zlatms.f - Man Page

TESTING/MATGEN/zlatms.f

Synopsis

Functions/Subroutines

subroutine zlatms (m, n, dist, iseed, sym, d, mode, cond, dmax, kl, ku, pack, a, lda, work, info)
ZLATMS

Function/Subroutine Documentation

subroutine zlatms (integer m, integer n, character dist, integer, dimension( 4 ) iseed, character sym, double precision, dimension( * ) d, integer mode, double precision cond, double precision dmax, integer kl, integer ku, character pack, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) work, integer info)

ZLATMS

Purpose:

    ZLATMS generates random matrices with specified singular values
    (or hermitian with specified eigenvalues)
    for testing LAPACK programs.

    ZLATMS operates by applying the following sequence of
    operations:

      Set the diagonal to D, where D may be input or
         computed according to MODE, COND, DMAX, and SYM
         as described below.

      Generate a matrix with the appropriate band structure, by one
         of two methods:

      Method A:
          Generate a dense M x N matrix by multiplying D on the left
              and the right by random unitary matrices, then:

          Reduce the bandwidth according to KL and KU, using
              Householder transformations.

      Method B:
          Convert the bandwidth-0 (i.e., diagonal) matrix to a
              bandwidth-1 matrix using Givens rotations, 'chasing'
              out-of-band elements back, much as in QR; then convert
              the bandwidth-1 to a bandwidth-2 matrix, etc.  Note
              that for reasonably small bandwidths (relative to M and
              N) this requires less storage, as a dense matrix is not
              generated.  Also, for hermitian or symmetric matrices,
              only one triangle is generated.

      Method A is chosen if the bandwidth is a large fraction of the
          order of the matrix, and LDA is at least M (so a dense
          matrix can be stored.)  Method B is chosen if the bandwidth
          is small (< 1/2 N for hermitian or symmetric, < .3 N+M for
          non-symmetric), or LDA is less than M and not less than the
          bandwidth.

      Pack the matrix if desired. Options specified by PACK are:
         no packing
         zero out upper half (if hermitian)
         zero out lower half (if hermitian)
         store the upper half columnwise (if hermitian or upper
               triangular)
         store the lower half columnwise (if hermitian or lower
               triangular)
         store the lower triangle in banded format (if hermitian or
               lower triangular)
         store the upper triangle in banded format (if hermitian or
               upper triangular)
         store the entire matrix in banded format
      If Method B is chosen, and band format is specified, then the
         matrix will be generated in the band format, so no repacking
         will be necessary.
Parameters

M

          M is INTEGER
           The number of rows of A. Not modified.

N

          N is INTEGER
           The number of columns of A. N must equal M if the matrix
           is symmetric or hermitian (i.e., if SYM is not 'N')
           Not modified.

DIST

          DIST is CHARACTER*1
           On entry, DIST specifies the type of distribution to be used
           to generate the random eigen-/singular values.
           'U' => UNIFORM( 0, 1 )  ( 'U' for uniform )
           'S' => UNIFORM( -1, 1 ) ( 'S' for symmetric )
           'N' => NORMAL( 0, 1 )   ( 'N' for normal )
           Not modified.

ISEED

          ISEED is INTEGER array, dimension ( 4 )
           On entry ISEED specifies the seed of the random number
           generator. They should lie between 0 and 4095 inclusive,
           and ISEED(4) should be odd. The random number generator
           uses a linear congruential sequence limited to small
           integers, and so should produce machine independent
           random numbers. The values of ISEED are changed on
           exit, and can be used in the next call to ZLATMS
           to continue the same random number sequence.
           Changed on exit.

SYM

          SYM is CHARACTER*1
           If SYM='H', the generated matrix is hermitian, with
             eigenvalues specified by D, COND, MODE, and DMAX; they
             may be positive, negative, or zero.
           If SYM='P', the generated matrix is hermitian, with
             eigenvalues (= singular values) specified by D, COND,
             MODE, and DMAX; they will not be negative.
           If SYM='N', the generated matrix is nonsymmetric, with
             singular values specified by D, COND, MODE, and DMAX;
             they will not be negative.
           If SYM='S', the generated matrix is (complex) symmetric,
             with singular values specified by D, COND, MODE, and
             DMAX; they will not be negative.
           Not modified.

D

          D is DOUBLE PRECISION array, dimension ( MIN( M, N ) )
           This array is used to specify the singular values or
           eigenvalues of A (see SYM, above.)  If MODE=0, then D is
           assumed to contain the singular/eigenvalues, otherwise
           they will be computed according to MODE, COND, and DMAX,
           and placed in D.
           Modified if MODE is nonzero.

MODE

          MODE is INTEGER
           On entry this describes how the singular/eigenvalues are to
           be specified:
           MODE = 0 means use D as input
           MODE = 1 sets D(1)=1 and D(2:N)=1.0/COND
           MODE = 2 sets D(1:N-1)=1 and D(N)=1.0/COND
           MODE = 3 sets D(I)=COND**(-(I-1)/(N-1))
           MODE = 4 sets D(i)=1 - (i-1)/(N-1)*(1 - 1/COND)
           MODE = 5 sets D to random numbers in the range
                    ( 1/COND , 1 ) such that their logarithms
                    are uniformly distributed.
           MODE = 6 set D to random numbers from same distribution
                    as the rest of the matrix.
           MODE < 0 has the same meaning as ABS(MODE), except that
              the order of the elements of D is reversed.
           Thus if MODE is positive, D has entries ranging from
              1 to 1/COND, if negative, from 1/COND to 1,
           If SYM='H', and MODE is neither 0, 6, nor -6, then
              the elements of D will also be multiplied by a random
              sign (i.e., +1 or -1.)
           Not modified.

COND

          COND is DOUBLE PRECISION
           On entry, this is used as described under MODE above.
           If used, it must be >= 1. Not modified.

DMAX

          DMAX is DOUBLE PRECISION
           If MODE is neither -6, 0 nor 6, the contents of D, as
           computed according to MODE and COND, will be scaled by
           DMAX / max(abs(D(i))); thus, the maximum absolute eigen- or
           singular value (which is to say the norm) will be abs(DMAX).
           Note that DMAX need not be positive: if DMAX is negative
           (or zero), D will be scaled by a negative number (or zero).
           Not modified.

KL

          KL is INTEGER
           This specifies the lower bandwidth of the  matrix. For
           example, KL=0 implies upper triangular, KL=1 implies upper
           Hessenberg, and KL being at least M-1 means that the matrix
           has full lower bandwidth.  KL must equal KU if the matrix
           is symmetric or hermitian.
           Not modified.

KU

          KU is INTEGER
           This specifies the upper bandwidth of the  matrix. For
           example, KU=0 implies lower triangular, KU=1 implies lower
           Hessenberg, and KU being at least N-1 means that the matrix
           has full upper bandwidth.  KL must equal KU if the matrix
           is symmetric or hermitian.
           Not modified.

PACK

          PACK is CHARACTER*1
           This specifies packing of matrix as follows:
           'N' => no packing
           'U' => zero out all subdiagonal entries (if symmetric
                  or hermitian)
           'L' => zero out all superdiagonal entries (if symmetric
                  or hermitian)
           'C' => store the upper triangle columnwise (only if the
                  matrix is symmetric, hermitian, or upper triangular)
           'R' => store the lower triangle columnwise (only if the
                  matrix is symmetric, hermitian, or lower triangular)
           'B' => store the lower triangle in band storage scheme
                  (only if the matrix is symmetric, hermitian, or
                  lower triangular)
           'Q' => store the upper triangle in band storage scheme
                  (only if the matrix is symmetric, hermitian, or
                  upper triangular)
           'Z' => store the entire matrix in band storage scheme
                      (pivoting can be provided for by using this
                      option to store A in the trailing rows of
                      the allocated storage)

           Using these options, the various LAPACK packed and banded
           storage schemes can be obtained:
           GB                    - use 'Z'
           PB, SB, HB, or TB     - use 'B' or 'Q'
           PP, SP, HB, or TP     - use 'C' or 'R'

           If two calls to ZLATMS differ only in the PACK parameter,
           they will generate mathematically equivalent matrices.
           Not modified.

A

          A is COMPLEX*16 array, dimension ( LDA, N )
           On exit A is the desired test matrix.  A is first generated
           in full (unpacked) form, and then packed, if so specified
           by PACK.  Thus, the first M elements of the first N
           columns will always be modified.  If PACK specifies a
           packed or banded storage scheme, all LDA elements of the
           first N columns will be modified; the elements of the
           array which do not correspond to elements of the generated
           matrix are set to zero.
           Modified.

LDA

          LDA is INTEGER
           LDA specifies the first dimension of A as declared in the
           calling program.  If PACK='N', 'U', 'L', 'C', or 'R', then
           LDA must be at least M.  If PACK='B' or 'Q', then LDA must
           be at least MIN( KL, M-1) (which is equal to MIN(KU,N-1)).
           If PACK='Z', LDA must be large enough to hold the packed
           array: MIN( KU, N-1) + MIN( KL, M-1) + 1.
           Not modified.

WORK

          WORK is COMPLEX*16 array, dimension ( 3*MAX( N, M ) )
           Workspace.
           Modified.

INFO

          INFO is INTEGER
           Error code.  On exit, INFO will be set to one of the
           following values:
             0 => normal return
            -1 => M negative or unequal to N and SYM='S', 'H', or 'P'
            -2 => N negative
            -3 => DIST illegal string
            -5 => SYM illegal string
            -7 => MODE not in range -6 to 6
            -8 => COND less than 1.0, and MODE neither -6, 0 nor 6
           -10 => KL negative
           -11 => KU negative, or SYM is not 'N' and KU is not equal to
                  KL
           -12 => PACK illegal string, or PACK='U' or 'L', and SYM='N';
                  or PACK='C' or 'Q' and SYM='N' and KL is not zero;
                  or PACK='R' or 'B' and SYM='N' and KU is not zero;
                  or PACK='U', 'L', 'C', 'R', 'B', or 'Q', and M is not
                  N.
           -14 => LDA is less than M, or PACK='Z' and LDA is less than
                  MIN(KU,N-1) + MIN(KL,M-1) + 1.
            1  => Error return from DLATM1
            2  => Cannot scale to DMAX (max. sing. value is 0)
            3  => Error return from ZLAGGE, CLAGHE or CLAGSY
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 330 of file zlatms.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

The man page zlatms(3) is an alias of zlatms.f(3).

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK