ZCHKGG  checks the nonsymmetric generalized eigenvalue problem
 routines.
                                H          H        H
 ZGGHRD factors A and B as U H V  and U T V , where   means conjugate
 transpose, H is hessenberg, T is triangular and U and V are unitary.
                                 H          H
 ZHGEQZ factors H and T as  Q S Z  and Q P Z , where P and S are upper
 triangular and Q and Z are unitary.  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
 ZTGEVC 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 ZCHKGG 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, 13
 tests will be performed.  The first twelve 'test ratios' should be
 small -- O(1).  They will be compared with the threshold THRESH:
                  H
 (1)   | A - U H V  | / ( |A| n ulp )
                  H
 (2)   | B - U T V  | / ( |B| n ulp )
               H
 (3)   | I - UU  | / ( n ulp )
               H
 (4)   | I - VV  | / ( n ulp )
                  H
 (5)   | H - Q S Z  | / ( |H| n ulp )
                  H
 (6)   | T - Q P Z  | / ( |T| n ulp )
               H
 (7)   | I - QQ  | / ( n ulp )
               H
 (8)   | I - ZZ  | / ( n ulp )
 (9)   max over all left eigenvalue/-vector pairs (beta/alpha,l) of
                           H
       | (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )
 (10)  max over all left eigenvalue/-vector pairs (beta/alpha,l') of
                           H
       | (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )
       where the eigenvectors l' are the result of passing Q to
       DTGEVC and back transforming (JOB='B').
 (11)  max over all right eigenvalue/-vector pairs (beta/alpha,r) of
       | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
 (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
       DTGEVC and back transforming (JOB='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 P*D1, P is a random unitary diagonal
                       matrix (i.e., with random magnitude 1 entries
                       on the diagonal), and D1=diag( 0, 1,..., N-1 )
                       (i.e., a diagonal matrix with D1(1,1)=0,
                       D1(2,2)=1, ..., D1(N,N)=N-1.)
 (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=P*diag( 0, 0, 1, ..., N-3, 0 ) and
                        D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and
                        P and Q are random unitary diagonal matrices.
           t   t
 (16) U ( J , J ) V     where U and V are random unitary 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) =
                        P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
                        Q*( 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) = P*( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
 (23) U ( small*T1, big*T2 ) V   diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
 (24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
                                 diag(T2) = ( 0, 1, ..., 1, 0, 0 )
 (25) U ( big*T1, big*T2 ) V     diag(T1) = P*( 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.

NSIZES

          NSIZES is INTEGER
          The number of sizes of matrices to use.  If it is zero,
          ZCHKGG 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, ZCHKGG
          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 ZCHKGG to continue the same random number
          sequence.

THRESH

          THRESH is DOUBLE PRECISION
          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 DOUBLE PRECISION
          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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDA, max(NN))
          The upper Hessenberg matrix computed from A by ZGGHRD.

T

          T is COMPLEX*16 array, dimension (LDA, max(NN))
          The upper triangular matrix computed from B by ZGGHRD.

S1

          S1 is COMPLEX*16 array, dimension (LDA, max(NN))
          The Schur (upper triangular) matrix computed from H by ZHGEQZ
          when Q and Z are also computed.

S2

          S2 is COMPLEX*16 array, dimension (LDA, max(NN))
          The Schur (upper triangular) matrix computed from H by ZHGEQZ
          when Q and Z are not computed.

P1

          P1 is COMPLEX*16 array, dimension (LDA, max(NN))
          The upper triangular matrix computed from T by ZHGEQZ
          when Q and Z are also computed.

P2

          P2 is COMPLEX*16 array, dimension (LDA, max(NN))
          The upper triangular matrix computed from T by ZHGEQZ
          when Q and Z are not computed.

U

          U is COMPLEX*16 array, dimension (LDU, max(NN))
          The (left) unitary matrix computed by ZGGHRD.

LDU

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

V

          V is COMPLEX*16 array, dimension (LDU, max(NN))
          The (right) unitary matrix computed by ZGGHRD.

Q

          Q is COMPLEX*16 array, dimension (LDU, max(NN))
          The (left) unitary matrix computed by ZHGEQZ.

Z

          Z is COMPLEX*16 array, dimension (LDU, max(NN))
          The (left) unitary matrix computed by ZHGEQZ.

ALPHA1

          ALPHA1 is COMPLEX*16 array, dimension (max(NN))

BETA1

          BETA1 is COMPLEX*16 array, dimension (max(NN))
          The generalized eigenvalues of (A,B) computed by ZHGEQZ
          when Q, Z, and the full Schur matrices are computed.

ALPHA3

          ALPHA3 is COMPLEX*16 array, dimension (max(NN))

BETA3

          BETA3 is COMPLEX*16 array, dimension (max(NN))
          The generalized eigenvalues of (A,B) computed by ZHGEQZ
          when neither Q, Z, nor the Schur matrices are computed.

EVECTL

          EVECTL is COMPLEX*16 array, dimension (LDU, max(NN))
          The (lower triangular) left eigenvector matrix for the
          matrices in S1 and P1.

EVECTR

          EVECTR is COMPLEX*16 array, dimension (LDU, max(NN))
          The (upper triangular) right eigenvector matrix for the
          matrices in S1 and P1.

WORK

          WORK is COMPLEX*16 array, dimension (LWORK)

LWORK

          LWORK is INTEGER
          The number of entries in WORK.  This must be at least
          max( 4*N, 2 * N**2, 1 ), for all N=NN(j).

RWORK

          RWORK is DOUBLE PRECISION array, dimension (2*max(NN))

LLWORK

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

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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.