Hermitian eigenproblem: bisection and inverse iteration

Complex number λ and complex vector z are called an eigen pair of complex matrix A, if Az = λz. If matrix A of size NxN is Hermitian, 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 less than N).

For a Hermitian 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 indexes). 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. For more information see description of the similar algorithm for real symmetric matrices.

Subroutine description

To find the eigenvalues (and their corresponding eigenvectors) from the given half-interval (A, B], use the HMatrixEVDR subroutine. The HMatrixEVDI 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 3.0