Jeudi 9 novembre 2000 Séminaire d'analyse numérique et mécanique, Université de Rennes 1


Henk van der VORST

Mathematical Institute
Utrecht University
The Netherlands

Matrix notation was unknown in 1846. This did not prevent Jacobi from inventing an algorithm for the computation of eigenvalues [1].
He did this in the context of linear systems related to the stability of the orbits of the then known 7 planets.
His method was reinvented 100 years later by Von Neumann and his colleagues, and part of Jacobi's method became popular for linear eigenproblems Ax=cx. The method was overtaken some thirty years later by more successful methods: QR and the methods of Lanczos and Arnoldi.

In 1975, the chemist Davidson proposed a new method that became very popular, specially for applications in Chemistry [2]. Numerical analysts all over the world largely ignored his method, mainly because of lack of understanding.

In 1993 our numerical pride was triggered by chemists from Utrecht, who impressed us with the apparant superiority of the Davidson method for their problems. At the same time, a student at our institute did her Master thesis on the old publications of Jacobi. This led to some remarkable observations and eventually it led to a happy marriage between parts of the methods of Jacobi and Davidson [3].
Impressive results could be obtained for unusual eigenproblems associated with aircraft design [4,5].

[1] C.G.J. Jacobi, `Ueber ein leichtes Verfahren, die in der Theorie der Saecularstoerungen vorkommenden Gleichungen numerisch aufzuloesen', J. fuer die reine und Angew. Math., 1846, p.51-94
[2] E.R. Davidson, `The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real symmetric matrices', J. Comp. Phys, 17, 1975, p.87-94