GENEIG Call

CALL GENEIG (eval, evecs, sym-matrix1, sym-matrix2) ;

The GENEIG subroutine computes eigenvalues and eigenvectors of a generalized eigenproblem.

The input arguments to the GENEIG subroutine are as follows:

sym-matrix1

is a symmetric numeric matrix.

sym-matrix2

is a positive definite symmetric matrix.

The subroutine returns the following output arguments:

evals

names a vector in which the eigenvalues are returned.

evecs

names a matrix in which the corresponding eigenvectors are returned.

The GENEIG subroutine computes eigenvalues and eigenvectors of the generalized eigenproblem. If $\mb {A}$ and $\mb {B}$ are symmetric and $\mb {B}$ is positive definite, then the vector $\mb {M}$ and the matrix $\mb {E}$ solve the generalized eigenproblem provided that

\[  \mb {A}*\mb {E} = \mb {B}*\mb {E}*\mbox{diag}(\mb {M})  \]

The vector $\mb {M}$ contains the eigenvalues arranged in descending order, and the matrix $\mb {E}$ contains the corresponding eigenvectors in the columns.

The following example is from Wilkinson and Reinsch (1971):

A = {10   2   3   1   1,
      2  12   1   2   1,
      3   1  11   1  -1,
      1   2   1   9   1,
      1   1  -1   1  15};

B = {12   1  -1   2    1,
      1  14   1  -1    1,
     -1   1  16  -1    1,
      2  -1  -1  12   -1,
      1   1   1  -1   11};

call geneig(M, E, A, B);
print M, E;

Figure 24.142: Solution of a Generalized Eigenproblem

M
1.4923532
1.1092845
0.943859
0.6636627
0.4327872

E
-0.076387 0.142012 0.19171 -0.08292 -0.134591
0.017098 0.14242 -0.158991 -0.153148 0.0612947
-0.066665 0.1209976 0.0748391 0.1186037 0.1579026
0.086048 0.125531 -0.137469 0.182813 -0.109466
0.2894334 0.0076922 0.0889779 -0.003562 0.041473