Real number λ and vector z are called an eigen pair of matrix A, if Az = λz. For a real matrix A there could be both the problem of finding the eigenvalues and the problem of finding the eigenvalues and eigenvectors. There could also be the problem of finding the eigen pairs which meet given conditions (say, only finding the biggest by absolute value eigenvalues).

If matrix A of size NxN is symmetric, it has N eigenvalues (not necessarily distinctive) and N corresponding eigenvectors which form an orthonormal basis (generally, eigenvectors are not orthogonal, and their number could be lower than N).

This algorithm finds all the eigenvalues (and, if needed, the eigenvectors) of a symmetric tridiagonal matrix. It is also used for finding eigen pairs of an arbitrary symmetric matrix, as each symmetric matrix could be reduced to tridiagonal form by the orthogonal transformation of similarity which doesn't change the matrix eigenvalues.

Subroutine description

The SMatrixTDEVD subroutine finds the eigenvalues (and, if needed, the eigenvectors) of a tridiagonal matrix which is given by its main diagonal D and secondary diagonal E.

The ZNeeded parameter controls whether the eigenvectors are required or not. It is possible to get the eigenvectors of a tridiagonal matrix or pass a transformation matrix which reduces the symmetric matrix to tridiagonal form to the algorithm and get the eigenvalues of an original symmetric matrix.

The algorithm is iterative, so, theoretically, it may not converge. In this case, it returns False.

This algorithm is transferred from the LAPACK library.

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C#

C# 1.0 source.
tdevd.csharp.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix

C++

C++ source.
tdevd.cpp.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix
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.
tdevd.mpfr.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix
mpfr.zip - precompiled Win32 MPFR/GMP binaries

Delphi

Delphi source.
Can be compiled under FPC (in Delphi compatibility mode).
tdevd.delphi.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix

Visual Basic 6

Visual Basic 6 source.
tdevd.vb6.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix

Zonnon ^beta

Zonnon source.
Zonnon is an experimental language developed at ETH Zurich.
See www.zonnon.ethz.ch for more information.
tdevd.zonnon.zip - Eigenvalues and eigenvectors of a real symmetric tridiagonal matrix