Theoretically elegant and ubiquitous in practice, the Lanczos method can approximate f(A)x for any symmetric matrix A ∈ R, vector x ∈ R, and function f . In exact arithmetic, the method’s error after k iterations is bounded by the error of the best degree-k polynomial uniformly approximating the scalar function f(x) on the range [λmin(A), λmax(A)]. However, despite decades of work, it has been ...

For a given real n n matrix A and initial vectors v 1 and w 1 , we examine the sensitivity of the tridiagonal matrix T and the biorthogonal sets of vectors of the Lanczos reduction to small changes in A, v 1 and w 1. We also consider the sensitivity of the developing Krylov subspaces.

Abstract We show that a proposal by Vishwakarma to realize conformal-covariance for the Weyl-Lanczos equation is nonviable.

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We prove that a Lanczos potential Labc for the Weyl candidate tensor Wabcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for Labc...

Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say ...

Abstract. Many problems in science and engineering require the evaluation of functionals of the form Fu(A) = uT f(A)u, where A is a large symmetric matrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector...

The numerical and computational aspects of the overlap formalism in lattice quantum chromodynamics are extremely demanding due to a matrix-vector product that involves the sign function of the hermitian Wilson matrix. In this paper we investigate several methods to compute the product of the matrix sign-function with a vector, in particular Lanczos based methods and partial fraction expansion m...