CDRGEV(3)

📦 version 3.12.0, in opensuse-leap 16.0

🌸 CDRGEV(3) 🌸

okay bestie, welcome to the Library Functions Manual page for cdrgev.f, bestie. The full path is /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/cdrgev.f, and this page is all about the CDRGEV subroutine. 💅

✨ NAME ✨

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

🦋 SYNOPSIS 🦋

Functions/Subroutines

subroutine CDRGEV (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)

CDRGEV

💖 Function/Subroutine Documentation 💖

subroutine CDRGEV (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    real thresh, integer nounit, complex, dimension( lda, * ) a, integer
    lda, complex, dimension( lda, * ) b, complex, dimension( lda, * ) s,
    complex, dimension( lda, * ) t, complex, dimension( ldq, * ) q, integer
    ldq, complex, dimension( ldq, * ) z, complex, dimension( ldqe, * ) qe,
    integer ldqe, complex, dimension( * ) alpha, complex, dimension( * )
    beta, complex, dimension( * ) alpha1, complex, dimension( * ) beta1,
    complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork,
    real, dimension( * ) result, integer info)

CDRGEV

🎯 Purpose 🎯

okay so CDRGEV checks the nonsymmetric generalized eigenvalue problem driver routine CGGEV.

CGGEV 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 CDRGEV 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 CGGEV 🌈

  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 right 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.

  1. 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.

  1. 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)  ( 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 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 )
          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

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

NN

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

NTYPES

NTYPES is INTEGER. It's the number of elements in DOTYPE. If it is zero, CDRGEV 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. 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 CDRGES to continue the same random number sequence.

THRESH

THRESH is REAL. A test will count as failed 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

A is COMPLEX 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 — INTEGER. The leading dimension of A, B, S, and T. It has to be at least 1 and at least max( NN ). Like, both. Don't skimp. 💅
  • B — COMPLEX array, dimension (LDA, max(NN)). This one's holding the original B matrix. It's used as input only if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and DOTYPE(MAXTYP+1)=.TRUE. — yeah, all three, iykyk. 🫶
  • S — COMPLEX array, dimension (LDA, max(NN)). The Schur form matrix computed from A by CGGEV. On exit, S contains the Schur form matrix corresponding to the matrix in A. It's giving glow-up arc. ✨
  • T — COMPLEX array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by CGGEV. Simple as that, no cap. 💖
  • Q — COMPLEX array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by CGGEV. Iconic behaviour, honestly. 👑
  • LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). Both, bestie. 🎀
  • Z — COMPLEX array, dimension ( LDQ, max(NN) ). The (right) orthogonal matrix computed by CGGEV. She's doing her own thing over here. 🦋
  • QE — COMPLEX array, dimension ( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. Either way, she ate. 🔥
  • LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). Write it down, bestie. 🌸
  • ALPHA — COMPLEX array, dimension (max(NN)). Just vibes, no extra lore here. 💜
  • BETA — COMPLEX array, dimension (max(NN)). The generalized eigenvalues of (A,B) computed by CGGEV. And like, ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. Math is mathing. 😭
  • ALPHA1 — COMPLEX array, dimension (max(NN)). Sibling energy to ALPHA. 🍓
  • BETA1 — COMPLEX array, dimension (max(NN)). Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when CGGEV only computes a partial eigendecomposition — i.e. not the eigenvalues and left and right eigenvectors. She's doing a half-shift, lowkey. 🙈
  • WORK — COMPLEX array, dimension (LWORK). The workspace bestie herself. 🧚
  • LWORK — INTEGER. The number of entries in WORK. LWORK >= N*(N+1). No, you can't make it smaller. Yes, the formula really is that. 💀
  • RWORK — REAL array, dimension (8*N). Real workspace. Eight whole N, babes. ⭐
  • RESULT — REAL array, dimension (2). The values computed by the tests described above. The values are currently limited to 1/ulp, to avoid overflow. Protective queen behaviour. 🛡️
  • INFO — INTEGER. The receipts:
    • = 0: successful exit. slay. 🎉
    • < 0: if INFO = -i, the i-th argument had an illegal value. not slay. 😬
    • > 0: a routine returned an error code. INFO is the absolute value of the INFO value returned. So like, the error just took off its minus sign and went "I'm fine". 💀

👑 Author

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

Definition at line 395 of file cdrgev.f. Main character line number, if we're being honest. 🫶

🧚 Author (again, the meta one)

Generated automatically by Doxygen for LAPACK from the source code. The bot did its little job and honestly? Periodt. 🌈

That's the whole page, bestie — this was CDRGEV(3) from LAPACK Version 3.12.0, /home/abui...IG/cdrgev.f(3). Go forth and eigendecompose responsibly. 💖✨🎀

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