ZCHKHS  checks the nonsymmetric eigenvalue problem routines.


            ZGEHRD factors A as  U H U' , where ' means conjugate


            transpose, H is hessenberg, and U is unitary.


            ZUNGHR generates the unitary matrix U.


            ZUNMHR multiplies a matrix by the unitary matrix U.


            ZHSEQR factors H as  Z T Z' , where Z is unitary and T


            is upper triangular.  It also computes the eigenvalues,


            w(1), ..., w(n); we define a diagonal matrix W whose


            (diagonal) entries are the eigenvalues.


            ZTREVC computes the left eigenvector matrix L and the


            right eigenvector matrix R for the matrix T.  The


            columns of L are the complex conjugates of the left


            eigenvectors of T.  The columns of R are the right


            eigenvectors of T.  L is lower triangular, and R is


            upper triangular.


            ZHSEIN computes the left eigenvector matrix Y and the


            right eigenvector matrix X for the matrix H.  The


            columns of Y are the complex conjugates of the left


            eigenvectors of H.  The columns of X are the right


            eigenvectors of H.  Y is lower triangular, and X is


            upper triangular.


            ZTREVC3 computes left and right eigenvector matrices


            from a Schur matrix T and backtransforms them with Z


            to eigenvector matrices L and R for A. L and R are


            GE matrices.


    When ZCHKHS 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, 16


    tests will be performed:


    (1)     | A - U H U**H | / ( |A| n ulp )


    (2)     | I - UU**H | / ( n ulp )


    (3)     | H - Z T Z**H | / ( |H| n ulp )


    (4)     | I - ZZ**H | / ( n ulp )


    (5)     | A - UZ H (UZ)**H | / ( |A| n ulp )


    (6)     | I - UZ (UZ)**H | / ( n ulp )


    (7)     | T(Z computed) - T(Z not computed) | / ( |T| ulp )


    (8)     | W(Z computed) - W(Z not computed) | / ( |W| ulp )


    (9)     | TR - RW | / ( |T| |R| ulp )


    (10)    | L**H T - W**H L | / ( |T| |L| ulp )


    (11)    | HX - XW | / ( |H| |X| ulp )


    (12)    | Y**H H - W**H Y | / ( |H| |Y| ulp )


    (13)    | AX - XW | / ( |A| |X| ulp )


    (14)    | Y**H A - W**H Y | / ( |A| |Y| ulp )


    (15)    | AR - RW | / ( |A| |R| ulp )


    (16)    | LA - WL | / ( |A| |L| ulp )


    The 'sizes' 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)  The zero matrix.


    (2)  The identity matrix.


    (3)  A (transposed) Jordan block, with 1's on the diagonal.


    (4)  A diagonal matrix with evenly spaced entries


         1, ..., ULP  and random complex angles.


         (ULP = (first number larger than 1) - 1 )


    (5)  A diagonal matrix with geometrically spaced entries


         1, ..., ULP  and random complex angles.


    (6)  A diagonal matrix with 'clustered' entries 1, ULP, ..., ULP


         and random complex angles.


    (7)  Same as (4), but multiplied by SQRT( overflow threshold )


    (8)  Same as (4), but multiplied by SQRT( underflow threshold )


    (9)  A matrix of the form  U' T U, where U is unitary and


         T has evenly spaced entries 1, ..., ULP with random complex


         angles on the diagonal and random O(1) entries in the upper


         triangle.


    (10) A matrix of the form  U' T U, where U is unitary and


         T has geometrically spaced entries 1, ..., ULP with random


         complex angles on the diagonal and random O(1) entries in


         the upper triangle.


    (11) A matrix of the form  U' T U, where U is unitary and


         T has 'clustered' entries 1, ULP,..., ULP with random


         complex angles on the diagonal and random O(1) entries in


         the upper triangle.


    (12) A matrix of the form  U' T U, where U is unitary and


         T has complex eigenvalues randomly chosen from


         ULP < |z| < 1   and random O(1) entries in the upper


         triangle.


    (13) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP


         with random complex angles on the diagonal and random O(1)


         entries in the upper triangle.


    (14) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has geometrically spaced entries


         1, ..., ULP with random complex angles on the diagonal


         and random O(1) entries in the upper triangle.


    (15) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has 'clustered' entries 1, ULP,..., ULP


         with random complex angles on the diagonal and random O(1)


         entries in the upper triangle.


    (16) A matrix of the form  X' T X, where X has condition


         SQRT( ULP ) and T has complex eigenvalues randomly chosen


         from   ULP < |z| < 1   and random O(1) entries in the upper


         triangle.


    (17) Same as (16), but multiplied by SQRT( overflow threshold )


    (18) Same as (16), but multiplied by SQRT( underflow threshold )


    (19) Nonsymmetric matrix with random entries chosen from |z| < 1


    (20) Same as (19), but multiplied by SQRT( overflow threshold )


    (21) Same as (19), but multiplied by SQRT( underflow threshold )

  NSIZES - INTEGER


           The number of sizes of matrices to use.  If it is zero,


           ZCHKHS does nothing.  It must be at least zero.


           Not modified.


  NN     - 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.


           Not modified.


  NTYPES - INTEGER


           The number of elements in DOTYPE.   If it is zero, ZCHKHS


           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. .


           Not modified.


  DOTYPE - 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.


           Not modified.


  ISEED  - 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 ZCHKHS to continue the same random number


           sequence.


           Modified.


  THRESH - 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.


           Not modified.


  NOUNIT - INTEGER


           The FORTRAN unit number for printing out error messages


           (e.g., if a routine returns IINFO not equal to 0.)


           Not modified.


  A      - COMPLEX*16 array, dimension (LDA,max(NN))


           Used to hold the matrix whose eigenvalues are to be


           computed.  On exit, A contains the last matrix actually


           used.


           Modified.


  LDA    - INTEGER


           The leading dimension of A, H, T1 and T2.  It must be at


           least 1 and at least max( NN ).


           Not modified.


  H      - COMPLEX*16 array, dimension (LDA,max(NN))


           The upper hessenberg matrix computed by ZGEHRD.  On exit,


           H contains the Hessenberg form of the matrix in A.


           Modified.


  T1     - COMPLEX*16 array, dimension (LDA,max(NN))


           The Schur (='quasi-triangular') matrix computed by ZHSEQR


           if Z is computed.  On exit, T1 contains the Schur form of


           the matrix in A.


           Modified.


  T2     - COMPLEX*16 array, dimension (LDA,max(NN))


           The Schur matrix computed by ZHSEQR when Z is not computed.


           This should be identical to T1.


           Modified.


  LDU    - INTEGER


           The leading dimension of U, Z, UZ and UU.  It must be at


           least 1 and at least max( NN ).


           Not modified.


  U      - COMPLEX*16 array, dimension (LDU,max(NN))


           The unitary matrix computed by ZGEHRD.


           Modified.


  Z      - COMPLEX*16 array, dimension (LDU,max(NN))


           The unitary matrix computed by ZHSEQR.


           Modified.


  UZ     - COMPLEX*16 array, dimension (LDU,max(NN))


           The product of U times Z.


           Modified.


  W1     - COMPLEX*16 array, dimension (max(NN))


           The eigenvalues of A, as computed by a full Schur


           decomposition H = Z T Z'.  On exit, W1 contains the


           eigenvalues of the matrix in A.


           Modified.


  W3     - COMPLEX*16 array, dimension (max(NN))


           The eigenvalues of A, as computed by a partial Schur


           decomposition (Z not computed, T only computed as much


           as is necessary for determining eigenvalues).  On exit,


           W3 contains the eigenvalues of the matrix in A, possibly


           perturbed by ZHSEIN.


           Modified.


  EVECTL - COMPLEX*16 array, dimension (LDU,max(NN))


           The conjugate transpose of the (upper triangular) left


           eigenvector matrix for the matrix in T1.


           Modified.


  EVEZTR - COMPLEX*16 array, dimension (LDU,max(NN))


           The (upper triangular) right eigenvector matrix for the


           matrix in T1.


           Modified.


  EVECTY - COMPLEX*16 array, dimension (LDU,max(NN))


           The conjugate transpose of the left eigenvector matrix


           for the matrix in H.


           Modified.


  EVECTX - COMPLEX*16 array, dimension (LDU,max(NN))


           The right eigenvector matrix for the matrix in H.


           Modified.


  UU     - COMPLEX*16 array, dimension (LDU,max(NN))


           Details of the unitary matrix computed by ZGEHRD.


           Modified.


  TAU    - COMPLEX*16 array, dimension (max(NN))


           Further details of the unitary matrix computed by ZGEHRD.


           Modified.


  WORK   - COMPLEX*16 array, dimension (NWORK)


           Workspace.


           Modified.


  NWORK  - INTEGER


           The number of entries in WORK.  NWORK >= 4*NN(j)*NN(j) + 2.


  RWORK  - DOUBLE PRECISION array, dimension (max(NN))


           Workspace.  Could be equivalenced to IWORK, but not SELECT.


           Modified.


  IWORK  - INTEGER array, dimension (max(NN))


           Workspace.


           Modified.


  SELECT - LOGICAL array, dimension (max(NN))


           Workspace.  Could be equivalenced to IWORK, but not RWORK.


           Modified.


  RESULT - DOUBLE PRECISION array, dimension (16)


           The values computed by the fourteen tests described above.


           The values are currently limited to 1/ulp, to avoid


           overflow.


           Modified.


  INFO   - INTEGER


           If 0, then everything ran OK.


            -1: NSIZES < 0


            -2: Some NN(j) < 0


            -3: NTYPES < 0


            -6: THRESH < 0


            -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).


           -14: LDU < 1 or LDU < NMAX.


           -26: NWORK too small.


           If  ZLATMR, CLATMS, or CLATME returns an error code, the


               absolute value of it is returned.


           If 1, then ZHSEQR could not find all the shifts.


           If 2, then the EISPACK code (for small blocks) failed.


           If >2, then 30*N iterations were not enough to find an


               eigenvalue or to decompose the problem.


           Modified.
-----------------------------------------------------------------------


     Some Local Variables and Parameters:


     ---- ----- --------- --- ----------


     ZERO, ONE       Real 0 and 1.


     MAXTYP          The number of types defined.


     MTEST           The number of tests defined: care must be taken


                     that (1) the size of RESULT, (2) the number of


                     tests actually performed, and (3) MTEST agree.


     NTEST           The number of tests performed on this matrix


                     so far.  This should be less than MTEST, and


                     equal to it by the last test.  It will be less


                     if any of the routines being tested indicates


                     that it could not compute the matrices that


                     would be tested.


     NMAX            Largest value in NN.


     NMATS           The number of matrices generated so far.


     NERRS           The number of tests which have exceeded THRESH


                     so far (computed by DLAFTS).


     COND, CONDS,


     IMODE           Values to be passed to the matrix generators.


     ANORM           Norm of A; passed to matrix generators.


     OVFL, UNFL      Overflow and underflow thresholds.


     ULP, ULPINV     Finest relative precision and its inverse.


     RTOVFL, RTUNFL,


     RTULP, RTULPI   Square roots of the previous 4 values.


             The following four arrays decode JTYPE:


     KTYPE(j)        The general type (1-10) for type 'j'.


     KMODE(j)        The MODE value to be passed to the matrix


                     generator for type 'j'.


     KMAGN(j)        The order of magnitude ( O(1),


                     O(overflow^(1/2) ), O(underflow^(1/2) )


     KCONDS(j)       Selects whether CONDS is to be 1 or


                     1/sqrt(ulp).  (0 means irrelevant.)

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