TESTING/MATGEN/zlatmt.f(3)

📦
version 3.12.1 · 2 versions
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🌸 NAME 🌸

TESTING/MATGEN/zlatmt.f — the ZLATMT routine, from the Library Functions Manual, part 1, bestie 💖

✨ SYNOPSIS ✨

Functions/Subroutines

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

ZLATMT

💖 Function/Subroutine Documentation 💖

subroutine zlatmt (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 rank,
    integer kl, integer ku, character pack, complex*16, dimension( lda, * )
    a, integer lda, complex*16, dimension( * ) work, integer info)

ZLATMT

Purpose

Okay so ZLATMT generates random matrices with specified singular values (or hermitian with specified eigenvalues) for testing LAPACK programs. 💅 It 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, 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 ZLATMT 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:RANK)=1.0/COND
    • MODE = 2 sets D(1:RANK-1)=1 and D(RANK)=1.0/COND
    • MODE = 3 sets D(I)=COND**(-(I-1)/(RANK-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.
  • RANK — RANK is INTEGER. The rank of matrix to be generated for modes 1,2,3 only. D( RANK+1:N ) = 0. 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 — okay so this is the packing selector, and it decides how the matrix gets tucked into storage. Here are the exact flavors, bestie:

  • '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 ZLATMT differ only in the PACK parameter, they will generate mathematically equivalent matrices. Not modified.

A is a COMPLEX*16 array, dimension ( LDA, N ). On exit, A is the desired test matrix. It 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 is INTEGER — it 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 is a COMPLEX*16 array, dimension ( 3*MAX( N, M ) ) — workspace. Modified.

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 DLATM7
  • 2 => Cannot scale to DMAX (max. sing. value is 0)
  • 3 => Error return from ZLAGGE, ZLAGHE or ZLAGSY

👑 Author 👑

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 338 of file zlatmt.f.

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