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 '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 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 '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,
              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 '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.  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

Generated automatically by Doxygen for LAPACK from the source code.