Applications and Computation of Orthogonal Polynomials: by Bernhard Beckermann, Edward B. Saff (auth.), Walter

By Bernhard Beckermann, Edward B. Saff (auth.), Walter Gautschi, Gerhard Opfer, Gene H. Golub (eds.)

The workshop on functions and Computation of Orthogonal Polynomials happened March 22-28, 1998 on the Oberwolfach Mathematical learn Institute. It used to be the 1st workshop in this subject ever held at Oberwolfach. there have been forty six contributors from thirteen international locations, greater than part coming from Germany and the USA, and a considerable quantity from Italy. a complete of 23 plenary lectures have been offered and four brief casual talks. Open difficulties have been mentioned in the course of a night consultation. This quantity includes refereed models of 18 papers offered at, or submitted to, the convention. the idea of orthogonal polynomials, as a department of classical research, is easily verified. yet orthogonal polynomials play additionally an enormous position in lots of components of medical computing, similar to least squares becoming, numerical integration, and fixing linear algebraic platforms. although the elemental tenets have their roots in nineteenth­ century arithmetic, using smooth pcs has required the improvement and examine of recent algorithms which are exact and powerful. The computational equipment and functions represented during this quantity, of necessity, are incomplete, but sufficiently assorted to show an effect of present actions during this area.

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Extra info for Applications and Computation of Orthogonal Polynomials: Conference at the Mathematical Research Institute Oberwolfach, Germany March 22–28, 1998

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R Graves-Morris, A "look-around Lanczos" algorithm for solving a system of linear equations, Numer. Algorithms, 15 (1997), 247-274. R Graves-Morris and A. Salam, Avoiding breakdown in van der Vorst's method, Numer. Algorithms, to appear. H. Gutknecht, The unsymmetric Lanczos algorithms and their relations to Pade approximation, continued fractions, and the qd algorithm, in: Preliminary Proceedings of the Copper Mountain Conference on Iterative Methods, April 1-5, 1990. H. Gutknecht, A completed theory of the unsymmetric Lanczos process and related algorithms.

131, © 1999 Birkhiiuser Verlag BaseVSwitzerland Applications of Anti-Gauss Quadrature Rules in Linear Algebra Daniela Calvetti, Lothar Reichel, and Fiorella Sgallari Abstract. The need to inexpensively determine upper and lower bounds for matrix functionals of the form F(A) := uT f(A)u arises in a large number of applications. Here A denotes a large symmetric matrix and u is a vector. Golub and collaborators have described how such bounds can be computed by using Gauss and Gauss-Radau quadrature rules when the derivatives ::, f(t), j = 1,2, ...

1. Introduction Lanczos' method [38] for solving a system of linear equations Ax = b is implemented via three-term or coupled two-term recurrence relationships. In these algorithms, an important problem is division by zero (called breakdown) or by a quantity close to zero (called near-breakdown). In the case of breakdown, the algorithm has to be stopped while, in the case of near-breakdown, it can become numerically unstable. A second problem is due to the necessity of computing products of the form AT v.

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