TESTING/EIG/zchkgg.f(3)

NAME
TESTING/EIG/zchkgg.f

SYNOPSIS

  Functions/Subroutines
subroutine zchkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif, thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z, alpha1, beta1, alpha3, beta3, evectl, evectr, work, lwork, rwork, llwork, result, info)
ZCHKGG

Function/Subroutine Documentation

  subroutine zchkgg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, double precision thresh, logical tstdif, double precision thrshn, integer nounit, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( lda, * ) b, complex*16, dimension( lda, * ) h, complex*16, dimension( lda, * ) t, complex*16, dimension( lda, * ) s1, complex*16, dimension( lda, * ) s2, complex*16, dimension( lda, * ) p1, complex*16, dimension( lda, * ) p2, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldu, * ) v, complex*16, dimension( ldu, * ) q, complex*16, dimension( ldu, * ) z, complex*16, dimension( * ) alpha1, complex*16, dimension( * ) beta1, complex*16, dimension( * ) alpha3, complex*16, dimension( * ) beta3, complex*16, dimension( ldu, * ) evectl, complex*16, dimension( ldu, * ) evectr, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) llwork, double precision, dimension( 15 ) result, integer info)

ZCHKGG

Purpose:

    !>
    !> ZCHKGG  checks the nonsymmetric generalized eigenvalue problem
    !> routines.
    !>                                H          H        H
    !> ZGGHRD factors A and B as U H V  and U T V , where   means conjugate
    !> transpose, H is hessenberg, T is triangular and U and V are unitary.
    !>
    !>                                 H          H
    !> ZHGEQZ factors H and T as  Q S Z  and Q P Z , where P and S are upper
    !> triangular and Q and Z are unitary.  It also computes the generalized
    !> eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where
    !> alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus, w(j) = alpha(j)/beta(j)
    !> is a root of the generalized eigenvalue problem
    !>
    !>     det( A - w(j) B ) = 0
    !>
    !> and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
    !> problem
    !>
    !>     det( m(j) A - B ) = 0
    !>
    !> ZTGEVC computes the matrix L of left eigenvectors and the matrix R
    !> of right eigenvectors for the matrix pair ( S, P ).  In the
    !> description below,  l and r are left and right eigenvectors
    !> corresponding to the generalized eigenvalues (alpha,beta).
    !>
    !> When ZCHKGG is called, a number of matrix  () and a
    !> number of matrix  are specified.  For each size ()
    !> and each type of matrix, one matrix will be generated and used
    !> to test the nonsymmetric eigenroutines.  For each matrix, 13
    !> tests will be performed.  The first twelve  should be
    !> small -- O(1).  They will be compared with the threshold THRESH:
    !>
    !>                  H
    !> (1)   | A - U H V  | / ( |A| n ulp )
    !>
    !>                  H
    !> (2)   | B - U T V  | / ( |B| n ulp )
    !>
    !>               H
    !> (3)   | I - UU  | / ( n ulp )
    !>
    !>               H
    !> (4)   | I - VV  | / ( n ulp )
    !>
    !>                  H
    !> (5)   | H - Q S Z  | / ( |H| n ulp )
    !>
    !>                  H
    !> (6)   | T - Q P Z  | / ( |T| n ulp )
    !>
    !>               H
    !> (7)   | I - QQ  | / ( n ulp )
    !>
    !>               H
    !> (8)   | I - ZZ  | / ( n ulp )
    !>
    !> (9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
    !>                           H
    !>       | (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )
    !>
    !> (10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
    !>                           H
    !>       | (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )
    !>
    !>       where the eigenvectors l' are the result of passing Q to
    !>       DTGEVC and back transforming (JOB='B').
    !>
    !> (11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of
    !>
    !>       | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
    !>
    !> (12)  max over all right eigenvalue/-vector pairs (beta/alpha,r') of
    !>
    !>       | (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )
    !>
    !>       where the eigenvectors r' are the result of passing Z to
    !>       DTGEVC and back transforming (JOB='B').
    !>
    !> The last three test ratios will usually be small, but there is no
    !> mathematical requirement that they be so.  They are therefore
    !> compared with THRESH only if TSTDIF is .TRUE.
    !>
    !> (13)  | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )
    !>
    !> (14)  | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )
    !>
    !> (15)  max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
    !>            |beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp
    !>
    !> In addition, the normalization of L and R are checked, and compared
    !> with the threshold THRSHN.
    !>
    !> 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 P*D1, P is a random unitary diagonal
    !>                       matrix (i.e., with random magnitude 1 entries
    !>                       on the diagonal), and D1=diag( 0, 1,..., N-1 )
    !>                       (i.e., a diagonal matrix with D1(1,1)=0,
    !>                       D1(2,2)=1, ..., D1(N,N)=N-1.)
    !> (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=P*diag( 0, 0, 1, ..., N-3, 0 ) and
    !>                        D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and
    !>                        P and Q are random unitary diagonal matrices.
    !>           t   t
    !> (16) U ( J , J ) V     where U and V are random unitary matrices.
    !>
    !> (17) U ( T1, T2 ) V    where T1 and T2 are upper triangular matrices
    !>                        with random O(1) entries above the diagonal
    !>                        and diagonal entries diag(T1) =
    !>                        P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
    !>                        Q*( 0, N-3, N-4,..., 1, 0, 0 )
    !>
    !> (18) U ( T1, T2 ) V    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
    !>                        diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
    !>                        s = machine precision.
    !>
    !> (19) U ( T1, T2 ) V    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) U ( T1, T2 ) V    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )
    !>                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
    !>
    !> (21) U ( T1, T2 ) V    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) U ( big*T1, small*T2 ) V   diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
    !>                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (23) U ( small*T1, big*T2 ) V   diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
    !>                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
    !>                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (25) U ( big*T1, big*T2 ) V     diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
    !>                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (26) U ( T1, T2 ) V     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,
    !>          ZCHKGG does nothing.  It must be at least zero.
    !>

    NN

    !>          NN is INTEGER array, dimension (NSIZES)
    !>          An array containing the sizes to be used for the matrices.
    !>          Zero values will be skipped.  The values must be at least
    !>          zero.
    !>

    NTYPES

    !>          NTYPES is INTEGER
    !>          The number of elements in DOTYPE.   If it is zero, ZCHKGG
    !>          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 ZCHKGG 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.
    !>

    TSTDIF

    !>          TSTDIF is LOGICAL
    !>          Specifies whether test ratios 13-15 will be computed and
    !>          compared with THRESH.
    !>          = .FALSE.: Only test ratios 1-12 will be computed and tested.
    !>                     Ratios 13-15 will be set to zero.
    !>          = .TRUE.:  All the test ratios 1-15 will be computed and
    !>                     tested.
    !>

    THRSHN

    !>          THRSHN is DOUBLE PRECISION
    !>          Threshold for reporting eigenvector normalization error.
    !>          If the normalization of any eigenvector differs from 1 by
    !>          more than THRSHN*ulp, then a special error message will be
    !>          printed.  (This is handled separately from the other tests,
    !>          since only a compiler or programming error should cause an
    !>          error message, at least if THRSHN is at least 5--10.)
    !>

    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))
    !>          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, H, T, S1, P1, S2, and P2.
    !>          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.
    !>

    H

    !>          H is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The upper Hessenberg matrix computed from A by ZGGHRD.
    !>

    T

    !>          T is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The upper triangular matrix computed from B by ZGGHRD.
    !>

    S1

    !>          S1 is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The Schur (upper triangular) matrix computed from H by ZHGEQZ
    !>          when Q and Z are also computed.
    !>

    S2

    !>          S2 is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The Schur (upper triangular) matrix computed from H by ZHGEQZ
    !>          when Q and Z are not computed.
    !>

    P1

    !>          P1 is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The upper triangular matrix computed from T by ZHGEQZ
    !>          when Q and Z are also computed.
    !>

    P2

    !>          P2 is COMPLEX*16 array, dimension (LDA, max(NN))
    !>          The upper triangular matrix computed from T by ZHGEQZ
    !>          when Q and Z are not computed.
    !>

    U

    !>          U is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (left) unitary matrix computed by ZGGHRD.
    !>

    LDU

    !>          LDU is INTEGER
    !>          The leading dimension of U, V, Q, Z, EVECTL, and EVEZTR.  It
    !>          must be at least 1 and at least max( NN ).
    !>

    V

    !>          V is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (right) unitary matrix computed by ZGGHRD.
    !>

    Q

    !>          Q is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (left) unitary matrix computed by ZHGEQZ.
    !>

    Z

    !>          Z is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (left) unitary matrix computed by ZHGEQZ.
    !>

    ALPHA1

    !>          ALPHA1 is COMPLEX*16 array, dimension (max(NN))
    !>

    BETA1

    !>          BETA1 is COMPLEX*16 array, dimension (max(NN))
    !>          The generalized eigenvalues of (A,B) computed by ZHGEQZ
    !>          when Q, Z, and the full Schur matrices are computed.
    !>

    ALPHA3

    !>          ALPHA3 is COMPLEX*16 array, dimension (max(NN))
    !>

    BETA3

    !>          BETA3 is COMPLEX*16 array, dimension (max(NN))
    !>          The generalized eigenvalues of (A,B) computed by ZHGEQZ
    !>          when neither Q, Z, nor the Schur matrices are computed.
    !>

    EVECTL

    !>          EVECTL is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (lower triangular) left eigenvector matrix for the
    !>          matrices in S1 and P1.
    !>

    EVECTR

    !>          EVECTR is COMPLEX*16 array, dimension (LDU, max(NN))
    !>          The (upper triangular) right eigenvector matrix for the
    !>          matrices in S1 and P1.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (LWORK)
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The number of entries in WORK.  This must be at least
    !>          max( 4*N, 2 * N**2, 1 ), for all N=NN(j).
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION array, dimension (2*max(NN))
    !>

    LLWORK

    !>          LLWORK is LOGICAL array, dimension (max(NN))
    !>

    RESULT

    !>          RESULT is DOUBLE PRECISION array, dimension (15)
    !>          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 498 of file zchkgg.f.

Author

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