DDRGVX checks the nonsymmetric generalized eigenvalue problem


 expert driver DGGEVX.


 DGGEVX computes the generalized eigenvalues, (optionally) the left


 and/or right eigenvectors, (optionally) computes a balancing


 transformation to improve the conditioning, and (optionally)


 reciprocal condition numbers for the eigenvalues and eigenvectors.


 When DDRGVX is called with NSIZE > 0, two types of test matrix pairs


 are generated by the subroutine DLATM6 and test the driver DGGEVX.


 The test matrices have the known exact condition numbers for


 eigenvalues. For the condition numbers of the eigenvectors


 corresponding the first and last eigenvalues are also know


 ``exactly'' (see DLATM6).


 For each matrix pair, the following tests will be performed and


 compared with the threshold THRESH.


 (1) max over all left eigenvalue/-vector pairs (beta/alpha,l) of


    | l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) )


     where l**H is the conjugate transpose of l.


 (2) max over all right eigenvalue/-vector pairs (beta/alpha,r) of


       | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )


 (3) The condition number S(i) of eigenvalues computed by DGGEVX


     differs less than a factor THRESH from the exact S(i) (see


     DLATM6).


 (4) DIF(i) computed by DTGSNA differs less than a factor 10*THRESH


     from the exact value (for the 1st and 5th vectors only).


 Test Matrices


 =============


 Two kinds of test matrix pairs


          (A, B) = inverse(YH) * (Da, Db) * inverse(X)


 are used in the tests:


 1: Da = 1+a   0    0    0    0    Db = 1   0   0   0   0


          0   2+a   0    0    0         0   1   0   0   0


          0    0   3+a   0    0         0   0   1   0   0


          0    0    0   4+a   0         0   0   0   1   0


          0    0    0    0   5+a ,      0   0   0   0   1 , and


 2: Da =  1   -1    0    0    0    Db = 1   0   0   0   0


          1    1    0    0    0         0   1   0   0   0


          0    0    1    0    0         0   0   1   0   0


          0    0    0   1+a  1+b        0   0   0   1   0


          0    0    0  -1-b  1+a ,      0   0   0   0   1 .


 In both cases the same inverse(YH) and inverse(X) are used to compute


 (A, B), giving the exact eigenvectors to (A,B) as (YH, X):


 YH:  =  1    0   -y    y   -y    X =  1   0  -x  -x   x


         0    1   -y    y   -y         0   1   x  -x  -x


         0    0    1    0    0         0   0   1   0   0


         0    0    0    1    0         0   0   0   1   0


         0    0    0    0    1,        0   0   0   0   1 , where


 a, b, x and y will have all values independently of each other from


 { sqrt(sqrt(ULP)),  0.1,  1,  10,  1/sqrt(sqrt(ULP)) }.

NSIZE

          NSIZE is INTEGER


          The number of sizes of matrices to use.  NSIZE must be at


          least zero. If it is zero, no randomly generated matrices


          are tested, but any test matrices read from NIN will be


          tested.

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.

NIN

          NIN is INTEGER


          The FORTRAN unit number for reading in the data file of


          problems to solve.

NOUT

          NOUT 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 DOUBLE PRECISION array, dimension (LDA, NSIZE)


          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, B, AI, BI, Ao, and Bo.


          It must be at least 1 and at least NSIZE.

B

          B is DOUBLE PRECISION array, dimension (LDA, NSIZE)


          Used to hold the matrix whose eigenvalues are to be


          computed.  On exit, B contains the last matrix actually used.

AI

          AI is DOUBLE PRECISION array, dimension (LDA, NSIZE)


          Copy of A, modified by DGGEVX.

BI

          BI is DOUBLE PRECISION array, dimension (LDA, NSIZE)


          Copy of B, modified by DGGEVX.

ALPHAR

          ALPHAR is DOUBLE PRECISION array, dimension (NSIZE)

ALPHAI

          ALPHAI is DOUBLE PRECISION array, dimension (NSIZE)

BETA

          BETA is DOUBLE PRECISION array, dimension (NSIZE)


          On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.

VL

          VL is DOUBLE PRECISION array, dimension (LDA, NSIZE)


          VL holds the left eigenvectors computed by DGGEVX.

VR

          VR is DOUBLE PRECISION array, dimension (LDA, NSIZE)


          VR holds the right eigenvectors computed by DGGEVX.

ILO

                ILO is INTEGER

IHI

                IHI is INTEGER

LSCALE

                LSCALE is DOUBLE PRECISION array, dimension (N)

RSCALE

                RSCALE is DOUBLE PRECISION array, dimension (N)

S

                S is DOUBLE PRECISION array, dimension (N)

DTRU

                DTRU is DOUBLE PRECISION array, dimension (N)

DIF

                DIF is DOUBLE PRECISION array, dimension (N)

DIFTRU

                DIFTRU is DOUBLE PRECISION array, dimension (N)

WORK

          WORK is DOUBLE PRECISION array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          Leading dimension of WORK.  LWORK >= 2*N*N+12*N+16.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)

LIWORK

          LIWORK is INTEGER


          Leading dimension of IWORK.  Must be at least N+6.

RESULT

                RESULT is DOUBLE PRECISION array, dimension (4)

BWORK

          BWORK is LOGICAL array, dimension (N)

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.

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