Stability of barycentric interpolation formulas

The barycentric interpolation formula defines a stable algorithm for evaluation at points in [−1, 1] of polynomial interpolants through data on Chebyshev grids. Here it is shown that for evaluation at points in the complex plane outside [−1, 1], the algorithm becomes unstable and should be replaced by the alternative modified Lagrange or "first barycentric" formula dating to Jacobi in 1825. This difference in stability confirms the theory published by N. J. Higham in 2004 (IMA J. Numer. Anal., v. 24) and has practical consequences for computation with rational functions

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Computing eigenvalues of real symmetric matrices\ud with rational filters in real arithmetic

Powerful algorithms have recently been proposed for computing eigenvalues of large matrices by methods related to contour integrals; best known are the works of Sakurai and coauthors and Polizzi and coauthors. Even if the matrices are real symmetric, most such methods rely on complex arithmetic, leading to expensive linear systems to solve. An appealing technique for overcoming this starts from the observation that certain discretized contour integrals are equivalent to rational interpolation problems, for which there is no need to leave the real axis. Investigation shows that using rational interpolation per se suffers from instability; however, related techniques involving real rational filters can be very effective. This article presents a technique of this kind that is related to previous work published in Japanese by Murakami

Eigenvalue Methods for Interpolation Bases

This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series ofpoints. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with polynomials expressed directly in the interpolation basis in which they were posed, avoiding the often ill-conditioned transformation between bases. Working within this framework, this thesis demonstrates that computing the roots of polynomials via these companion matrices is numerically stable, and the matrices involved can be reduced in such a way as to significantly lower the number of operations required to obtain the roots. Through examination of these various techniques, this thesis offers insight into the speed, stability, and accuracy of rootfinding algorithms for polynomials expressed in alternative bases

A QR based approach for the nonlinear eigenvalue problem

We describe a fast and numerically robust approach based on the structured QR eigenvalue algorithm for computing approximations of the eigenvalues of a holomorphic matrix-valued function inside the unit circle. Numerical experiments confirm the effectiveness of the proposed method