SDRGVX(3)

📦 version 3.12.0, in opensuse-leap 16.0

✨ NAME ✨

Okay so bestie, this page is from the Library Functions Manual — the header path is /home/abui...IG/sdrgvx.f(3) — and the NAME section is literally the full path:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/sdrgvx.f

That's the whole vibe. Slay. 💅

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine SDRGVX (nsize, thresh, nin, nout, a, lda, b, ai, bi, alphar,
    alphai, beta, vl, vr, ilo, ihi, lscale, rscale, s, stru, dif,
    diftru, work, lwork, iwork, liwork, result, bwork, info)

SDRGVX — that's the subroutine name, no cap. ✨

💅 Function/Subroutine Documentation 💅

Here's the full signature, copied exactly because we do not mess with Fortran:

subroutine SDRGVX (integer nsize, real thresh, integer nin, integer nout,
    real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b,
    real, dimension( lda, * ) ai, real, dimension( lda, * ) bi, real,
    dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * )
    beta, real, dimension( lda, * ) vl, real, dimension( lda, * ) vr,
    integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension(
    * ) rscale, real, dimension( * ) s, real, dimension( * ) stru, real,
    dimension( * ) dif, real, dimension( * ) diftru, real, dimension( * )
    work, integer lwork, integer, dimension( * ) iwork, integer liwork,
    real, dimension( 4 ) result, logical, dimension( * ) bwork, integer
    info)

SDRGVX — again, iconic. 👑

🎀 Purpose 🎀

Okay so SDRGVX is the vibe-checker for the nonsymmetric generalized eigenvalue problem expert driver SGGEVX. Fr fr.

SGGEVX computes the generalized eigenvalues, (optionally) the left and/or right eigenvectors, (optionally) computes a balancing transformation to improve the conditioning, and (optionally) reciprocal condition numbers for the eigenvalues and eigenvectors. It's giving multitasking main character energy. ✨

When SDRGVX is called with NSIZE > 0, two types of test matrix pairs are generated by the subroutine SLATM6 and test the driver SGGEVX. The test matrices have the known exact condition numbers for eigenvalues. For the condition numbers of the eigenvectors corresponding the first and last eigenvalues are also know ``exactly'' (see SLATM6). (Yes, "know" is what the page says, bestie — I'm not fixing it. 🫶)

For each matrix pair, the following tests will be performed and compared with the threshold THRESH.

  1. max over all left eigenvalue/-vector pairs (beta/alpha,l) of
| l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) )

where l**H is the conjugate transpose of l.

  1. max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
  1. The condition number S(i) of eigenvalues computed by SGGEVX differs less than a factor THRESH from the exact S(i) (see SLATM6).
  2. DIF(i) computed by STGSNA differs less than a factor 10*THRESH from the exact value (for the 1st and 5th vectors only).

💖 Test Matrices 💖

Two kinds of test matrix pairs

(A, B) = inverse(YH) * (Da, Db) * inverse(X)

are used in the tests:

1: Da = 1+a   0    0    0    0    Db = 1   0   0   0   0
         0   2+a   0    0    0         0   1   0   0   0
         0    0   3+a   0    0         0   0   1   0   0
         0    0    0   4+a   0         0   0   0   1   0
         0    0    0    0   5+a ,      0   0   0   0   1 , and
2: Da =  1   -1    0    0    0    Db = 1   0   0   0   0
         1    1    0    0    0         0   1   0   0   0
         0    0    1    0    0         0   0   1   0   0
         0    0    0   1+a  1+b        0   0   0   1   0
         0    0    0  -1-b  1+a ,      0   0   0   0   1 .

In both cases the same inverse(YH) and inverse(X) are used to compute (A, B), giving the exact eigenvectors to (A,B) as (YH, X):

YH:  =  1    0   -y    y   -y    X =  1   0  -x  -x   x
        0    1   -y    y   -y         0   1   x  -x  -x
        0    0    1    0    0         0   0   1   0   0
        0    0    0    1    0         0   0   0   1   0
        0    0    0    0    1,        0   0   0   0   1 , where

a, b, x and y will have all values independently of each other from { sqrt(sqrt(ULP)), 0.1, 1, 10, 1/sqrt(sqrt(ULP)) }.

🌸 Parameters 🌸

  • NSIZE — NSIZE is INTEGER. The number of sizes of matrices to use. NSIZE must be at least zero. If it is zero, no randomly generated matrices are tested, but any test matrices read from NIN will be tested.
  • THRESH — THRESH is REAL. A test will count as if the , computed as described above, exceeds THRESH. Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, e.g., 10 or 100. In particular, it should not depend on the precision (single vs. double) or the size of the matrix. It must be at least zero.
  • NIN — NIN is INTEGER. The FORTRAN unit number for reading in the data file of problems to solve.
  • NOUT — NOUT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0.)
  • A — A is REAL array, dimension (LDA, NSIZE). Used to hold the matrix whose eigenvalues are to be computed. On exit, A contains the last matrix actually used.
  • LDA — LDA is INTEGER. The leading dimension of A, B, AI, BI, Ao, and Bo. It must be at least 1 and at least NSIZE.
  • B — B is REAL array, dimension (LDA, NSIZE). Used to hold the matrix whose eigenvalues are to be computed. On exit, B contains the last matrix actually used.
  • AI — AI is REAL array, dimension (LDA, NSIZE). Copy of A, modified by SGGEVX.
  • BI — BI is REAL array, dimension (LDA, NSIZE). Copy of B, modified by SGGEVX.
  • ALPHAR — ALPHAR is REAL array, dimension (NSIZE).
  • ALPHAI — ALPHAI is REAL array, dimension (NSIZE).
  • BETA — BETA is REAL array, dimension (NSIZE). On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.
  • VL — VL is REAL array, dimension (LDA, NSIZE). VL holds the left eigenvectors computed by SGGEVX.
  • VR — VR is REAL array, dimension (LDA, NSIZE). VR holds the right eigenvectors computed by SGGEVX.
  • ILO — ILO is INTEGER.
  • IHI — IHI is INTEGER.
  • LSCALE — LSCALE is REAL array, dimension (N).
  • RSCALE — RSCALE is REAL array, dimension (N).
  • S — S is REAL array, dimension (N).
  • STRU — STRU is REAL array, dimension (N).
  • DIF — DIF is REAL array, dimension (N).
  • DIFTRU — DIFTRU is REAL array, dimension (N).
  • WORK — WORK is REAL array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. Leading dimension of WORK. LWORK >= 2*N*N+12*N+16.
  • IWORK — IWORK is INTEGER array, dimension (LIWORK).
  • LIWORK — LIWORK is INTEGER. Leading dimension of IWORK. Must be at least N+6.
  • RESULT — RESULT is REAL array, dimension (4).
  • BWORK — BWORK is LOGICAL array, dimension (N).
  • INFO — INFO is INTEGER.
    • = 0: successful exit
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: A routine returned an error code.

🫶 Author 🫶

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

Definition at line 297 of file sdrgvx.f.

🖋️ Author 🖋️

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Cute sign-off: this page is LAPACK Version 3.12.0, footer path /home/abui...IG/sdrgvx.f(3) — and no date is printed in that footer, bestie, just version + path. 💖✨

📦 where this page comes from