Applications and Computation of Orthogonal Polynomials: - download pdf or read online

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

ISBN-10: 3034886853

ISBN-13: 9783034886857

ISBN-10: 3034897286

ISBN-13: 9783034897280

The workshop on functions and Computation of Orthogonal Polynomials happened March 22-28, 1998 on the Oberwolfach Mathematical examine Institute. It used to be the 1st workshop in this subject ever held at Oberwolfach. there have been forty six individuals from thirteen nations, greater than part coming from Germany and the U.S., 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 comprises refereed types of 18 papers provided at, or submitted to, the convention. the speculation of orthogonal polynomials, as a department of classical research, is definitely proven. yet orthogonal polynomials play additionally a major position in lots of components of medical computing, akin to least squares becoming, numerical integration, and fixing linear algebraic platforms. notwithstanding the elemental tenets have their roots in nineteenth­ century arithmetic, using smooth desktops has required the advance and research of latest algorithms which are actual and strong. The computational equipment and purposes represented during this quantity, of necessity, are incomplete, but sufficiently assorted to exhibit an influence 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

Sample text

The method of Lanczos [38) consists in constructing the sequence of iterates Xk E IR n defined by Xk - Xo E Kk(A, ro) = span(ro, Aro, ... 1 Kk(A T , y) = span(y, AT y, ... ,ATk - 1 y), where Xo is an arbitrary vector, y an almost arbitrary one, and ro = b - Axo. These two conditions define entirely the vector Xk, and we have rk = ro + alArO + ... + akAkro = Pk(A)ro. From the orthogonality conditions for rk, we see that the polynomial Pk satisfies c(~i Pk) = 0 for i = 0, ... ,k - 1, Transpose-Free Look-Ahead Algorithms for Lanczos' Method 23 where c is the linear functional on the vector space of polynomials defined by c(~i) = (y, Airo).

Indeed, if we substitute the relation for Pk+I in that of P~~I' we obtain P~~I (~) = (->'~+18k+I~+f3kH)P~I) (~) +t5kHPk(~) = (~+{jkH)P~I) (~)+8kHPk(~). 4. Numerical examples We now compare Lanczos/Orthomin and Lanczos/Orthosym to show that their stability properties are quite similar. This comparison is made necessary because, as we will explain later, Lanczos/Orthomin cannot be extended to cure nearbreakdowns. All the examples have dimension 100. The solution was chosen randomly, and then the right-hand side was computed accordingly.

Fig. 5; Fig. 2 for the matrix minij, whose condition number is 16370; and Fig. 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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Applications and Computation of Orthogonal Polynomials: Conference at the Mathematical Research Institute Oberwolfach, Germany March 22–28, 1998 by Bernhard Beckermann, Edward B. Saff (auth.), Walter Gautschi, Gerhard Opfer, Gene H. Golub (eds.)


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