SDRGEV checks the nonsymmetric generalized eigenvalue problem driver


 routine SGGEV.


 SGGEV computes for a pair of n-by-n nonsymmetric matrices (A,B) the


 generalized eigenvalues and, optionally, the left and right


 eigenvectors.


 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w


 or a ratio  alpha/beta = w, such that A - w*B is singular.  It is


 usually represented as the pair (alpha,beta), as there is reasonable


 interpretation for beta=0, and even for both being zero.


 A right generalized eigenvector corresponding to a generalized


 eigenvalue  w  for a pair of matrices (A,B) is a vector r  such that


 (A - wB) * r = 0.  A left generalized eigenvector is a vector l such


 that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.


 When SDRGEV 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, a pair of matrices (A, B) will be generated


 and used for testing.  For each matrix pair, the following tests


 will be performed and compared with the threshold THRESH.


 Results from SGGEV:


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


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


      where VL**H is the conjugate-transpose of VL.


 (2)  | |VL(i)| - 1 | / ulp and whether largest component real


      VL(i) denotes the i-th column of VL.


 (3)  max over all left eigenvalue/-vector pairs (alpha/beta,r) of


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


 (4)  | |VR(i)| - 1 | / ulp and whether largest component real


      VR(i) denotes the i-th column of VR.


 (5)  W(full) = W(partial)


      W(full) denotes the eigenvalues computed when both l and r


      are also computed, and W(partial) denotes the eigenvalues


      computed when only W, only W and r, or only W and l are


      computed.


 (6)  VL(full) = VL(partial)


      VL(full) denotes the left eigenvectors computed when both l


      and r are computed, and VL(partial) denotes the result


      when only l is computed.


 (7)  VR(full) = VR(partial)


      VR(full) denotes the right eigenvectors computed when both l


      and r are also computed, and VR(partial) denotes the result


      when only l is computed.


 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) Q ( J , J ) Z     where Q and Z are random orthogonal matrices.


 (17) Q ( T1, T2 ) Z    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) Q ( T1, T2 ) Z    diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )


                        diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )


                        s = machine precision.


 (19) Q ( T1, T2 ) Z    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) Q ( T1, T2 ) Z    diag(T1)=( 0, 0, 1, 1, a, ..., a   =s, 0 )


                        diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )


 (21) Q ( T1, T2 ) Z    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) Q ( big*T1, small*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )


                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )


 (23) Q ( small*T1, big*T2 ) Z    diag(T1) = ( 0, 0, 1, ..., N-3, 0 )


                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )


 (24) Q ( small*T1, small*T2 ) Z  diag(T1) = ( 0, 0, 1, ..., N-3, 0 )


                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )


 (25) Q ( big*T1, big*T2 ) Z      diag(T1) = ( 0, 0, 1, ..., N-3, 0 )


                                  diag(T2) = ( 0, 1, ..., 1, 0, 0 )


 (26) Q ( T1, T2 ) Z     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,


          SDRGES does nothing.  NSIZES >= 0.

NN

          NN is INTEGER array, dimension (NSIZES)


          An array containing the sizes to be used for the matrices.


          Zero values will be skipped.  NN >= 0.

NTYPES

          NTYPES is INTEGER


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


          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 SDRGES 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.

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


          (e.g., if a routine returns IERR 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, S, and T.


          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.

S

          S is REAL array,


                                 dimension (LDA, max(NN))


          The Schur form matrix computed from A by SGGES.  On exit, S


          contains the Schur form matrix corresponding to the matrix


          in A.

T

          T is REAL array,


                                 dimension (LDA, max(NN))


          The upper triangular matrix computed from B by SGGES.

Q

          Q is REAL array,


                                 dimension (LDQ, max(NN))


          The (left) eigenvectors matrix computed by SGGEV.

LDQ

          LDQ is INTEGER


          The leading dimension of Q and Z. It must


          be at least 1 and at least max( NN ).

Z

          Z is REAL array, dimension( LDQ, max(NN) )


          The (right) orthogonal matrix computed by SGGES.

QE

          QE is REAL array, dimension( LDQ, max(NN) )


          QE holds the computed right or left eigenvectors.

LDQE

          LDQE is INTEGER


          The leading dimension of QE. LDQE >= max(1,max(NN)).

ALPHAR

          ALPHAR is REAL array, dimension (max(NN))

ALPHAI

          ALPHAI is REAL array, dimension (max(NN))

BETA

          BETA is REAL array, dimension (max(NN))


 \verbatim


          The generalized eigenvalues of (A,B) computed by SGGEV.


          ( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th


          generalized eigenvalue of A and B.

ALPHR1

          ALPHR1 is REAL array, dimension (max(NN))

ALPHI1

          ALPHI1 is REAL array, dimension (max(NN))

BETA1

          BETA1 is REAL array, dimension (max(NN))


          Like ALPHAR, ALPHAI, BETA, these arrays contain the


          eigenvalues of A and B, but those computed when SGGEV only


          computes a partial eigendecomposition, i.e. not the


          eigenvalues and left and right eigenvectors.

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          The number of entries in WORK.  LWORK >= MAX( 8*N, N*(N+1) ).

RESULT

          RESULT is REAL array, dimension (2)


          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.

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