Lanczos research project

Sponsored by

FWO project

Short description

The Lanczos algorithm is an iterative method to solve linear systems or eigenvalue problems, usually for large sparse matrices. For the case of a non-symmetric matrix, look-ahead steps are needed to avoid breakdown in the iteration steps. The resulting recurrence relation is precisely the one which is used in the classical Euclidean algorithm for the computation of the greatest common divisor which, modulo the numerical aspects, can be seen as the oldest and most natural look-ahead method. This algorithm also computes an inverse block triangular factorization of indefinite Hankel matrices (Gram-Schmidt) and thus it is linked with formal orthogonal polynomials. This connection between formal orthogonal polynomials and the basic iterative methods is emphasized in the research. Generalizations of classical methods are investigated Toeplitz structured and unstructured metric.

The project resulted in the publication of the book

A. Bultheel and M. Van Barel. Linear Algebra, Rational Approximation and Orthogonal Polynomials. Studies in Computational Mathematics. Elsevier, Amsterdam, 1997. (446 pages).

Researchers

• Adhemar Bultheel
• Marc Van Barel