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

The workshop on functions and Computation of Orthogonal Polynomials came about March 22-28, 1998 on the Oberwolfach Mathematical examine Institute. It was once the 1st workshop in this subject ever held at Oberwolfach. there have been forty six individuals 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 awarded and four brief casual talks. Open difficulties have been mentioned in the course of a night consultation. This quantity includes refereed types of 18 papers provided at, or submitted to, the convention. the idea of orthogonal polynomials, as a department of classical research, is definitely verified. yet orthogonal polynomials play additionally a massive position in lots of parts of medical computing, reminiscent of least squares becoming, numerical integration, and fixing linear algebraic structures. notwithstanding the fundamental tenets have their roots in nineteenth­ century arithmetic, using sleek pcs has required the improvement and learn of latest algorithms which are exact and strong. The computational tools and purposes represented during this quantity, of necessity, are incomplete, but sufficiently diverse to show an impact of present actions during this area.

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

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Unfortunately, as explained in [8], although the algorithm can be extended for treating breakdowns, near-breakdowns cannot be cured by the technique which consists of jumping over the polynomials which do not exist and also over the existing ones which are badly computed. This is due to the nonexistence of the corresponding recurrence relationship for the orthogonal polynomials. The same is true for Lanczos / Orthobal. Thus, the first step was to find another algorithm, almost as stable as Lanczos/Orthomin, and where the treatment of near-breakdowns was possible by our technique.

15 (1994), 313-337. 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.

5. LTPM and transpose-free algorithms We now explain why, in order to obtain a cheap algorithm, it is necessary to make use of AT. For example, to compute the numerator of >'kH with the choice Uk(~) = ~k in Lanczos/Orthomin, we need to compute (y, Akrk ). Since the vector 26 C. Brezinski and M. Redivo-Zaglia Comparison of iterative residuals 1014 1011 \08 \0' 102 \0-1 \0-4 " \0-7 " /\_ ... ''.. - 80 90 100 Comparison of actual residuals \010 \07 \0' \0 1 \0-2 -, \0-' " \0-8 ", 10- 11 0 FIGURE l.

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