Bernhard Beckermann

A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants

Recently, a uniform approach was given by B. Beckermann and G. Labahn [Numer. Algorithms, 3 (1992), pp. 45-54] for different concepts of matrix-type Pade approximants, such as descriptions of vector and matrix Pade approximants along with generalizations of simultaneous and Hermite Pade approximants. The considerations in this paper are based on this generalized form of the classical scalar Hermite Pade approximation problem, power Hermite Pade approximation. In particular, this paper studies the problem of computing these new approximants. A recurrence relation is presented for the computation of a basis for the corresponding linear solution space of these approximants. This recurrence also provides bases for particular subproblems. This generalizes previous work by Van Barel and Bultheel and, in a more general form, by Beckermann. The computation of the bases has complexity

Superlinear Convergence of Conjugate Gradients

{\cal O}(\sigma^{2})

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

, where

Error Estimates and Evaluation of Matrix Functions via the Faber Transform

\sigma

Fraction-free row reduction of matrices of Ore polynomials

is the order of the desired approximant and requires no conditions on the input data. A second algorithm using the same recurrence relation along with divide-and-conquer methods is also presented. When the coefficient field allows for fast polynomial multiplication, this second algorithm computes a basis in the superfast complexity

Complex Jacobi matrices

{\cal O}(\sigma \log^{2})

Shifted normal forms of polynomial matrices

. In both cases the algorithms are reliable in exact arithmetic. That is, they never break down, and the complexity depends neither on any normality assumptions nor on the singular structure of the corresponding solution table. As a further application, these methods result in fast (and superfast) reliable algorithms for the inversion of striped Hankel, layered Hankel, and (rectangular) block-Hankel matrices.

When are Two Numerical Polynomials Relatively Prime

We give a theoretical explanation for superlinear convergence behavior observed while solving large symmetric systems of equations using the conjugate gradient method or other Krylov subspace methods. We present a new bound on the relative error after

Some Remarks on the Elman Estimate for GMRES

n

Recursiveness in matrix rational interpolation problems

iterations. This bound is valid in an asymptotic sense when the size