The generalized symmetric positive-definite eigenvalue problem is one of the following eigenproblems:Ax = λBx
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).Ax = λBx
so we get the following problem:Cy = λy
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.
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.
This article is intended for personal use only.
C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
Python version (CPython and IronPython are supported).
ALGLIB® - numerical analysis library, 1999-2013.