 0.

           NRHS

                     NRHS is INTEGER
                     The number of right hand sides, i.e., the number of columns
                     of the matrices B and X. NRHS >= 0.

           A

                     A is REAL array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, A has been destroyed.

           LDA

                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           B

                     B is REAL 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.

           LDB

                     LDB is INTEGER
                     The leading dimension of the array B. LDB >= max(1,max(M,N)).

           S

                     S is REAL 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)).

           RCOND

                     RCOND is REAL
                     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

                     RANK is INTEGER
                     The effective rank of A, i.e., the number of singular values
                     which are greater than RCOND*S(1).

           WORK

                     WORK is REAL array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     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
                         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 array WORK and the
                     minimum size of the array IWORK, and returns these values as
                     the first entries of the WORK and IWORK arrays, and no error
                     message related to LWORK is issued by XERBLA.

           IWORK

                     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.

           INFO

                     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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2011

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

       Definition at line 210 of file sgelsd.f.

Author

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