ZDRGES(3)

💅 zdrges(3) — NAME 💅

okay bestie, this is the page for zdrges(3) from the Library Functions Manual, and the file at the heart of it is:

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

That path is literally the whole NAME section in the source, so I’m folding the manual name up here where you can actually see it ✨

🌸 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 🦋

ZDRGES checks the nonsymmetric generalized eigenvalue (Schur form) problem driver ZGGES. no cap, that’s the whole vibe check of this routine.

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 reorders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal block of the Schur forms. main character energy, honestly.

When ZDRGES 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).

(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.

🌈 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 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
    ( J , J  )

(6) ( X, Y ) where:

                                    t                ( I   0  )
    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 big 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) random orthogonal matrices doing their thing:

          t   t
    Q ( J , J ) Z     where Q and Z are random orthogonal matrices.

(17) Q ( T1, T2 ) Z — 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) this one’s got the little stacked N-5 moment:

                                                       N-5
    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. The number of sizes of matrices to use. If it is zero, DDRGES 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, 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

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 DDRGES to continue the same random number sequence.

THRESH

THRESH is DOUBLE PRECISION. A test will count as a fail if the value, 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

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 COMPLEX*16 array, dimension(LDA, max(NN)). It's the storage for the original A matrix 💖. It's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — yes, that exact combo, bestie.

LDA

LDA is INTEGER. It's the leading dimension of A, B, S, and T, and it must be at least 1 and at least max( NN ). ✨

B

B is COMPLEX*16 array, dimension(LDA, max(NN)). It holds the original B matrix, and it's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. 🫶

S

S is COMPLEX*16 array, dimension (LDA, max(NN)). 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. 🌸

T

T is COMPLEX*16 array, dimension (LDA, max(NN)). It's the upper triangular matrix computed from B by ZGGES. 💅

Q

Q is COMPLEX*16 array, dimension (LDQ, max(NN)). It's the (left) orthogonal matrix computed by ZGGES. 🦋

LDQ

LDQ is INTEGER. It's the leading dimension of Q and Z. It must be at least 1 and at least max( NN ). ⭐

Z

Z is COMPLEX*16 array, dimension( LDQ, max(NN) ). It's the (right) orthogonal matrix computed by ZGGES. 🦋

ALPHA

ALPHA is COMPLEX*16 array, dimension (max(NN)). 👑

BETA

BETA is COMPLEX*16 array, dimension (max(NN)). These are the generalized eigenvalues of (A,B) computed by ZGGES. ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A and B. 🔥

WORK

WORK is COMPLEX*16 array, dimension (LWORK). 🧚

LWORK

LWORK is INTEGER. It's the dimension of the array WORK. LWORK >= 3*N*N. ✨

RWORK

RWORK is DOUBLE PRECISION array, dimension ( 8*N ). It's real workspace. 🌈

RESULT

RESULT is DOUBLE PRECISION array, dimension (15). These are the values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. 😭

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. 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

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That's the whole parameter party, bestie 🫶. Signed off by LAPACK Version 3.12.0, source /home/abui...IG/zdrges.f(3) — the footer didn't give a date, so I'm not inventing one 😭✨

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