TESTING/EIG/zdrges.f(3)

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💅 TESTING/EIG/zdrges.f 💅

okay bestie, this is the TESTING/EIG/zdrges.f(3) page from the Library Functions Manual — and it's giving “LAPACK test driver for the nonsymmetric generalized eigenvalue problem” in the best way. Let's read it like a group chat, no cap.

🌸 NAME 🌸

TESTING/EIG/zdrges.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

subroutine zdrges (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result,
    bwork, info)
    ZDRGES

💅 Function/Subroutine Documentation 💅

🌟 subroutine zdrges 🌟

subroutine zdrges (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer nounit, complex*16, dimension( lda, * )
    a, integer lda, complex*16, dimension( lda, * ) b, complex*16,
    dimension( lda, * ) s, complex*16, dimension( lda, * ) t, complex*16,
    dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldq, * ) z,
    complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta,
    complex*16, dimension( * ) work, integer lwork, double precision,
    dimension( * ) rwork, double precision, dimension( 13 ) result, logical,
    dimension( * ) bwork, integer info)
    ZDRGES

🎀 Purpose 🎀

Okay so ZDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver ZGGES.

ZGGES factors A and B as Q*S*Z' and Q*T*Z', where ' means conjugate transpose, S and T are upper triangular (i.e., in generalized Schur form), and Q and Z are unitary. 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 ZDRGES is called, a number of matrix sizes and a number of matrix types 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).

  • (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. triangular form) (no sorting of eigenvalues)
  • (6) — if eigenvalues = diagonal elements of the Schur form (S, T), i.e., test the maximum over j of D(j) where:
                    |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
          D(j) = ------------------------ + -----------------------
                 max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

(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 elements of the Schur form (S, T), i.e. test the maximum over j of D(j) where:
                    |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
          D(j) = ------------------------ + -----------------------
                 max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)

(with sorting of eigenvalues).

  • (12) — if sorting worked and SDIM is the number of eigenvalues which were CELECTed.
Yes, bestie, the page says 13 tests and then lists (1) through (12), and it says (13) is exempt. That's literally what the source says — we don't make the rules 💀

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

                                                       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 — INTEGER. The number of sizes of matrices to use. If it is zero, DDRGES does nothing. NSIZES >= 0.
The page literally says DDRGES in a few spots even though this is zdrges; I'm keeping the source's words exactly. iykyk
  • NN — INTEGER array, dimension (NSIZES). An array containing the sizes to be used for the matrices. Zero values will be skipped. NN >= 0.
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it is zero, DDRGES 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 — 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 — 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 DDRGES to continue the same random number sequence.
  • THRESH — DOUBLE PRECISION. A test will count as a failure if the error, 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. 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 — part 1 ends right here at A (the signature already told you it's complex*16, dimension( lda, * ) a); the next part picks up the description.

💅 A

A is a COMPLEX*16 array, dimension (LDA, max(NN)), and it's literally just here to hold the original A matrix 🫶. Tiny little plot twist tho: it's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE.. All three, bestie. No shortcuts ✨.

🎀 LDA

LDA is INTEGER, and it's giving "mom friend of the group" energy — it's the leading dimension of A, B, S, and T all at once 👑. It must be at least 1 and at least max( NN ). Both. Not optional 💅.

🦋 B

B is a COMPLEX*16 array, dimension (LDA, max(NN)), holding the original B matrix 🌸. Same exact mood as A: it's only used as input if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — yes, all of them again, iykyk 😌.

⭐ S

S is a COMPLEX*16 array, dimension (LDA, max(NN)), and it's the Schur form matrix computed from A by ZGGES 🔥. On exit, S contains the Schur form matrix corresponding to the matrix in A. It literally said "I'll take it from here" and took it 💅.

🌈 T

T is a COMPLEX*16 array, dimension (LDA, max(NN)) — the upper triangular matrix computed from B by ZGGES. Upper triangular and thriving ✨.

💜 Q

Q is a COMPLEX*16 array, dimension (LDQ, max(NN)), holding the (left) orthogonal matrix computed by ZGGES. Left-side main character energy 👑.

🍓 LDQ

LDQ is INTEGER, and it's the leading dimension of Q and Z — a two-for-one situation 😌. It must be at least 1 and at least max( NN ). Same rules, different letter fr fr.

🧚 Z

Z is a COMPLEX*16 array, dimension ( LDQ, max(NN) ) — the (right) orthogonal matrix computed by ZGGES. The right-hand counterpart to Q, and yes, they're matching 🎀.

🌸 ALPHA

ALPHA is a COMPLEX*16 array, dimension (max(NN)). That's it. That's the whole vibe. Short and iconic 💅.

✨ BETA

BETA is a COMPLEX*16 array, dimension (max(NN)) 🦋.

Together with ALPHA, these are the generalized eigenvalues of (A,B) computed by ZGGES, and listen — ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A and B. That division is not a suggestion, it's the whole point 👑.

🔥 WORK

WORK is a COMPLEX*16 array, dimension (LWORK). It works hard so you don't have to 💀.

💖 LWORK

LWORK is INTEGER — the dimension of the array WORK — and it has one rule, bestie: LWORK >= 3*N*N. Respect it 🙈.

🎀 RWORK

RWORK is a DOUBLE PRECISION array, dimension ( 8*N ), and it's the real workspace. Real in the literal type sense and the "no drama" sense 🌸.

⭐ RESULT

RESULT is a DOUBLE PRECISION array, dimension (15), and it holds the values computed by the tests described above ✨. Important caveat tho: those values are currently limited to 1/ulp, to avoid overflow. Like, they're capped on purpose. Not a bug, a boundary 🫶.

🦋 BWORK

BWORK is a LOGICAL array, dimension (N). Yes ma'am, logical 🧚.

💅 INFO

INFO is INTEGER and it's the group chat status update:

  • = 0: successful exit — we ate 👑
  • < 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 378 of file zdrges.f ⭐.

💜 Author

Generated automatically by Doxygen for LAPACK from the source code 🎀.

bestie this whole page was auto-generated, so if you're out here hand-editing it, maybe don't 💀

okay that's the whole page!! 💖 From LAPACK Version 3.12.0, the TESTING/EIG/zdrges.f(3) page — slay, you read the whole thing, I'm so proud of you 🫶✨🎀

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