TESTING/EIG/schkgg.f(3)

NAME
TESTING/EIG/schkgg.f

SYNOPSIS

  Functions/Subroutines
subroutine schkgg (nsizes, nn, ntypes, dotype, iseed, thresh, tstdif, thrshn, nounit, a, lda, b, h, t, s1, s2, p1, p2, u, ldu, v, q, z, alphr1, alphi1, beta1, alphr3, alphi3, beta3, evectl, evectr, work, lwork, llwork, result, info)
SCHKGG

Function/Subroutine Documentation

  subroutine schkgg (integer nsizes, integer, dimension( * ) nn, integer ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed, real thresh, logical tstdif, real thrshn, integer nounit, real, dimension( lda, * ) a, integer lda, real, dimension( lda, * ) b, real, dimension( lda, * ) h, real, dimension( lda, * ) t, real, dimension( lda, * ) s1, real, dimension( lda, * ) s2, real, dimension( lda, * ) p1, real, dimension( lda, * ) p2, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldu, * ) v, real, dimension( ldu, * ) q, real, dimension( ldu, * ) z, real, dimension( * ) alphr1, real, dimension( * ) alphi1, real, dimension( * ) beta1, real, dimension( * ) alphr3, real, dimension( * ) alphi3, real, dimension( * ) beta3, real, dimension( ldu, * ) evectl, real, dimension( ldu, * ) evectr, real, dimension( * ) work, integer lwork, logical, dimension( * ) llwork, real, dimension( 15 ) result, integer info)

SCHKGG

Purpose:

    !>
    !> SCHKGG  checks the nonsymmetric generalized eigenvalue problem
    !> routines.
    !>                                T          T        T
    !> SGGHRD factors A and B as U H V  and U T V , where   means
    !> transpose, H is hessenberg, T is triangular and U and V are
    !> orthogonal.
    !>                                 T          T
    !> SHGEQZ factors H and T as  Q S Z  and Q P Z , where P 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(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
    !>
    !> STGEVC 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 SCHKGG 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, 15
    !> tests will be performed.  The first twelve  should be
    !> small -- O(1).  They will be compared with the threshold THRESH:
    !>
    !>                  T
    !> (1)   | A - U H V  | / ( |A| n ulp )
    !>
    !>                  T
    !> (2)   | B - U T V  | / ( |B| n ulp )
    !>
    !>               T
    !> (3)   | I - UU  | / ( n ulp )
    !>
    !>               T
    !> (4)   | I - VV  | / ( n ulp )
    !>
    !>                  T
    !> (5)   | H - Q S Z  | / ( |H| n ulp )
    !>
    !>                  T
    !> (6)   | T - Q P Z  | / ( |T| n ulp )
    !>
    !>               T
    !> (7)   | I - QQ  | / ( n ulp )
    !>
    !>               T
    !> (8)   | I - ZZ  | / ( n ulp )
    !>
    !> (9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
    !>
    !>    | l**H * (beta S - alpha P) | / ( ulp max( |beta S|, |alpha P| ) )
    !>
    !> (10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
    !>                           T
    !>   | l'**H * (beta H - alpha T) | / ( ulp max( |beta H|, |alpha T| ) )
    !>
    !>       where the eigenvectors l' are the result of passing Q to
    !>       STGEVC and back transforming (HOWMNY='B').
    !>
    !> (11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of
    !>
    !>       | (beta S - alpha T) r | / ( ulp max( |beta S|, |alpha T| ) )
    !>
    !> (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
    !>       STGEVC and back transforming (HOWMNY='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 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) U ( J , J ) V     where U and V are random orthogonal 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) =
    !>                        ( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
    !>                        ( 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) = ( 0, 0, 1, ..., N-3, 0 )
    !>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (23) U ( small*T1, big*T2 ) V    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
    !>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (24) U ( small*T1, small*T2 ) V  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
    !>                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )
    !>
    !> (25) U ( big*T1, big*T2 ) V      diag(T1) = ( 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,
    !>          SCHKGG 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, SCHKGG
    !>          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 SCHKGG 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.
    !>

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

    H

    !>          H is REAL array, dimension (LDA, max(NN))
    !>          The upper Hessenberg matrix computed from A by SGGHRD.
    !>

    T

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

    S1

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

    S2

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

    P1

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

    P2

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

    U

    !>          U is REAL array, dimension (LDU, max(NN))
    !>          The (left) orthogonal matrix computed by SGGHRD.
    !>

    LDU

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

    V

    !>          V is REAL array, dimension (LDU, max(NN))
    !>          The (right) orthogonal matrix computed by SGGHRD.
    !>

    Q

    !>          Q is REAL array, dimension (LDU, max(NN))
    !>          The (left) orthogonal matrix computed by SHGEQZ.
    !>

    Z

    !>          Z is REAL array, dimension (LDU, max(NN))
    !>          The (left) orthogonal matrix computed by SHGEQZ.
    !>

    ALPHR1

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

    ALPHI1

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

    BETA1

    !>          BETA1 is REAL array, dimension (max(NN))
    !>
    !>          The generalized eigenvalues of (A,B) computed by SHGEQZ
    !>          when Q, Z, and the full Schur matrices are computed.
    !>          On exit, ( ALPHR1(k)+ALPHI1(k)*i ) / BETA1(k) is the k-th
    !>          generalized eigenvalue of the matrices in A and B.
    !>

    ALPHR3

    !>          ALPHR3 is REAL array, dimension (max(NN))
    !>

    ALPHI3

    !>          ALPHI3 is REAL array, dimension (max(NN))
    !>

    BETA3

    !>          BETA3 is REAL array, dimension (max(NN))
    !>

    EVECTL

    !>          EVECTL is REAL array, dimension (LDU, max(NN))
    !>          The (block lower triangular) left eigenvector matrix for
    !>          the matrices in S1 and P1.  (See STGEVC for the format.)
    !>

    EVECTR

    !>          EVECTR is REAL array, dimension (LDU, max(NN))
    !>          The (block upper triangular) right eigenvector matrix for
    !>          the matrices in S1 and P1.  (See STGEVC for the format.)
    !>

    WORK

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

    LWORK

    !>          LWORK is INTEGER
    !>          The number of entries in WORK.  This must be at least
    !>          max( 2 * N**2, 6*N, 1 ), for all N=NN(j).
    !>

    LLWORK

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

    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.
    !>

    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 506 of file schkgg.f.

Author

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