Abstrakt
In theory, the Lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic, the orthogonality and linear independence of the computed Lanczos vectors is usually lost quickly.
In this presentation the speaker will parametrize a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the Lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results will be shown for a variant of the conjugate gradient method that produces almost exact results.
Finally, we will discuss the usage of the obtained results in the analysis of theoretical as well as finite precision behaviour of the considered algorithms