TESTING/EIG/cdrges.f(3)

NAME
TESTING/EIG/cdrges.f

SYNOPSIS

  Functions/Subroutines
subroutine cdrges (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, alpha, beta, work, lwork, rwork, result, bwork, info)
CDRGES

Function/Subroutine Documentation

  subroutine cdrges (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( * ) alpha, complex, dimension( * ) beta, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, real, dimension( 13 ) result, logical, dimension( * ) bwork, integer info)

CDRGES

Purpose:

     CDRGES checks the nonsymmetric generalized eigenvalue (Schur form)
     problem driver CGGES.
     CGGES 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 reorder the eigenvalues so that a selected
     cluster of eigenvalues appears in the leading diagonal block of the
     Schur forms.
     When CDRGES is called, a number of matrix 'sizes' ('N's') and a
     number of matrix 'TYPES' are specified.  For each size ('N')
     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 '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
              The number of sizes of matrices to use.  If it is zero,
              SDRGES 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, SDRGES
              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 SDRGES 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.  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 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 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 array, dimension (LDA, max(NN))
              The Schur form matrix computed from A by CGGES.  On exit, S
              contains the Schur form matrix corresponding to the matrix
              in A.

    T

              T is COMPLEX array, dimension (LDA, max(NN))
              The upper triangular matrix computed from B by CGGES.

    Q

              Q is COMPLEX array, dimension (LDQ, max(NN))
              The (left) orthogonal matrix computed by CGGES.

    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 array, dimension( LDQ, max(NN) )
              The (right) orthogonal matrix computed by CGGES.

    ALPHA

              ALPHA is COMPLEX array, dimension (max(NN))

    BETA

              BETA is COMPLEX array, dimension (max(NN))
              The generalized eigenvalues of (A,B) computed by CGGES.
              ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A
              and B.

    WORK

              WORK is COMPLEX array, dimension (LWORK)

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.  LWORK >= 3*N*N.

    RWORK

              RWORK is REAL array, dimension ( 8*N )
              Real workspace.

    RESULT

              RESULT is REAL array, dimension (15)
              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 cdrges.f.

Author

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