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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C#

C# 1.0 source.
spdgevd.csharp.zip - Generalized symmetric positive definite eigenproblem

C++

C++ source.
spdgevd.cpp.zip - Generalized symmetric positive definite eigenproblem
ablas.zip - optimized basic linear algebra subroutines with SSE2 support (for C++ sources only)

C++, multiple precision arithmetic

C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
spdgevd.mpfr.zip - Generalized symmetric positive definite eigenproblem
mpfr.zip - precompiled Win32 MPFR/GMP binaries

Delphi

Delphi source.
Can be compiled under FPC (in Delphi compatibility mode).
spdgevd.delphi.zip - Generalized symmetric positive definite eigenproblem

Visual Basic 6

Visual Basic 6 source.
spdgevd.vb6.zip - Generalized symmetric positive definite eigenproblem

Zonnon ^beta

Zonnon source.
Zonnon is an experimental language developed at ETH Zurich.
See www.zonnon.ethz.ch for more information.
spdgevd.zonnon.zip - Generalized symmetric positive definite eigenproblem