TESTING/EIG/ddrges3.f(3)

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💅 TESTING/EIG/ddrges3.f — Library Functions Manual ✨

okay so bestie, welcome to the group chat for TESTING/EIG/ddrges3.f. This page is giving serious numerical linear algebra energy, and I'm obsessed. It's the manual page for the test routine that checks the nonsymmetric generalized eigenvalue (Schur form) driver, so grab a strawberry 🍓 and let's go.

🎀 NAME 🎀

TESTING/EIG/ddrges3.f

✨ SYNOPSIS ✨

Functions/Subroutines

  • subroutine ddrges3 (...) — the one and only, DDRGES3.
subroutine ddrges3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit,
    a, lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork,
    result, bwork, info)

DDRGES3

🦋 Function/Subroutine Documentation 🦋

okay, deep dive time, bestie.

subroutine ddrges3 (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer nounit, double precision, dimension(
    lda, * ) a, integer lda, double precision, dimension( lda, * ) b, double
    precision, dimension( lda, * ) s, double precision, dimension( lda, * )
    t, double precision, dimension( ldq, * ) q, integer ldq, double
    precision, dimension( ldq, * ) z, double precision, dimension( * )
    alphar, double precision, dimension( * ) alphai, double precision,
    dimension( * ) beta, double precision, dimension( * ) work, integer
    lwork, double precision, dimension( 13 ) result, logical, dimension( * )
    bwork, integer info)

DDRGES3

🎀 Purpose

okay so DDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form) problem driver DGGES3.

DGGES3 factors A and B as Q S Z' and Q T Z', where ' means transpose, T is upper triangular, S is in generalized Schur form (block upper triangular, with 1x1 and 2x2 blocks on the diagonal, the 2x2 blocks corresponding to complex conjugate pairs of generalized eigenvalues), and Q and Z are orthogonal. It also computes the generalized eigenvalues (alpha(j),beta(j)), j=1,...,n, Thus, w(j) = alpha(j)/beta(j) is a root of the characteristic equation

det( A - w(j) B ) = 0

Optionally it also reorder the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms.

When DDRGES3 is called, a number of matrix () and a number of matrix are specified. For each size () and each TYPE of matrix, a pair of matrices (A, B) will be generated and used for testing. For each matrix pair, the following 13 tests will be performed and compared with the threshold THRESH except the tests (5), (11) and (13).

bestie, the source text literally has a couple of blanks here — I'm keeping them because inventing details is not the vibe.
  • (1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
  • (2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
  • (3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
  • (4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
  • (5) if A is in Schur form (i.e. quasi-triangular form) (no sorting of eigenvalues)
  • (6) if eigenvalues = diagonal blocks of the Schur form (S, T), i.e., test the maximum over j of D(j) where:
if alpha(j) is real:
              |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
      D(j) = ------------------------ + -----------------------
             max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
                            | det( s S - w T ) |
      D(j) = ---------------------------------------------------
             ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the j-th and j+1-th eigenvalues. (no sorting of eigenvalues)

  • (7) | (A,B) - Q (S,T) Z' | / ( | (A,B) | n ulp ) (with sorting of eigenvalues).
  • (8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
  • (9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
  • (10) if A is in Schur form (i.e. quasi-triangular form) (with sorting of eigenvalues).
  • (11) if eigenvalues = diagonal blocks of the Schur form (S, T), i.e. test the maximum over j of D(j) where:
if alpha(j) is real:
              |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
      D(j) = ------------------------ + -----------------------
             max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

if alpha(j) is complex:
                            | det( s S - w T ) |
      D(j) = ---------------------------------------------------
             ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )

and S and T are here the 2 x 2 diagonal blocks of S and T corresponding to the j-th and j+1-th eigenvalues. (with sorting of eigenvalues).

  • (12) if sorting worked and SDIM is the number of eigenvalues which were SELECTed.

🧪 Test Matrices

The sizes of the test matrices are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size. The (blank) are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type (blank) will be generated. Currently, the list of possible types is:

  • (1) ( 0, 0 ) — a pair of zero matrices.
  • (2) ( I, 0 ) — an identity and a zero matrix.
  • (3) ( 0, I ) — an identity and a zero matrix.
  • (4) ( I, I ) — a pair of identity matrices.
  • (5) — a pair of transposed Jordan blocks:
        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)
  • (6) ( X, Y ) where:
                                    t                ( I   0  )
(6)  ( X, Y )         where  X = ( J   0  )  and Y = (      t )
                                 ( 0   I  )          ( 0   J  )
                      and I is a k x k identity and J a (k+1)x(k+1)
                      Jordan block; k=(N-1)/2
  • (7) ( D, I ) — where D is diag( 0, 1,..., N-1 ) (a diagonal matrix with those diagonal entries.)
  • (8) ( I, D )
  • (9) ( big*D, small*I ) — where (the source has a blank here) is near overflow and small=1/big
  • (10) ( small*D, big*I )
  • (11) ( big*I, small*D )
  • (12) ( small*I, big*D )
  • (13) ( big*D, big*I )
  • (14) ( small*D, small*I )
  • (15) ( D1, D2 ) — where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
  • (16) — where Q and Z are random orthogonal matrices:
          t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.
  • (17) Q ( T1, T2 ) Z — where T1 and T2 are upper triangular matrices with random O(1) entries above the diagonal and diagonal entries diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, N-3, N-4,..., 1, 0, 0 )
  • (18) Q ( T1, T2 ) Z — diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), and s = machine precision.
  • (19) Q ( T1, T2 ) Z — diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ), diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
  • (20) Q ( T1, T2 ) Z:
                                                       N-5
(20) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
  • (21) Q ( T1, T2 ) Z — diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 ), diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ), where r1,..., r(N-4) are random.
  • (22) Q ( big*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (23) Q ( small*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (24) Q ( small*T1, small*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (25) Q ( big*T1, big*T2 ) Z — diag(T1) = ( 0, 0, 1, ..., N-3, 0 ), diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  • (26) Q ( T1, T2 ) Z — where T1 and T2 are random upper-triangular matrices.

🌸 Parameters

NSIZES

NSIZES is INTEGER. It's the number of sizes of matrices to use. If it is zero, DDRGES3 does nothing. NSIZES >= 0.

NN

NN is INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.

NTYPES

NTYPES is INTEGER. The number of elements in DOTYPE. If it is zero, DDRGES3 does nothing. It must be at least zero. If it is MAXTYP+1 and NSIZES is 1, then an additional type, MAXTYP+1 is defined, which is to use whatever matrix is in A on input. This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .

DOTYPE

DOTYPE is LOGICAL array, dimension (NTYPES). If DOTYPE(j) is .TRUE., then for each size in NN a matrix of that size and of type j will be generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP will not be generated. If NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) will be ignored.

ISEED

and that's exactly where this part of the source cuts off, bestie — part 2 has the rest.
  • ISEED — INTEGER array, dimension (4). On entry, ISEED is the seed for the random number generator, bestie. The array elements should be between 0 and 4095; if they aren’t, they get reduced mod 4096. Also, ISEED(4) must be odd. The generator uses a linear congruential sequence limited to small integers, so it should produce machine-independent random numbers. On exit, the values of ISEED are changed, and you can use them in the next call to DDRGES3 to continue the same random number sequence. ✨
  • THRESH — DOUBLE PRECISION. A test counts as a fail if the error, computed as described above, exceeds THRESH. 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 precision (single vs. double) or the size of the matrix. THRESH >= 0.
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IINFO not equal to 0).
  • A — DOUBLE PRECISION array, dimension(LDA, max(NN)). Used to hold the original A matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE..
  • LDA — INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ).
  • B — DOUBLE PRECISION array, dimension(LDA, max(NN)). Used to hold the original B matrix. Used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE..
  • S — DOUBLE PRECISION array, dimension (LDA, max(NN)). The Schur form matrix computed from A by DGGES3. On exit, S contains the Schur form matrix corresponding to the matrix in A.
  • T — DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by DGGES3.
  • Q — DOUBLE PRECISION array, dimension (LDQ, max(NN)). The (left) orthogonal matrix computed by DGGES3.
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ).
  • Z — DOUBLE PRECISION array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by DGGES3.
  • ALPHAR — DOUBLE PRECISION array, dimension (max(NN)).
  • ALPHAI — DOUBLE PRECISION array, dimension (max(NN)).
  • BETA — DOUBLE PRECISION array, dimension (max(NN)). Together with ALPHAR and ALPHAI, these are the generalized eigenvalues of (A,B) computed by DGGES3. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B.
  • WORK — DOUBLE PRECISION array, dimension (LWORK).
  • LWORK — INTEGER. The dimension of the array WORK. LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest matrix dimension.
  • RESULT — DOUBLE PRECISION array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
  • BWORK — LOGICAL array, dimension (N).
  • INFO — INTEGER.
    • = 0: successful exit.
    • < 0: if INFO = -i, the i-th argument had an illegal value.
    • > 0: a routine returned an error code. INFO is the absolute value of the INFO value returned.

👑 Author 👑

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

Definition at line 399 of file ddrges3.f.

🌈 Author 🌈

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and that’s the whole page, bestie 🫶 — footer vibe check: source TESTING/EIG/ddrges3.f(3), LAPACK Version 3.12.0; no date is printed on this footer, so don’t let anyone gaslight you into inventing one 💅✨

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