TESTING/MATGEN/zlatmr.f(3)

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version 3.12.1 · 2 versions
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💖 NAME 💖

okay so this is the TESTING/MATGEN/zlatmr.f(3) page from the Library Functions Manual, and it's literally all about ZLATMR. bestie, we're in the random-matrix generator corner of LAPACK testing and it's giving main character energy. ✨

TESTING/MATGEN/zlatmr.f

🌈 SYNOPSIS 🌈

✨ Functions/Subroutines ✨

subroutine zlatmr (m, n, dist, iseed, sym, d, mode, cond, dmax, rsign,
           grade, dl, model, condl, dr, moder, condr, pivtng, ipivot, kl, ku,
           sparse, anorm, pack, a, lda, iwork, info)

ZLATMR

🦋 Function/Subroutine Documentation 🦋

subroutine zlatmr (integer m, integer n, character dist, integer, dimension(
       4 ) iseed, character sym, complex*16, dimension( * ) d, integer mode,
       double precision cond, complex*16 dmax, character rsign, character
       grade, complex*16, dimension( * ) dl, integer model, double precision
       condl, complex*16, dimension( * ) dr, integer moder, double precision
       condr, character pivtng, integer, dimension( * ) ipivot, integer kl,
       integer ku, double precision sparse, double precision anorm, character
       pack, complex*16, dimension( lda, * ) a, integer lda, integer,
       dimension( * ) iwork, integer info)

ZLATMR

💅 Purpose 💅

okay so ZLATMR generates random matrices of various types for testing LAPACK programs. no cap, that's the whole gig. 🫶

ZLATMR operates by applying the following sequence of operations:

  • Generate a matrix A with random entries of distribution DIST which is symmetric if SYM='S', Hermitian if SYM='H', and nonsymmetric if SYM='N'.
  • Set the diagonal to D, where D may be input or computed according to MODE, COND, DMAX and RSIGN as described below.
  • Grade the matrix, if desired, from the left and/or right as specified by GRADE. The inputs DL, MODEL, CONDL, DR, MODER and CONDR also determine the grading as described below.
  • Permute, if desired, the rows and/or columns as specified by PIVTNG and IPIVOT.
  • Set random entries to zero, if desired, to get a random sparse matrix as specified by SPARSE.
  • Make A a band matrix, if desired, by zeroing out the matrix outside a band of lower bandwidth KL and upper bandwidth KU.
  • Scale A, if desired, to have maximum entry ANORM.
  • Pack the matrix if desired. Options specified by PACK are:
    • no packing
    • zero out upper half (if symmetric or Hermitian)
    • zero out lower half (if symmetric or Hermitian)
    • store the upper half columnwise (if symmetric or Hermitian or square upper triangular)
    • store the lower half columnwise (if symmetric or Hermitian or square lower triangular)
      • same as upper half rowwise if symmetric
      • same as conjugate upper half rowwise if Hermitian
    • store the lower triangle in banded format (if symmetric or Hermitian)
    • store the upper triangle in banded format (if symmetric or Hermitian)
    • store the entire matrix in banded format
Note: If two calls to ZLATMR differ only in the PACK parameter, they will generate mathematically equivalent matrices. If two calls to ZLATMR both have full bandwidth (KL = M-1 and KU = N-1), and differ only in the PIVTNG and PACK parameters, then the matrices generated will differ only in the order of the rows and/or columns, and otherwise contain the same data. This consistency cannot be and is not maintained with less than full bandwidth. 💀

🎀 Parameters 🎀

🍓 M 🍓

M is INTEGER

Number of rows of A. Not modified.

🍓 N 🍓

N is INTEGER

Number of columns of A. Not modified.

🍓 DIST 🍓

DIST is CHARACTER*1

On entry, DIST specifies the type of distribution to be used to generate a random matrix. the vibes are:

  • 'U' => real and imaginary parts are independent UNIFORM( 0, 1 ) ( 'U' for uniform )
  • 'S' => real and imaginary parts are independent UNIFORM( -1, 1 ) ( 'S' for symmetric )
  • 'N' => real and imaginary parts are independent NORMAL( 0, 1 ) ( 'N' for normal )
  • 'D' => uniform on interior of unit disk ( 'D' for disk )

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 ZLATMR to continue the same random number sequence. Changed on exit. 🔥

🍓 SYM 🍓

SYM is CHARACTER*1

  • If SYM='S', generated matrix is symmetric.
  • If SYM='H', generated matrix is Hermitian.
  • If SYM='N', generated matrix is nonsymmetric.

Not modified.

🍓 D 🍓

D is COMPLEX*16 array, dimension (min(M,N))

On entry this array specifies the diagonal entries of the diagonal of A. D may either be specified on entry, or set according to MODE and COND as described below. If the matrix is Hermitian, the real part of D will be taken. May be changed on exit if MODE is nonzero.

🍓 MODE 🍓

MODE is INTEGER

On entry describes how D is to be used:

  • 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. Not modified.

🍓 COND 🍓

COND is DOUBLE PRECISION

On entry, used as described under MODE above. If used, it must be >= 1. Not modified.

🍓 DMAX 🍓

DMAX is COMPLEX*16

If MODE neither -6, 0 nor 6, the diagonal is scaled by DMAX / max(abs(D(i))), so that maximum absolute entry of diagonal is abs(DMAX). If DMAX is complex (or zero), diagonal will be scaled by a complex number (or zero). 👑

🍓 RSIGN 🍓

RSIGN is CHARACTER*1

If MODE neither -6, 0 nor 6, specifies sign of diagonal as follows:

  • 'T' => diagonal entries are multiplied by a random complex number uniformly distributed with absolute value 1
  • 'F' => diagonal unchanged

Not modified.

🍓 GRADE 🍓

GRADE is CHARACTER*1

Specifies grading of matrix as follows:

  • 'N' => no grading
  • 'L' => matrix premultiplied by diag( DL ) (only if matrix nonsymmetric)
  • 'R' => matrix postmultiplied by diag( DR ) (only if matrix nonsymmetric)
  • 'B' => matrix premultiplied by diag( DL ) and postmultiplied by diag( DR ) (only if matrix nonsymmetric)
  • 'H' => matrix premultiplied by diag( DL ) and postmultiplied by diag( CONJG(DL) ) (only if matrix Hermitian or nonsymmetric)
  • 'S' => matrix premultiplied by diag( DL ) and postmultiplied by diag( DL ) (only if matrix symmetric or nonsymmetric)
  • 'E' => matrix premultiplied by diag( DL ) and postmultiplied by inv( diag( DL ) ) ( 'S' for similarity ) (only if matrix nonsymmetric)
Note: if GRADE='S', then M must equal N.

Not modified.

🍓 DL 🍓

DL is COMPLEX*16 array, dimension (M)

If MODEL=0, then on entry this array specifies the diagonal entries of a diagonal matrix used as described under GRADE above. If MODEL is not zero, then DL will be set according to MODEL and CONDL, analogous to the way D is set according to MODE and COND (except there is no DMAX parameter for DL). If GRADE='E', then DL cannot have zero entries. Not referenced if GRADE = 'N' or 'R'. Changed on exit.

🍓 MODEL 🍓

MODEL is INTEGER

This specifies how the diagonal array DL is to be computed, just as MODE specifies how D is to be computed. Not modified.

🍓 CONDL 🍓

CONDL is DOUBLE PRECISION

When MODEL is not zero, this specifies the condition number of the computed DL. Not modified.

🌸 DR 🌸

  • DR — okay bestie, this is a COMPLEX*16 array, dimension (N). If MODER=0, then on entry this array specifies the diagonal entries of a diagonal matrix used as described under GRADE above. If MODER is not zero, then DR will be set according to MODER and CONDR, analogous to the way D is set according to MODE and COND (except there is no DMAX parameter for DR). Not referenced if GRADE = 'N', 'L', 'H' or 'S'. Changed on exit.
  • MODER — INTEGER. This specifies how the diagonal array DR is to be computed, just as MODE specifies how D is to be computed. Not modified.
  • CONDR — DOUBLE PRECISION. When MODER is not zero, this specifies the condition number of the computed DR. Not modified.
  • PIVTNG — CHARACTER*1. On entry specifies pivoting permutations as follows:
    • 'N' or ' ' => none.
    • 'L' => left or row pivoting (matrix must be nonsymmetric).
    • 'R' => right or column pivoting (matrix must be nonsymmetric).
    • 'B' or 'F' => both or full pivoting, i.e., on both sides. In this case, M must equal N.

If two calls to ZLATMR both have full bandwidth (KL = M-1 and KU = N-1), and differ only in the PIVTNG and PACK parameters, then the matrices generated will differ only in the order of the rows and/or columns, and otherwise contain the same data. This consistency cannot be maintained with less than full bandwidth.

  • IPIVOT — INTEGER array, dimension (N or M). This array specifies the permutation used. After the basic matrix is generated, the rows, columns, or both are permuted. If, say, row pivoting is selected, ZLATMR starts with the last row and interchanges the M-th and IPIVOT(M)-th rows, then moves to the next-to-last row, interchanging the (M-1)-th and the IPIVOT(M-1)-th rows, and so on. The permutation is (1 IPIVOT(1)) (2 IPIVOT(2)) ... (M IPIVOT(M)) where the rightmost cycle is applied first. This is the inverse of the effect of pivoting in LINPACK. The idea is that factoring (with pivoting) an identity matrix which has been inverse-pivoted in this way should result in a pivot vector identical to IPIVOT. Not referenced if PIVTNG = 'N'. Not modified.
  • KL — INTEGER. On entry specifies the lower bandwidth of the matrix. For example, KL=0 implies upper triangular, KL=1 implies upper Hessenberg, and KL at least M-1 implies the matrix is not banded. Must equal KU if matrix is symmetric or Hermitian. Not modified.
  • KU — INTEGER. On entry specifies the upper bandwidth of the matrix. For example, KU=0 implies lower triangular, KU=1 implies lower Hessenberg, and KU at least N-1 implies the matrix is not banded. Must equal KL if matrix is symmetric or Hermitian. Not modified.
  • SPARSE — DOUBLE PRECISION. On entry specifies the sparsity of the matrix if a sparse matrix is to be generated. SPARSE should lie between 0 and 1. To generate a sparse matrix, for each matrix entry a uniform ( 0, 1 ) random number x is generated and compared to SPARSE; if x is larger the matrix entry is unchanged and if x is smaller the entry is set to zero. Thus on the average a fraction SPARSE of the entries will be set to zero. Not modified.
  • ANORM — DOUBLE PRECISION. On entry specifies maximum entry of output matrix (output matrix will by multiplied by a constant so that its largest absolute entry equal ANORM) if ANORM is nonnegative. If ANORM is negative no scaling is done. Not modified.
  • PACK — CHARACTER*1. On entry specifies packing of the 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 matrix symmetric or Hermitian or square upper triangular)
    • 'R' => store the lower triangle columnwise (only if matrix symmetric or Hermitian or square lower triangular) (same as upper half rowwise if symmetric) (same as conjugate upper half rowwise if Hermitian)
    • 'B' => store the lower triangle in band storage scheme (only if matrix symmetric or Hermitian)
    • 'Q' => store the upper triangle in band storage scheme (only if matrix symmetric or Hermitian)
    • '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, HB or TB — use 'B' or 'Q'
  • PP, HP or TP — use 'C' or 'R'

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

  • A — COMPLEX*16 array, dimension (LDA,N). On exit A is the desired test matrix. Only those entries of A which are significant on output will be referenced (even if A is in packed or band storage format). The 'unoccupied corners' of A in band format will be zeroed out.
  • LDA — INTEGER. On entry LDA specifies the first dimension of A as declared in the calling program.
    • If PACK='N', 'U' or 'L', LDA must be at least max ( 1, M ).
    • If PACK='C' or 'R', LDA must be at least 1.
    • If PACK='B', or 'Q', LDA must be MIN ( KU+1, N ).
    • If PACK='Z', LDA must be at least KUU+KLL+1, where KUU = MIN ( KU, N-1 ) and KLL = MIN ( KL, M-1 ).

Not modified.

  • IWORK — INTEGER array, dimension (N or M). Workspace. Not referenced if PIVTNG = 'N'. Changed on exit.
  • INFO — INTEGER. Error parameter on exit:
    • 0 => normal return
    • -1 => M negative or unequal to N and SYM='S' or 'H'
    • -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 => MODE neither -6, 0 nor 6 and RSIGN illegal string
    • -11 => GRADE illegal string, or GRADE='E' and M not equal to N, or GRADE='L', 'R', 'B', 'S' or 'E' and SYM = 'H', or GRADE='L', 'R', 'B', 'H' or 'E' and SYM = 'S'
    • -12 => GRADE = 'E' and DL contains zero
    • -13 => MODEL not in range -6 to 6 and GRADE= 'L', 'B', 'H', 'S' or 'E'
    • -14 => CONDL less than 1.0, GRADE='L', 'B', 'H', 'S' or 'E', and MODEL neither -6, 0 nor 6
    • -16 => MODER not in range -6 to 6 and GRADE= 'R' or 'B'
    • -17 => CONDR less than 1.0, GRADE='R' or 'B', and MODER neither -6, 0 nor 6
    • -18 => PIVTNG illegal string, or PIVTNG='B' or 'F' and M not equal to N, or PIVTNG='L' or 'R' and SYM='S' or 'H'
    • -19 => IPIVOT contains out of range number and PIVTNG not equal to 'N'
    • -20 => KL negative
    • -21 => KU negative, or SYM='S' or 'H' and KU not equal to KL
    • -22 => SPARSE not in range 0. to 1.
    • -24 => PACK illegal string, or PACK='U', 'L', 'B' or 'Q' and SYM='N', or PACK='C' and SYM='N' and either KL not equal to 0 or N not equal to M, or PACK='R' and SYM='N', and either KU not equal to 0 or N not equal to M
    • -26 => LDA too small
    • 1 => Error return from ZLATM1 (computing D)
    • 2 => Cannot scale diagonal to DMAX (max. entry is 0)
    • 3 => Error return from ZLATM1 (computing DL)
    • 4 => Error return from ZLATM1 (computing DR)
    • 5 => ANORM is positive, but matrix constructed prior to attempting to scale it to have norm ANORM, is zero

💖 Author 💖

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

Definition at line 486 of file zlatmr.f.

Author

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