TESTING/EIG/zdrgev3.f(3)

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💖 TESTING/EIG/zdrgev3.f(3) — Library Functions Manual 💖

✨ NAME ✨

TESTING/EIG/zdrgev3.f — yes bestie, that’s the page, and it’s all about the zdrgev3 test routine 🧚

✨ SYNOPSIS ✨

🛠️ Functions/Subroutines 🛠️

subroutine zdrgev3 (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)
    ZDRGEV3

🌸 Function/Subroutine Documentation 🌸

subroutine zdrgev3 (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)
    ZDRGEV3

💫 Purpose 💫

okay so, ZDRGEV3 checks the nonsymmetric generalized eigenvalue problem driver routine ZGGEV3. 💅

ZGGEV3 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 ZDRGEV3 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 tests will be performed and compared with the threshold THRESH. 👑

Results from ZGGEV3:

(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.

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

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

(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of

| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )

(4) | |VR(i)| - 1 | / ulp and whether largest component real

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

(5) 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.

(6) 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.

(7) 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 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) 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. The number of sizes of matrices to use. If it is zero, ZDRGEV3 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, ZDRGEV3 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 ZDRGES to continue the same random number sequence. 🌈

`THRESH`

THRESH is DOUBLE PRECISION. A test will count as if the , 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*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`

okay bestie, we're still in the argument docs and it's giving typed parameters only — but every single one matters, so let's go 💅

LDA — INTEGER. The leading dimension of A, B, S, and T. It must be at least 1 and at least max( NN ). ✨
B — 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 — COMPLEX*16 array, dimension (LDA, max(NN)). The Schur form matrix computed from A by ZGGEV3. On exit, S contains the Schur form matrix corresponding to the matrix in A. 🎀
T — COMPLEX*16 array, dimension (LDA, max(NN)). The upper triangular matrix computed from B by ZGGEV3. 🌸
Q — COMPLEX*16 array, dimension (LDQ, max(NN)). The (left) eigenvectors matrix computed by ZGGEV3. 💜
LDQ — INTEGER. The leading dimension of Q and Z. It must be at least 1 and at least max( NN ). 👑
Z — COMPLEX*16 array, dimension( LDQ, max(NN) ). The (right) orthogonal matrix computed by ZGGEV3. 🌈
QE — COMPLEX*16 array, dimension( LDQ, max(NN) ). QE holds the computed right or left eigenvectors. 🫶
LDQE — INTEGER. The leading dimension of QE. LDQE >= max(1,max(NN)). 🧚
ALPHA — COMPLEX*16 array, dimension (max(NN)). ⭐
BETA — COMPLEX*16 array, dimension (max(NN)).

The generalized eigenvalues of (A,B) computed by ZGGEV3. ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th generalized eigenvalue of A and B. 🔥

ALPHA1 — COMPLEX*16 array, dimension (max(NN)). 🌟
BETA1 — COMPLEX*16 array, dimension (max(NN)).

Like ALPHAR, ALPHAI, BETA, these arrays contain the eigenvalues of A and B, but those computed when ZGGEV3 only computes a partial eigendecomposition, i.e. not the eigenvalues and left and right eigenvectors. 💫

WORK — COMPLEX*16 array, dimension (LWORK). 🍓
LWORK — INTEGER. The number of entries in WORK. LWORK >= N*(N+1). 💅
RWORK — DOUBLE PRECISION array, dimension (8*N). Real workspace. 😭
RESULT — 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 — 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 👑

The academic royalty behind this one:

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

Definition at line 395 of file zdrgev3.f. 🦋

🌸 Author 🌸

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