In this paper the new theoretical error bounds on the convergence of the Lanczos and the block-Lanczos methods are established based on results given by Saad. Similar further inequalities are found for the eigenele-ments by using bounds on the acute angle between the exact eigenvectors and the Krylov subspace spanned by x0; Ax0; ; A n?1 x0, where x0 is the initial starting vector of the process...

We establish rigourously the scaling properties of the Lanczos process applied to an arbitrary extensive Many-Body System which is carried to convergence n → ∞ and the thermodynamic limit N → ∞ taken. In this limit the solution for the limiting Lanczos coefficients are found exactly and generally through two equivalent sets of equations, given initial knowledge of the exact cumulant generating ...

For real symmetric eigenvalue problems, there are a number of algorithms that are mathematically equivalent, for example, the Lanczos algorithm, the Arnoldi method and the unpreconditioned Davidson method. The Lanczos algorithm is often preferred because it uses signiicantly fewer arithmetic operations per iteration. To limit the maximum memory usage, these algorithms are often restarted. In re...

In 1950 Lanczos [22] proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrice...

The Lanczos process is a well known technique for computing a few, say k, eigenvalues and associated eigenvectors of a large symmetric n×n matrix. However, loss of orthogonality of the computed Krylov subspace basis can reduce the accuracy of the computed approximate eigenvalues. In the implicitly restarted Lanczos method studied in the present paper, this problem is addressed by fixing the num...

First several algorithms based on the unsymmetric Lanczos process are surveyed the biorthog onalization BO algorithm for constructing a tridiagonal matrix T similar to a given matrix A whose ex treme spectrum is sought typically the BOBC algorithm which generates directly the LU factors of T the Biores Lanczos Orthores Biomin Lanczos Orthomin or biconjugate gradient BCG and the Biodir Lanczos O...

Lanczos-type product methods for solving large sparse non-Hermitian linear systems have as residual polynomials either the squares of the Lanczos polynomials or the products of the latter with another sequence of polynomials, which is normally chosen to enforce some local minimization of the residual norm. In either case, these methods inherit from the underlying Lanczos process the danger of b...

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. We present an implementation of a look-ahead version of the Lanczos algorithm that|except for the very special situation of an in...

this paper describes the application of approximate methods to invert airborne magnetic data as well as helicopter-borne frequency domain electromagnetic data in order to retrieve a joint model of magnetic susceptibility and electrical resistivity. the study area located in semnan province of iran consists of an arc-shaped porphyry andesite covered by sedimentary units which may have potential ...