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 'sizes' ('n's') and a
     number of matrix 'types' are specified.  For each size ('n')
     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 'test ratios' 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 '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) 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 '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.

    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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