GENEIG Call :: SAS/IML(R) 13.1 User's Guide

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