TESTING/EIG/sdrgev3.f(3)

NAME
TESTING/EIG/sdrgev3.f

SYNOPSIS

  Functions/Subroutines
subroutine sdrgev3 (nsizes, nn, ntypes, dotype, iseed, thresh, nounit, a, lda, b, s, t, q, ldq, z, qe, ldqe, alphar, alphai, beta, alphr1, alphi1, beta1, work, lwork, result, info)
SDRGEV3

Function/Subroutine Documentation

  subroutine sdrgev3 (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) s, real, dimension( lda, * ) t, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldq, * ) z, real, dimension( ldqe, * ) qe, integer ldqe, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( * ) alphr1, real, dimension( * ) alphi1, real, dimension( * ) beta1, real, dimension( * ) work, integer lwork, real, dimension( * ) result, integer info)

SDRGEV3

Purpose:

    !>
    !> SDRGEV3 checks the nonsymmetric generalized eigenvalue problem driver
    !> routine SGGEV3.
    !>
    !> SGGEV3 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 SDRGEV3 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 SGGEV3:
    !>
    !> (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,
    !>          SDRGEV3 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, SDRGEV3
    !>          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 SDRGEV3 to continue the same random number
    !>          sequence.
    !>

    THRESH

    !>          THRESH is REAL
    !>          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 REAL 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 REAL 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 REAL array,
    !>                                 dimension (LDA, max(NN))
    !>          The Schur form matrix computed from A by SGGEV3.  On exit, S
    !>          contains the Schur form matrix corresponding to the matrix
    !>          in A.
    !>

    T

    !>          T is REAL array,
    !>                                 dimension (LDA, max(NN))
    !>          The upper triangular matrix computed from B by SGGEV3.
    !>

    Q

    !>          Q is REAL array,
    !>                                 dimension (LDQ, max(NN))
    !>          The (left) eigenvectors matrix computed by SGGEV3.
    !>

    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 REAL array, dimension( LDQ, max(NN) )
    !>          The (right) orthogonal matrix computed by SGGEV3.
    !>

    QE

    !>          QE is REAL array, dimension( LDQ, max(NN) )
    !>          QE holds the computed right or left eigenvectors.
    !>

    LDQE

    !>          LDQE is INTEGER
    !>          The leading dimension of QE. LDQE >= max(1,max(NN)).
    !>

    ALPHAR

    !>          ALPHAR is REAL array, dimension (max(NN))
    !>

    ALPHAI

    !>          ALPHAI is REAL array, dimension (max(NN))
    !>

    BETA

    !>          BETA is REAL array, dimension (max(NN))
    !> \verbatim
    !>          The generalized eigenvalues of (A,B) computed by SGGEV3.
    !>          ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
    !>          generalized eigenvalue of A and B.
    !>

    ALPHR1

    !>          ALPHR1 is REAL array, dimension (max(NN))
    !>

    ALPHI1

    !>          ALPHI1 is REAL array, dimension (max(NN))
    !>

    BETA1

    !>          BETA1 is REAL array, dimension (max(NN))
    !>
    !>          Like ALPHAR, ALPHAI, BETA, these arrays contain the
    !>          eigenvalues of A and B, but those computed when SGGEV3 only
    !>          computes a partial eigendecomposition, i.e. not the
    !>          eigenvalues and left and right eigenvectors.
    !>

    WORK

    !>          WORK is REAL array, dimension (LWORK)
    !>

    LWORK

    !>          LWORK is INTEGER
    !>          The number of entries in WORK.  LWORK >= MAX( 8*N, N*(N+1) ).
    !>

    RESULT

    !>          RESULT is REAL array, dimension (2)
    !>          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 404 of file sdrgev3.f.

Author

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