TESTING/EIG/dchkgg.f(3)

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version 3.12.1 · 2 versions
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🎀 TESTING/EIG/dchkgg.f — Library Functions Manual 🎀

okay so bestie, this is the manual page for TESTING/EIG/dchkgg.f, and it's literally the vibe check for the whole nonsymmetric generalized eigenvalue problem squad. pull up a chair 💅

✨ NAME ✨

TESTING/EIG/dchkgg.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine dchkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif,
    thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z,
    alphr1, alphi1, beta1, alphr3, alphi3, beta3, evectl, evectr, work,
    lwork, llwork, result, info)

DCHKGG

🌸 Function/Subroutine Documentation 🌸

💖 subroutine dchkgg 💖

subroutine dchkgg (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, logical tstdif, double precision thrshn,
    integer nounit, double precision, dimension( lda, * ) a, integer lda,
    double precision, dimension( lda, * ) b, double precision, dimension(
    lda, * ) h, double precision, dimension( lda, * ) t, double precision,
    dimension( lda, * ) s1, double precision, dimension( lda, * ) s2, double
    precision, dimension( lda, * ) p1, double precision, dimension( lda, * )
    p2, double precision, dimension( ldu, * ) u, integer ldu, double
    precision, dimension( ldu, * ) v, double precision, dimension( ldu, * )
    q, double precision, dimension( ldu, * ) z, double precision, dimension(
    * ) alphr1, double precision, dimension( * ) alphi1, double precision,
    dimension( * ) beta1, double precision, dimension( * ) alphr3, double
    precision, dimension( * ) alphi3, double precision, dimension( * )
    beta3, double precision, dimension( ldu, * ) evectl, double precision,
    dimension( ldu, * ) evectr, double precision, dimension( * ) work,
    integer lwork, logical, dimension( * ) llwork, double precision,
    dimension( 15 ) result, integer info)

DCHKGG

🧚 Purpose 🧚

okay so DCHKGG checks the nonsymmetric generalized eigenvalue problem routines. that's the whole job. it's giving quality assurance queen 👑

DGGHRD factors A and B as U H V<sup>T</sup> and U T V<sup>T</sup>, where <sup>T</sup> means transpose, H is hessenberg, T is triangular and U and V are orthogonal.

DHGEQZ factors H and T as Q S Z<sup>T</sup> and Q P Z<sup>T</sup>, where P 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(1),beta(1)),...,(alpha(n),beta(n)), where alpha(j)=S(j,j) and beta(j)=P(j,j) — thus, w(j) = alpha(j)/beta(j) is a root of the generalized eigenvalue problem

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

and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent problem

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

DTGEVC computes the matrix L of left eigenvectors and the matrix R of right eigenvectors for the matrix pair ( S, P ). in the description below, l and r are left and right eigenvectors corresponding to the generalized eigenvalues (alpha,beta).

when DCHKGG is called, a number of matrix () and a number of matrix are specified. for each size () and each type of matrix, one matrix will be generated and used to test the nonsymmetric eigenroutines. for each matrix, 15 tests will be performed. the first twelve should be small — O(1). they will be compared with the threshold THRESH:

💀 ngl the original text has a couple of blank spots right there (the "matrix ()" bits) — that's not me dropping words, that's how the page ships. iykyk.
                 T
(1)   | A - U H V  | / ( |A| n ulp )

                 T
(2)   | B - U T V  | / ( |B| n ulp )

              T
(3)   | I - UU  | / ( n ulp )

              T
(4)   | I - VV  | / ( n ulp )

                 T
(5)   | H - Q S Z  | / ( |H| n ulp )

                 T
(6)   | T - Q P Z  | / ( |T| n ulp )

              T
(7)   | I - QQ  | / ( n ulp )

              T
(8)   | I - ZZ  | / ( n ulp )

(9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of

   | l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )

(10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
                          T
  | l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )

      where the eigenvectors l' are the result of passing Q to
      DTGEVC and back transforming (HOWMNY='B').

(11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of

      | (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )

(12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of

      | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )

      where the eigenvectors r' are the result of passing Z to
      DTGEVC and back transforming (HOWMNY='B').

the last three test ratios will usually be small, but there is no mathematical requirement that they be so 😌. they are therefore compared with THRESH only if TSTDIF is .TRUE..

(13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )

(14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )

(15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
           |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp

in addition, the normalization of L and R are checked, and compared with the threshold THRSHN.

🌈 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 are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type 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)

        t   t
(5)  ( J , J  )       (a pair of transposed Jordan blocks)

                                    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  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 )
          t   t
(16) U ( J , J ) V     where U and V are random orthogonal matrices.

(17) U ( T1, T2 ) V    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) U ( T1, T2 ) V    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
                       diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
                       s = machine precision.

(19) U ( T1, T2 ) V    diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )

                                                       N-5
(20) U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
                       diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )

(21) U ( T1, T2 ) V    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) U ( big*T1, small*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(23) U ( small*T1, big*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(24) U ( small*T1, small*T2 ) V  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(25) U ( big*T1, big*T2 ) V      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )

(26) U ( T1, T2 ) V     where T1 and T2 are random upper-triangular
                        matrices.

🍓 Parameters 🍓

NSIZES

!>          NSIZES is INTEGER
!>          The number of sizes of matrices to use.  If it is zero,
!>          DCHKGG does nothing.  It must be at least zero.
!>

NN

!>          NN is INTEGER array, dimension (NSIZES)
!>          An array containing the sizes to be used for the matrices.
!>          Zero values will be skipped.  The values must be at least
!>          zero.
!>

NTYPES

!>          NTYPES is INTEGER
!>          The number of elements in DOTYPE.   If it is zero, DCHKGG
!>          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.  This
!>          is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
!>          DOTYPE(MAXTYP+1) is .TRUE. .
!>

DOTYPE

🫶 and that's exactly where this page cuts off — DOTYPE's description is rolling in on the next part, so hold that thought bestie 💜
  • 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 — ISEED is INTEGER array, dimension (4). On entry ISEED specifies the seed of the random number generator. The array elements should be between 0 and 4095; if not they will be reduced mod 4096. Also, ISEED(4) must 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 DCHKGG to continue the same random number sequence. ✨
  • THRESH — THRESH is DOUBLE PRECISION. A test will count as [blank] if the [blank], 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.
  • TSTDIF — TSTDIF is LOGICAL. Specifies whether test ratios 13-15 will be computed and compared with THRESH.
    • = .FALSE.: Only test ratios 1-12 will be computed and tested. Ratios 13-15 will be set to zero.
    • = .TRUE.: All the test ratios 1-15 will be computed and tested.
  • THRSHN — THRSHN is DOUBLE PRECISION. Threshold for reporting eigenvector normalization error. If the normalization of any eigenvector differs from 1 by more than THRSHN*ulp, then a special error message will be printed. (This is handled separately from the other tests, since only a compiler or programming error should cause an error message, at least if THRSHN is at least 5--10.)
  • NOUNIT — NOUNIT 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 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 — LDA is INTEGER. The leading dimension of A, B, H, T, S1, P1, S2, and P2. It must be at least 1 and at least max( NN ).
  • B — B is 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.
  • H — H is DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper Hessenberg matrix computed from A by DGGHRD.
  • T — T is DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by DGGHRD.
  • S1 — S1 is DOUBLE PRECISION array, dimension (LDA, max(NN)). The Schur (block upper triangular) matrix computed from H by DHGEQZ when Q and Z are also computed.
  • S2 — S2 is DOUBLE PRECISION array, dimension (LDA, max(NN)). The Schur (block upper triangular) matrix computed from H by DHGEQZ when Q and Z are not computed.
  • P1 — P1 is DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by DHGEQZ when Q and Z are also computed.
  • P2 — P2 is DOUBLE PRECISION array, dimension (LDA, max(NN)). The upper triangular matrix computed from T by DHGEQZ when Q and Z are not computed.
  • U — U is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by DGGHRD.
  • LDU — LDU is INTEGER. The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR. It must be at least 1 and at least max( NN ).
  • V — V is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (right) orthogonal matrix computed by DGGHRD.
  • Q — Q is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by DHGEQZ.
  • Z — Z is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (left) orthogonal matrix computed by DHGEQZ.
  • ALPHR1 — ALPHR1 is DOUBLE PRECISION array, dimension (max(NN)).
  • ALPHI1 — ALPHI1 is DOUBLE PRECISION array, dimension (max(NN)).
  • BETA1 — BETA1 is DOUBLE PRECISION array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by DHGEQZ when Q, Z, and the full Schur matrices are computed. On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th generalized eigenvalue of the matrices in A and B. 🔥
  • ALPHR3 — ALPHR3 is DOUBLE PRECISION array, dimension (max(NN)).
  • ALPHI3 — ALPHI3 is DOUBLE PRECISION array, dimension (max(NN)).
  • BETA3 — BETA3 is DOUBLE PRECISION array, dimension (max(NN)).
  • EVECTL — EVECTL is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (block lower triangular) left eigenvector matrix for the matrices in S1 and P1. (See DTGEVC for the format.)
  • EVECTR — EVECTR is DOUBLE PRECISION array, dimension (LDU, max(NN)). The (block upper triangular) right eigenvector matrix for the matrices in S1 and P1. (See DTGEVC for the format.)
  • WORK — WORK is DOUBLE PRECISION array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. The number of entries in WORK. This must be at least max( 2 * N**2, 6*N, 1 ), for all N=NN(j).
  • LLWORK — LLWORK is LOGICAL array, dimension (max(NN)).
  • RESULT — RESULT is DOUBLE PRECISION array, dimension (15). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow.
  • 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. 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 506 of file dchkgg.f.

Author

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💖 that’s the whole page, bestie — source/version is LAPACK Version 3.12.0, the page tag is TESTING/EIG/dchkgg.f(3), and the footer date isn’t actually spelled out in this snippet, so I’m not gonna invent one. Slay responsibly. 🎀

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