# Poster presented by James Sexton

Poster Session

Topic: Algorithms and Machines

### Numerical Stability of Lanczos Methods for Inner Product Calculation

Abstract: The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues. The mechanism by which this instability arises is well documented in the literature. A recent application of the Lanczos algorithm proposed by Bai, Fahey, and Golub allows quadrature evaluation of inner products of the form $(\psi, f(M) \psi)$. We prove that this quadrature evaluation is numerically stable, and explain how the the finite precision numerical errors which are such a fundamental element of the Lanczos tridiagonalisation procedure are automatically and exactly compensated in the Bai, Fahey and Golub algorithm. Application of the method to evaluate fermion determinants is discussed.

Massimo Campostrini,   campo@mailbox.difi.unipi.it