zgelsd(3S)

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

SYNOPSIS
     SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK,
                        LWORK, RWORK, IWORK, INFO )

         INTEGER        INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

         DOUBLE         PRECISION RCOND

         INTEGER        IWORK( * )

         DOUBLE         PRECISION RWORK( * ), S( * )

         COMPLEX*16     A( LDA, * ), B( LDB, * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZGELSD 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 tranformations, 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
     M       (input) INTEGER
             The number of rows of the matrix A. M >= 0.

     N       (input) INTEGER
             The number of columns of the matrix 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) COMPLEX*16 array, dimension (LDA,N)
             On entry, the M-by-N 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) COMPLEX*16 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     (input) INTEGER
             The leading dimension of the array B.  LDB >= max(1,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 2-norm = 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) COMPLEX*16 array, dimension (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 2 * N + N * NRHS if M is
             greater than or equal to N or 2 * M + M * NRHS if M is less than
             N, the code will execute correctly.  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.

     RWORK   (workspace) DOUBLE PRECISION array, dimension at least
             10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2 if M
             is greater than or equal to N or 10*M + 2*M*SMLSIZ + 8*M*NLVL +
             3*SMLSIZ*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 )

     IWORK   (workspace) INTEGER array, dimension (LIWORK)
             LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, where MINMN = MIN( M,N
             ).

     INFO    (output) 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.

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


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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