TESTING/EIG/ddrges3.f(3)

NAME
TESTING/EIG/ddrges3.f

SYNOPSIS

  Functions/Subroutines
subroutine ddrges3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, alphar, alphai, beta, work, lwork, result, bwork, info)
DDRGES3

Function/Subroutine Documentation

  subroutine ddrges3 (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, double precision thresh, integer nounit, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( lda, * ) b, double precision, dimension( lda, * ) s, double precision, dimension( lda, * ) t, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldq, * ) z, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( * ) work, integer lwork, double precision, dimension( 13 ) result, logical, dimension( * ) bwork, integer info)

DDRGES3

Purpose:

    !>
    !> DDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form)
    !> problem driver DGGES3.
    !>
    !> DGGES3 factors A and B as Q S Z'  and Q T Z' , where ' means
    !> transpose, T is upper triangular, S is in generalized Schur form
    !> (block upper triangular, with 1x1 and 2x2 blocks on the diagonal,
    !> the 2x2 blocks corresponding to complex conjugate pairs of
    !> generalized eigenvalues), and Q and Z are orthogonal. 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 DDRGES3 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 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. quasi-triangular form)
    !>       (no sorting of eigenvalues)
    !>
    !> (6)   if eigenvalues = diagonal blocks of the Schur form (S, T),
    !>       i.e., test the maximum over j of D(j)  where:
    !>
    !>       if alpha(j) is real:
    !>                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
    !>           D(j) = ------------------------ + -----------------------
    !>                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)
    !>
    !>       if alpha(j) is complex:
    !>                                 | det( s S - w T ) |
    !>           D(j) = ---------------------------------------------------
    !>                  ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
    !>
    !>       and S and T are here the 2 x 2 diagonal blocks of S and T
    !>       corresponding to the j-th and j+1-th eigenvalues.
    !>       (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 blocks of the Schur form (S, T),
    !>       i.e. test the maximum over j of D(j)  where:
    !>
    !>       if alpha(j) is real:
    !>                     |alpha(j) - S(j,j)|        |beta(j) - T(j,j)|
    !>           D(j) = ------------------------ + -----------------------
    !>                  max(|alpha(j)|,|S(j,j)|)   max(|beta(j)|,|T(j,j)|)
    !>
    !>       if alpha(j) is complex:
    !>                                 | det( s S - w T ) |
    !>           D(j) = ---------------------------------------------------
    !>                  ulp max( s norm(S), |w| norm(T) )*norm( s S - w T )
    !>
    !>       and S and T are here the 2 x 2 diagonal blocks of S and T
    !>       corresponding to the j-th and j+1-th eigenvalues.
    !>       (with sorting of eigenvalues).
    !>
    !> (12)  if sorting worked and SDIM is the number of eigenvalues
    !>       which were SELECTed.
    !>
    !> 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,
    !>          DDRGES3 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, DDRGES3
    !>          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 DDRGES3 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.  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDA, max(NN))
    !>          The Schur form matrix computed from A by DGGES3.  On exit, S
    !>          contains the Schur form matrix corresponding to the matrix
    !>          in A.
    !>

    T

    !>          T is DOUBLE PRECISION array, dimension (LDA, max(NN))
    !>          The upper triangular matrix computed from B by DGGES3.
    !>

    Q

    !>          Q is DOUBLE PRECISION array, dimension (LDQ, max(NN))
    !>          The (left) orthogonal matrix computed by DGGES3.
    !>

    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 DOUBLE PRECISION array, dimension( LDQ, max(NN) )
    !>          The (right) orthogonal matrix computed by DGGES3.
    !>

    ALPHAR

    !>          ALPHAR is DOUBLE PRECISION array, dimension (max(NN))
    !>

    ALPHAI

    !>          ALPHAI is DOUBLE PRECISION array, dimension (max(NN))
    !>

    BETA

    !>          BETA is DOUBLE PRECISION array, dimension (max(NN))
    !>
    !>          The generalized eigenvalues of (A,B) computed by DGGES3.
    !>          ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
    !>          generalized eigenvalue of A and B.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (LWORK)
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The dimension of the array WORK.
    !>          LWORK >= MAX( 10*(N+1), 3*N*N ), where N is the largest
    !>          matrix dimension.
    !>

    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.
    !>

    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 399 of file ddrges3.f.

Author

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