SGET52  does an eigenvector check for the generalized eigenvalue


 problem.


 The basic test for right eigenvectors is:


                           | b(j) A E(j) -  a(j) B E(j) |


         RESULT(1) = max   -------------------------------


                      j    n ulp max( |b(j) A|, |a(j) B| )


 using the 1-norm.  Here, a(j)/b(j) = w is the j-th generalized


 eigenvalue of A - w B, or, equivalently, b(j)/a(j) = m is the j-th


 generalized eigenvalue of m A - B.


 For real eigenvalues, the test is straightforward.  For complex


 eigenvalues, E(j) and a(j) are complex, represented by


 Er(j) + i*Ei(j) and ar(j) + i*ai(j), resp., so the test for that


 eigenvector becomes


                 max( |Wr|, |Wi| )


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


     n ulp max( |b(j) A|, (|ar(j)|+|ai(j)|) |B| )


 where


     Wr = b(j) A Er(j) - ar(j) B Er(j) + ai(j) B Ei(j)


     Wi = b(j) A Ei(j) - ai(j) B Er(j) - ar(j) B Ei(j)


                         T   T  _


 For left eigenvectors, A , B , a, and b  are used.


 SGET52 also tests the normalization of E.  Each eigenvector is


 supposed to be normalized so that the maximum 'absolute value'


 of its elements is 1, where in this case, 'absolute value'


 of a complex value x is  |Re(x)| + |Im(x)| ; let us call this


 maximum 'absolute value' norm of a vector v  M(v).


 if a(j)=b(j)=0, then the eigenvector is set to be the jth coordinate


 vector.  The normalization test is:


         RESULT(2) =      max       | M(v(j)) - 1 | / ( n ulp )


                    eigenvectors v(j)

LEFT

          LEFT is LOGICAL


          =.TRUE.:  The eigenvectors in the columns of E are assumed


                    to be *left* eigenvectors.


          =.FALSE.: The eigenvectors in the columns of E are assumed


                    to be *right* eigenvectors.

N

          N is INTEGER


          The size of the matrices.  If it is zero, SGET52 does


          nothing.  It must be at least zero.

A

          A is REAL array, dimension (LDA, N)


          The matrix A.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at least 1


          and at least N.

B

          B is REAL array, dimension (LDB, N)


          The matrix B.

LDB

          LDB is INTEGER


          The leading dimension of B.  It must be at least 1


          and at least N.

E

          E is REAL array, dimension (LDE, N)


          The matrix of eigenvectors.  It must be O( 1 ).  Complex


          eigenvalues and eigenvectors always come in pairs, the


          eigenvalue and its conjugate being stored in adjacent


          elements of ALPHAR, ALPHAI, and BETA.  Thus, if a(j)/b(j)


          and a(j+1)/b(j+1) are a complex conjugate pair of


          generalized eigenvalues, then E(,j) contains the real part


          of the eigenvector and E(,j+1) contains the imaginary part.


          Note that whether E(,j) is a real eigenvector or part of a


          complex one is specified by whether ALPHAI(j) is zero or not.

LDE

          LDE is INTEGER


          The leading dimension of E.  It must be at least 1 and at


          least N.

ALPHAR

          ALPHAR is REAL array, dimension (N)


          The real parts of the values a(j) as described above, which,


          along with b(j), define the generalized eigenvalues.


          Complex eigenvalues always come in complex conjugate pairs


          a(j)/b(j) and a(j+1)/b(j+1), which are stored in adjacent


          elements in ALPHAR, ALPHAI, and BETA.  Thus, if the j-th


          and (j+1)-st eigenvalues form a pair, ALPHAR(j+1)/BETA(j+1)


          is assumed to be equal to ALPHAR(j)/BETA(j).

ALPHAI

          ALPHAI is REAL array, dimension (N)


          The imaginary parts of the values a(j) as described above,


          which, along with b(j), define the generalized eigenvalues.


          If ALPHAI(j)=0, then the eigenvalue is real, otherwise it


          is part of a complex conjugate pair.  Complex eigenvalues


          always come in complex conjugate pairs a(j)/b(j) and


          a(j+1)/b(j+1), which are stored in adjacent elements in


          ALPHAR, ALPHAI, and BETA.  Thus, if the j-th and (j+1)-st


          eigenvalues form a pair, ALPHAI(j+1)/BETA(j+1) is assumed to


          be equal to  -ALPHAI(j)/BETA(j).  Also, nonzero values in


          ALPHAI are assumed to always come in adjacent pairs.

BETA

          BETA is REAL array, dimension (N)


          The values b(j) as described above, which, along with a(j),


          define the generalized eigenvalues.

WORK

          WORK is REAL array, dimension (N**2+N)

RESULT

          RESULT is REAL array, dimension (2)


          The values computed by the test described above.  If A E or


          B E is likely to overflow, then RESULT(1:2) is set to


          10 / ulp.

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