The Lanczos and conjugate gradient algorithms in finite precision arithmetic

Abstract

Dedicated to Chris Paige for his fundamental contributions to the rounding error analysis of the Lanczos algorithmThe Lanczos and conjugate gradient algorithms were introduced more than five decades ago as tools for numerical computation of dominant eigenvalues of symmetric matrices and for solving linear algebraic systems with symmetric positive definite matrices, respectively. Because of their fundamental relationship with the theory of orthogonal polynomials and Gauss quadrature of the Riemann-Stieltjes integral, the Lanczos and conjugate gradient algorithms represent very interesting general mathematical objects, with highly nonlinear properties which can be conveniently translated from algebraic language into the language of mathematical analysis, and vice versa. The algorithms are also very interesting numerically, since their numerical behaviour can be explained by an elegant mathematical theory, and the interplay between analysis and algebra is useful there too. Motivated by this view, the present contribution wishes to pay a tribute to those who have made an understanding of the Lanczos and conjugate gradient algorithms possible through their pioneering work, and to review recent solutions of several open problems that have also contributed to knowledge of the subject.

Static Analysis of Finite Precision Computations

“…this is a well-known phenomenon for well-behaved problems such as this one, called orthogonality defects, see [19].…”

An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process

“…The matrices U k and U k are defined in equations (5), u k+1 and u k+1 in equations (6), F k and F k in equations (7), R k and R k in equations (11), and the Sheffield augmentation P k and P k in equations (16). Then with A k in (22),…”

Experiences with a Lanczos Eigensolver in High-Precision Arithmetic

“…The Lanczos process is a long-established and well-known eigensolver [15] (see also [10,17,23,24,29]). It takes as input an n-by-n Hermitian matrix A and produces a sequence of matrices T (m) and Q (m) such that…”

“…To the best of our knowledge none of the Lanczos codes that have been published since 1985 have been classical Lanczos. Even though development of new codes has slowed down, there has been intense ongoing theoretical interest in classical Lanczos, resulting in a large body of results (see [17] and the numerous references therein). Experimental studies have also been published [13].…”