ZDRGEV(3)

📦 version 3.12.0, in opensuse-leap 16.0

🦋 ZDRGEV(3) 🦋

Library Functions Manual — header ref: /home/abui...IG/zdrgev.f(3). Manual name folded in near the top, exactly as requested, bestie. 💖

💅 NAME 💅

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/zdrgev.f — yes, that whole path is the name. Iconic. ✨

🌈 SYNOPSIS 🌈

Functions/Subroutines

subroutine ZDRGEV (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a,
    lda, b, s, t, q, ldq, z, qe, ldqe, alpha, beta, alpha1, beta1, work,
    lwork, rwork, result, info)

ZDRGEV

🌸 Function/Subroutine Documentation 🌸

subroutine ZDRGEV (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( ldqe, * ) qe, integer ldqe, complex*16,
    dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16,
    dimension( * ) alpha1, complex*16, dimension( * ) beta1, complex*16,
    dimension( * ) work, integer lwork, double precision, dimension( * )
    rwork, double precision, dimension( * ) result, integer info)

ZDRGEV

🎯 Purpose 🎯

Okay bestie, ZDRGEV is the vibe-checker for the nonsymmetric generalized eigenvalue problem driver routine ZGGEV. 💅

ZGGEV computes, for a pair of n-by-n nonsymmetric matrices (A,B), the generalized eigenvalues and, optionally, the left and right eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is reasonable interpretation for beta=0, and even for both being zero.

A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - wB) * r = 0. A left generalized eigenvector is a vector l such that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.

When ZDRGEV 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 tests will be performed and compared with the threshold THRESH.

Results from ZGGEV:

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

where VL**H is the conjugate-transpose of VL.

  1. | |VL(i)| - 1 | / ulp and whether largest component real

VL(i) denotes the i-th column of VL.

  1. max over all left eigenvalue/-vector pairs (alpha/beta,r) of
| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )
  1. | |VR(i)| - 1 | / ulp and whether largest component real

VR(i) denotes the i-th column of VR.

  1. W(full) = W(partial) W(full) denotes the eigenvalues computed when both l and r are also computed, and W(partial) denotes the eigenvalues computed when only W, only W and r, or only W and l are computed.
  2. VL(full) = VL(partial) VL(full) denotes the left eigenvectors computed when both l and r are computed, and VL(partial) denotes the result when only l is computed.
  3. VR(full) = VR(partial) VR(full) denotes the right eigenvectors computed when both l and r are also computed, and VR(partial) denotes the result when only l is computed.

🧪 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. Type 1 — ( 0, 0 ), a pair of zero matrices. Literally zeros, no cap.
  2. Type 2 — ( I, 0 ), an identity and a zero matrix.
  3. Type 3 — ( 0, I ), an identity and a zero matrix. (Yes, same little description as type 2 in the original, keep it.)
  4. Type 4 — ( I, I ), a pair of identity matrices.
  5. Type 5 — a pair of transposed Jordan blocks, exactly this layout:
        t   t
(5)  ( J , J  )
  1. Type 6 — ( X, Y ), where the X and Y situation is:
                                     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.

  1. Type 7 — ( D, I ), where D is diag( 0, 1,..., N-1 ), a diagonal matrix with those diagonal entries.
  2. Type 8 — ( I, D ).
  3. Type 9 — ( big*D, small*I ), where big is near overflow and small=1/big.
  4. Type 10 — ( small*D, big*I ).
  5. Type 11 — ( big*I, small*D ).
  6. Type 12 — ( small*I, big*D ).
  7. Type 13 — ( big*D, big*I ).
  8. Type 14 — ( small*D, small*I ).
  9. Type 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 ):
(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 )
  1. Type 16 — Q ( J^t, J^t ) Z, where Q and Z are random orthogonal matrices:
           t   t
(16) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.
  1. Type 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 ):
(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 )
  1. Type 18 — Q ( T1, T2 ) Z, with diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 ), diag(T2) = ( 0, 1, 0, 1,..., 1, 0 ), and s = machine precision.
(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.
  1. Type 19 — Q ( T1, T2 ) Z, with diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 ) and diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 ):
(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 )
  1. Type 20 — Q ( T1, T2 ) Z, with this exact diagonal energy:
                                                        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 )
  1. Type 21 — Q ( T1, T2 ) Z, with diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 ), diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 ), and where r1,..., r(N-4) are random:
(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.
  1. Type 22 — Q ( big*T1, small*T2 ) Z, with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) = ( 0, 1, ..., 1, 0, 0 ):
(22) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
  1. Type 23 — Q ( small*T1, big*T2 ) Z, with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and 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 )
  1. Type 24 — Q ( small*T1, small*T2 ) Z, with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and 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 )
  1. Type 25 — Q ( big*T1, big*T2 ) Z, with diag(T1) = ( 0, 0, 1, ..., N-3, 0 ) and 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 )
  1. Type 26 — Q ( T1, T2 ) Z, where T1 and T2 are random upper-triangular matrices. Periodt. 🫶

🎛️ Parameters 🎛️

NSIZES

NSIZES is INTEGER. The number of sizes of matrices to use. If it is zero, ZDRGES does nothing. NSIZES >= 0.

Yes, that says ZDRGES, exactly as written in the original. Not me being chaotic. 💀

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, ZDRGEV 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

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.

Real caveat, bestie: ISEED(4) must be odd. Do not skip that.

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 ZDRGES to continue the same random number sequence.

THRESH

THRESH is DOUBLE PRECISION. A test will count as 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. It must be at least zero.

NOUNIT

NOUNIT is INTEGER. The FORTRAN unit number for printing out error messages (e.g., if a routine returns IERR not equal to 0.).

A

okay so picking up mid-arg-list — here's the rest of the parameter cast, no skips, no cap 💅

  • A — COMPLEX*16 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, S, and T. It must be at least 1 and at least max( NN ).
  • B — B is COMPLEX*16 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 — S is COMPLEX*16 array, dimension (LDA, max(NN)). The Schur form matrix computed from A by ZGGEV. On exit, S contains the Schur form matrix corresponding to the matrix in A.
  • T — T is COMPLEX*16 array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by ZGGEV.
  • Q — Q is COMPLEX*16 array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by ZGGEV.
  • LDQ — LDQ is INTEGER. 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) ). The (right) orthogonal matrix computed by ZGGEV.
  • QE — QE is COMPLEX*16 array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors.
  • LDQE — LDQE is INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)).
  • ALPHA — ALPHA is COMPLEX*16 array, dimension (max(NN)).
  • BETA — BETA is COMPLEX*16 array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by ZGGEV. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B.
  • ALPHA1 — ALPHA1 is COMPLEX*16 array, dimension (max(NN)).
  • BETA1 — BETA1 is COMPLEX*16 array, dimension (max(NN)). Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when ZGGEV only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors.
  • WORK — WORK is COMPLEX*16 array, dimension (LWORK).
  • LWORK — LWORK is INTEGER. The number of entries in WORK. LWORK >= N*(N+1).
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (8*N). Real workspace.
  • RESULT — RESULT is DOUBLE PRECISION array, dimension (2). 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 395 of file zdrgev.f.

🤖 Author

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