SGET22 does an eigenvector check.


 The basic test is:


    RESULT(1) = | A E  -  E W | / ( |A| |E| ulp )


 using the 1-norm.  It also tests the normalization of E:


    RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )


                 j


 where E(j) is the j-th eigenvector, and m-norm is the max-norm of a


 vector.  If an eigenvector is complex, as determined from WI(j)


 nonzero, then the max-norm of the vector ( er + i*ei ) is the maximum


 of


    |er(1)| + |ei(1)|, ... , |er(n)| + |ei(n)|


 W is a block diagonal matrix, with a 1 by 1 block for each real


 eigenvalue and a 2 by 2 block for each complex conjugate pair.


 If eigenvalues j and j+1 are a complex conjugate pair, so that


 WR(j) = WR(j+1) = wr and WI(j) = - WI(j+1) = wi, then the 2 by 2


 block corresponding to the pair will be:


    (  wr  wi  )


    ( -wi  wr  )


 Such a block multiplying an n by 2 matrix ( ur ui ) on the right


 will be the same as multiplying  ur + i*ui  by  wr + i*wi.


 To handle various schemes for storage of left eigenvectors, there are


 options to use A-transpose instead of A, E-transpose instead of E,


 and/or W-transpose instead of W.

TRANSA

          TRANSA is CHARACTER*1


          Specifies whether or not A is transposed.


          = 'N':  No transpose


          = 'T':  Transpose


          = 'C':  Conjugate transpose (= Transpose)

TRANSE

          TRANSE is CHARACTER*1


          Specifies whether or not E is transposed.


          = 'N':  No transpose, eigenvectors are in columns of E


          = 'T':  Transpose, eigenvectors are in rows of E


          = 'C':  Conjugate transpose (= Transpose)

TRANSW

          TRANSW is CHARACTER*1


          Specifies whether or not W is transposed.


          = 'N':  No transpose


          = 'T':  Transpose, use -WI(j) instead of WI(j)


          = 'C':  Conjugate transpose, use -WI(j) instead of WI(j)

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          The matrix whose eigenvectors are in E.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

E

          E is REAL array, dimension (LDE,N)


          The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors


          are stored in the columns of E, if TRANSE = 'T' or 'C', the


          eigenvectors are stored in the rows of E.

LDE

          LDE is INTEGER


          The leading dimension of the array E.  LDE >= max(1,N).

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)


          The real and imaginary parts of the eigenvalues of A.


          Purely real eigenvalues are indicated by WI(j) = 0.


          Complex conjugate pairs are indicated by WR(j)=WR(j+1) and


          WI(j) = - WI(j+1) non-zero; the real part is assumed to be


          stored in the j-th row/column and the imaginary part in


          the (j+1)-th row/column.

WORK

          WORK is REAL array, dimension (N*(N+1))

RESULT

          RESULT is REAL array, dimension (2)


          RESULT(1) = | A E  -  E W | / ( |A| |E| ulp )


          RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )


                       j

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