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Monday, August 3, 2020 | History

3 edition of Optimal Chebyshev polynomials on ellipses in the complex plane found in the catalog.

Optimal Chebyshev polynomials on ellipses in the complex plane

Fischer, Bernd

Optimal Chebyshev polynomials on ellipses in the complex plane

by Fischer, Bernd

  • 311 Want to read
  • 38 Currently reading

Published by Research Institute for Advanced Computer Science, NASA Ames Research Center in [Moffett Field, CA] .
Written in English

    Subjects:
  • Chebyshev approximation.,
  • Complex variables.,
  • Ellipses.,
  • Iterative solution.,
  • Matrices (Mathematics),
  • Polynomials.

  • Edition Notes

    StatementBernd Fischer, Roland Freund.
    SeriesRIACS technical report -- 89-5., NASA contractor report -- NASA/CR-188836.
    ContributionsFreund, Roland W., Research Institute for Advanced Computer Science (U.S.)
    The Physical Object
    FormatMicroform
    Pagination1 v.
    ID Numbers
    Open LibraryOL16126789M

    Results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the Chebyshev series approximations. Advantages and disadvantages of the Chebyshev series approximation compared with other polynomial approximation methods, e.g., the tau-method approximations, are by: 3. The interest in this paper is the use of Chebyshev polynomials to approximate functions. Furthermore, various classes of mathematical functions will be analyzed in order to conclude which kinds of functions could best be modeled by Chebyshev polynomials. Scilab[1] .

    Sin(x) as a sum of Chebyshev polynomials. The first step is to re-express over the domain of interest as an infinite polynomial. One could use a Taylor series, but convergence is very slow. Instead, I use Chebyshev polynomials. For me, it helps to view these things using concepts from linear algebra.   This video is unavailable. Watch Queue Queue. Watch Queue Queue.

    Plots of Chebyshev polynomials of the first and second kind. By dividing the complex plane into a number of sectors, however, we may expect the powerful convergence of the Chebyshev series, which is defined as a real valued function on the interval [ - 1, 1], to be kept on each central ray J. ZHANG AND J. A. BELWARD on which the variable is by: 3.


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Optimal Chebyshev polynomials on ellipses in the complex plane by Fischer, Bernd Download PDF EPUB FB2

Get this from a library. Optimal Chebyshev polynomials on ellipses in the complex plane. [Bernd Fischer; Roland W Freund; Research Institute for Advanced Computer Science (U.S.)]. Chebyshev Polynomials John D.

Cook∗ February 9, Abstract The Chebyshev polynomials are both elegant and useful. This note summarizes some of their elementary properties with brief proofs. 1 Cosines We begin with the following identity for cosines. cos((n + 1)θ) = 2cos(θ)cos(nθ) − cos((n − 1)θ) (1) This may be proven by applying.

Polynomial interpolants defined using Chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval. However, the convergence rate can be arbitrarily close to unity if the function has a singularity Cited by: 4.

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set K in the complex plane are established. These estimates are exact (up to a constant factor) in the case where K consists of a finite number of quasiconformal curves or arcs.

The case where K is a uniformly perfect subset of the real line is also by: 5. Abstract: The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set $K$ in the complex plane are established.

These estimates are Author: Vladimir Andrievskii. Here H. denotes the set of all complex polynomials of degree at most n, £r:= {z _ ¢ I Iz - ll + lz + ll l_, (2) r is any ellipse (including its interior) in the complex plane with foci atand it is always assumed that is any ellipse (including its interior) in the complex plane c E ¢ \ Cr.

Since. $\begingroup$ I disagree to say that $|z-z_0|=1$ is the equation of an ellipse in the complex plane. This is a unit circle and nothing else. $\endgroup$ – Yves Daoust Jan 30 '15 at 1.

In this chapter we outline how to compute Chebyshev polynomials and certain closely related optimal polynomials for one interval and for the union of two disjoint intervals, respectively. This is a preview of subscription content, log in to check : Bernd Fischer.

For the case of ellipses, we introduce a new class of complex polynomials which are in general very good approximations to the best polynomials and even optimal in most cases. Read more Chapter. We observe that the Chebyshev polynomials form an orthogonal set on the interval 1 x 1 with the weighting function (1 x2) 1=2 Orthogonal Series of Chebyshev Polynomials An arbitrary function f(x) which is continuous and single-valued, de ned over the interval 1 x 1, can be expanded as a series of Chebyshev polynomials: f(x) = A 0T 0(x) + A 1T 1 File Size: KB.

Most widely held works about Research Institute for Advanced Computer Science (U.S.) RIACS annual report by Research Institute for Advanced Computer Science (U.S.) Optimal Chebyshev polynomials on ellipses in the complex plane by Bernd Fischer. If P k are the Chebyshev polynomials T k, then P coincides with CN+1:= (cos jkß N) N j;k=0.

This paper presents a new fast algorithm for the computation of the matrix-- vector product Pa in O. In fact the more "well behaved" Complex Functions are exactly the same as Conformal Mappings of a Plane. Given that, a Polynomial is just a specific way of distorting the regular Complex Plane.

First we take a look at the regular Complex Plane. Noting that everything is simply uniformly increasing in value as we move away from the Origin. We propose an algorithm for computing a class of least squares polynomials on polygonal regions of the complex plane. An important application of this technique to solving large sparse linear systems is considered.

The advantage of using general polygonal regions instead of ellipses as was done in previous work, is that elliptic regions may fail to accurately represent the convex hull of the Cited by: context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems.

In this note, we show that this is not true in er, we derive sufficient conditions which guarantee that Chebyshev polynomials aresome numerical examples are. When graphed, the Chebyshev polynomials pro-duce some interesting patterns.

Figure 1 shows the first four Chebyshev polynomials, and figure 2 shows the next four. The following patterns can be discerned by analyzing these graphs. Even-numbered Chebyshev polynomials yield even functions whose graphs have reflective symmetry across the y-axis. Odd. There are several kinds of Chebyshev polynomials.

In particular we shall in-troduce the first and second kind polynomials Tn(x)andUn(x), as well as a pair of related (Jacobi) polynomials Vn(x)andWn(x), which we call the ‘Chebyshev polynomials of the third and fourth kinds’; in addition we cover the shifted polynomials T ∗ n (x), Un (x File Size: 2MB.

The evaluation of Chebyshev polynomials by a three-term recurrence is known to be mixed forward-backward stable for x ∊ [-1, 1].

However, the author does not know of a similar result for x outside [-1, 1]. For this reason, evaluation of Chebyshev polynomials outside of [-1, 1] is strongly discouraged. Chebyshev series is efficiently solved by means of Clenshaw’s method, which is also pre-sented in this chapter. Before this, we give a very concise overview of well-known results in interpola-tion theory, followed by a brief summary of important properties satisfied by Chebyshev polynomials.

Basic results on interpolationFile Size: KB. polynomials are optimal for such constrained approximation problems. In this note, wc show that this is not true in general.

Moreover, we &-rive sufficient conditions which guarantee that Chebyshev polynomials arc optimal. Also, som_" numerical examples are presented. The work of the first author was supported by the Gernmn Research Association. For example, the first quadrant is covered by three 30 sectors corresponding to q = it/12, qr/4 and 51r/12, and the same Faber polynomials are used for each of the sectors; it should be noted that, as in sectiona 30 sector in the z plane corresponds J.

P. Coleman / Polynomial approximations in the complex plane Table 6 Results for Cited by: For a domain D of the complex plane we can ask what monic polynomial of degree n will minimize max|z" + an_xz"~l + +axz + a0\.

ze£> The polynomial %D(z) which satisfies this condition is called the Chebyshev polynomial of degree n for the domain D. The success of Chebyshev expansions on.Chebyshev polynomials are orthogonal w.r.t. weight function w(x) = p1 1 x2. Namely, Z 1 21 T n(x)T m(x) p 1 x2 dx= ˆ 0 if m6= n ˇ if n= m for each n 1 (1) Theorem (Roots of Chebyshev polynomials) The roots of T n(x) of degree n 1 has nsimple zeros in [ 1;1] at x k= cos 2k 1 2n ˇ; for each k= 1;2 n: Moreover, T n(x) assumes its absolute.