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

## 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