ZDRVVX  checks the nonsymmetric eigenvalue problem expert driver
    ZGEEVX.
    ZDRVVX uses both test matrices generated randomly depending on
    data supplied in the calling sequence, as well as on data
    read from an input file and including precomputed condition
    numbers to which it compares the ones it computes.
    When ZDRVVX is called, a number of matrix 'sizes' ('n's') and a
    number of matrix 'types' are specified in the calling sequence.
    For each size ('n') and each type of matrix, one matrix will be
    generated and used to test the nonsymmetric eigenroutines.  For
    each matrix, 9 tests will be performed:
    (1)     | A * VR - VR * W | / ( n |A| ulp )
      Here VR is the matrix of unit right eigenvectors.
      W is a diagonal matrix with diagonal entries W(j).
    (2)     | A**H  * VL - VL * W**H | / ( n |A| ulp )
      Here VL is the matrix of unit left eigenvectors, A**H is the
      conjugate transpose of A, and W is as above.
    (3)     | |VR(i)| - 1 | / ulp and largest component real
      VR(i) denotes the i-th column of VR.
    (4)     | |VL(i)| - 1 | / ulp and largest component real
      VL(i) denotes the i-th column of VL.
    (5)     W(full) = W(partial)
      W(full) denotes the eigenvalues computed when VR, VL, RCONDV
      and RCONDE are also computed, and W(partial) denotes the
      eigenvalues computed when only some of VR, VL, RCONDV, and
      RCONDE are computed.
    (6)     VR(full) = VR(partial)
      VR(full) denotes the right eigenvectors computed when VL, RCONDV
      and RCONDE are computed, and VR(partial) denotes the result
      when only some of VL and RCONDV are computed.
    (7)     VL(full) = VL(partial)
      VL(full) denotes the left eigenvectors computed when VR, RCONDV
      and RCONDE are computed, and VL(partial) denotes the result
      when only some of VR and RCONDV are computed.
    (8)     0 if SCALE, ILO, IHI, ABNRM (full) =
                 SCALE, ILO, IHI, ABNRM (partial)
            1/ulp otherwise
      SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.
      (full) is when VR, VL, RCONDE and RCONDV are also computed, and
      (partial) is when some are not computed.
    (9)     RCONDV(full) = RCONDV(partial)
      RCONDV(full) denotes the reciprocal condition numbers of the
      right eigenvectors computed when VR, VL and RCONDE are also
      computed. RCONDV(partial) denotes the reciprocal condition
      numbers when only some of VR, VL and RCONDE are computed.
    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 a constant near
         the overflow threshold
    (8)  Same as (4), but multiplied by a constant near
         the 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 a constant
         near the overflow threshold
    (18) Same as (16), but multiplied by a constant
         near the underflow threshold
    (19) Nonsymmetric matrix with random entries chosen from |z| < 1
         If N is at least 4, all entries in first two rows and last
         row, and first column and last two columns are zero.
    (20) Same as (19), but multiplied by a constant
         near the overflow threshold
    (21) Same as (19), but multiplied by a constant
         near the underflow threshold
    In addition, an input file will be read from logical unit number
    NIUNIT. The file contains matrices along with precomputed
    eigenvalues and reciprocal condition numbers for the eigenvalues
    and right eigenvectors. For these matrices, in addition to tests
    (1) to (9) we will compute the following two tests:
   (10)  |RCONDV - RCDVIN| / cond(RCONDV)
      RCONDV is the reciprocal right eigenvector condition number
      computed by ZGEEVX and RCDVIN (the precomputed true value)
      is supplied as input. cond(RCONDV) is the condition number of
      RCONDV, and takes errors in computing RCONDV into account, so
      that the resulting quantity should be O(ULP). cond(RCONDV) is
      essentially given by norm(A)/RCONDE.
   (11)  |RCONDE - RCDEIN| / cond(RCONDE)
      RCONDE is the reciprocal eigenvalue condition number
      computed by ZGEEVX and RCDEIN (the precomputed true value)
      is supplied as input.  cond(RCONDE) is the condition number
      of RCONDE, and takes errors in computing RCONDE into account,
      so that the resulting quantity should be O(ULP). cond(RCONDE)
      is essentially given by norm(A)/RCONDV.

NSIZES

          NSIZES is INTEGER
          The number of sizes of matrices to use.  NSIZES must be at
          least zero. If it is zero, no randomly generated matrices
          are tested, but any test matrices read from NIUNIT will be
          tested.

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. NTYPES must be at least
          zero. If it is zero, no randomly generated test matrices
          are tested, but and test matrices read from NIUNIT will be
          tested. 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 ZDRVVX 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.

NIUNIT

          NIUNIT is INTEGER
          The FORTRAN unit number for reading in the data file of
          problems to solve.

NOUNIT

          NOUNIT is INTEGER
          The FORTRAN unit number for printing out error messages
          (e.g., if a routine returns INFO not equal to 0.)

A

          A is COMPLEX*16 array, dimension (LDA, max(NN,12))
          Used to hold the matrix whose eigenvalues are to be
          computed.  On exit, A contains the last matrix actually used.

LDA

          LDA is INTEGER
          The leading dimension of A, and H. LDA must be at
          least 1 and at least max( NN, 12 ). (12 is the
          dimension of the largest matrix on the precomputed
          input file.)

H

          H is COMPLEX*16 array, dimension (LDA, max(NN,12))
          Another copy of the test matrix A, modified by ZGEEVX.

W

          W is COMPLEX*16 array, dimension (max(NN,12))
          Contains the eigenvalues of A.

W1

          W1 is COMPLEX*16 array, dimension (max(NN,12))
          Like W, this array contains the eigenvalues of A,
          but those computed when ZGEEVX only computes a partial
          eigendecomposition, i.e. not the eigenvalues and left
          and right eigenvectors.

VL

          VL is COMPLEX*16 array, dimension (LDVL, max(NN,12))
          VL holds the computed left eigenvectors.

LDVL

          LDVL is INTEGER
          Leading dimension of VL. Must be at least max(1,max(NN,12)).

VR

          VR is COMPLEX*16 array, dimension (LDVR, max(NN,12))
          VR holds the computed right eigenvectors.

LDVR

          LDVR is INTEGER
          Leading dimension of VR. Must be at least max(1,max(NN,12)).

LRE

          LRE is COMPLEX*16 array, dimension (LDLRE, max(NN,12))
          LRE holds the computed right or left eigenvectors.

LDLRE

          LDLRE is INTEGER
          Leading dimension of LRE. Must be at least max(1,max(NN,12))

RCONDV

          RCONDV is DOUBLE PRECISION array, dimension (N)
          RCONDV holds the computed reciprocal condition numbers
          for eigenvectors.

RCNDV1

          RCNDV1 is DOUBLE PRECISION array, dimension (N)
          RCNDV1 holds more computed reciprocal condition numbers
          for eigenvectors.

RCDVIN

          RCDVIN is DOUBLE PRECISION array, dimension (N)
          When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
          condition numbers for eigenvectors to be compared with
          RCONDV.

RCONDE

          RCONDE is DOUBLE PRECISION array, dimension (N)
          RCONDE holds the computed reciprocal condition numbers
          for eigenvalues.

RCNDE1

          RCNDE1 is DOUBLE PRECISION array, dimension (N)
          RCNDE1 holds more computed reciprocal condition numbers
          for eigenvalues.

RCDEIN

          RCDEIN is DOUBLE PRECISION array, dimension (N)
          When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
          condition numbers for eigenvalues to be compared with
          RCONDE.

SCALE

          SCALE is DOUBLE PRECISION array, dimension (N)
          Holds information describing balancing of matrix.

SCALE1

          SCALE1 is DOUBLE PRECISION array, dimension (N)
          Holds information describing balancing of matrix.

WORK

          WORK is COMPLEX*16 array, dimension (NWORK)

RESULT

          RESULT is DOUBLE PRECISION array, dimension (11)
          The values computed by the seven tests described above.
          The values are currently limited to 1/ulp, to avoid
          overflow.

NWORK

          NWORK is INTEGER
          The number of entries in WORK.  This must be at least
          max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) =
          max(    360     ,6*NN(j)+2*NN(j)**2)    for all j.

RWORK

          RWORK is DOUBLE PRECISION array, dimension (2*max(NN,12))

INFO

          INFO is INTEGER
          If 0,  then successful exit.
          If <0, then input parameter -INFO is incorrect.
          If >0, ZLATMR, CLATMS, CLATME or ZGET23 returned an error
                 code, and INFO is its absolute value.
-----------------------------------------------------------------------
     Some Local Variables and Parameters:
     ---- ----- --------- --- ----------
     ZERO, ONE       Real 0 and 1.
     MAXTYP          The number of types defined.
     NMAX            Largest value in NN or 12.
     NERRS           The number of tests which have exceeded THRESH
     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.
     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)       Selectw whether CONDS is to be 1 or
                     1/sqrt(ulp).  (0 means irrelevant.)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.