Generalized positive definite EVD

The generalized symmetric positive-definite eigenvalue problem is one of the following eigenproblems:

where A is a symmetric matrix, and B is a symmetric positive-definite matrix.

It is obvious that this problem is easily reduced to the problem of finding eigenvalues for a non-symmetric general matrix (we can perform this reduction by multiplying both sides of the system by B^-1 in the first case, and by multiplying together matrices A and B in the second and the third cases). However, the non-symmetric eigenvalue problem is much more complex, therefore it is reasonable to find a more effective way of solving the generalized symmetric problem. For instance, we can reduce this problem to a classic symmetric problem by using the Cholesky decomposition of matrix B (the example below applies to the first problem).

so we get the following problem:

The eigenvalues for both problems are the same, the eigenvectors for the initial problem could be found by solving a system of linear equations with a triangular matrix. Similar transformations could be performed for two other generalized problems.

Subroutine description

This module contains two subroutines for solving a generalized symmetric positive-definite eigenvalue problem. The first subroutine, SMatrixGEVDReduce, performs the reduction of the problem to a classic symmetric problem. It returns matrices C (problem matrix) and R (triangular matrix which is used to find the eigenvectors). The second subroutine, SMatrixGEVD, uses the first one to solve a generalized problem. It calls all the necessary subroutines by itself and transforms the obtained vectors.

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