Symmetric eigenproblem: bisection and inverse iteration

Real number λ and vector z are called an eigen pair of matrix A, if Az = λz. 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).

For a real matrix A, there might be both the problem of finding all the eigenvalues and eigenvectors (the so-called matrix spectrum) and the problem of finding part of a spectrum. If not all the eigenvalues are required, we can use the bisection method to find the eigenvalues from a given interval (or having given numbers). After that, we can find the eigenvectors by using the inverse iteration method. If we only have to find a small part of the spectrum, we can increase the performance considerably in comparison with QL/QR algorithm.

As when using QL/QR algorithm, the symmetric matrix is reduced to tridiagonal form and then the bisection and inverse iteration methods which are working with the tridiagonal matrix are called.

Convergence and precision

Let's consider the reliability of such a calculation procedure. This methods are rather reliable when dealing with matrices with well-separated eigenvalues. If some eigenvalues are close to each other, the inverse iteration method can return an inaccurate set of eigenvectors (the closer the eigenvalues are, the less accurate the eigenvectors will be).

It should be understood that such difficulties are typical of all methods of finding eigenvalues and no satisfactory solution has yet been found. Nevertheless, this method is applicable to the majority of real problems. If the algorithm is compared with the calculation problem which it couldn't solve, the subroutine returns False. In this case, it is reasonable to use QL/QR algorithm which is more stable than the inverse iteration method, and then select a required vector from the full set of eigenvectors found (it is rather a slow way but you will get a solution).

Subroutine description

To find the eigenvalues (and their corresponding eigenvectors) from the given half-interval (A, B], use the SMatrixEVDR subroutine. The SMatrixEVDI subroutine finds the eigen pairs having given numbers (the spectrum is considered as being sorted in ascending order).

It should be noted that the algorithm is effective only when finding a small part of the spectrum. If it is required to find all eigenvalues (or the majority of them), the QL/QR algorithm is more effective.

This algorithm is transferred from the LAPACK library.