A numerical method for the computation of Faber polynomials for starlike domains

Papamichael, N.; Soares, Maria Joana; Stylianopoulos, Nikos

doi:10.1093/imanum/13.2.181

Cited by 11 publications

(1 citation statement)

References 13 publications

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“…Gatermann et al [18] modified the algorithm of [4] to obtain a form involving only rational coefficients and therefore amenable to computer algebra systems; the algebraic forms in [ 19] allow the computation of the Faber polynomials of degree up to 20 for an arbitrary circular sector. In other cases, where explicit formulae were not available, numerical algorithms for conformai mapping have been used to generate Faber polynomials (see Ellacott [9], Starke and Varga [29] and Papamichael et al [25]). Virtues of suitably normalized Faber polynomials as residue polynomials for matrix iterative methods are described in [29].…”

Section: Introductionmentioning

confidence: 99%

The Faber polynomials for annular sectors

Coleman¹,

Myers²

1995

Math. Comp.

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Abstract. A conformai mapping of the exterior of the unit circle to the exterior of a region of the complex plane determines the Faber polynomials for that region. These polynomials are of interest in providing near-optimal polynomial approximations in a variety of contexts, including the construction of semiiterative methods for linear equations. The relevant conformai map for an annular sector {z : R < \z\ < 1, 0 < |argz| < it), with 0 < 9 < it, is derived here and a recurrence relation is established for the coefficients of its Laurent expansion about the point at infinity. The recursive evaluation of scaled Faber polynomials is formulated in such a way that an algebraic manipulation package may be used to generate explicit expressions for their coefficients, in terms of two parameters which are determined by the interior angle of the annular sector and the ratio of its radii. Properties of the coefficients of the scaled Faber polynomials are established, and those for polynomials of degree < 15 are tabulated in a Supplement at the end of this issue. A simple closed form is obtained for the coefficients of the Faber series for 1/z . Known results for an interval, a circular arc, and a circular sector are reproduced as special cases.

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Section: Introductionmentioning

confidence: 99%

The Faber polynomials for annular sectors

Coleman¹,

Myers²

1995

Math. Comp.

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The Zeros of Faber Polynomials for an m-Cusped Hypocycloid

Saff

1994

Journal of Approximation Theory

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Direct and Inverse Results on Row Sequences of Simultaneous Padé–Faber Approximants

Bosuwan

Lagomasino

2019

Mediterr. J. Math.

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Given a vector function F = (F 1 , . . . , F d ), analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components F k , k = 1, . . . , d, with respect to the sequence of Faber polynomials associated with E. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Padé approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions "closest" to E and their order.

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A numerical method for the computation of Faber polynomials for starlike domains

Abstract: We describe a simple numerical process (based on the Theodorsen method for conformal mapping ) for computing approximations to Faber polynomials for starlike domains.

Cited by 11 publications

References 13 publications

The Faber polynomials for annular sectors

The Faber polynomials for annular sectors

The Zeros of Faber Polynomials for an m-Cusped Hypocycloid

Direct and Inverse Results on Row Sequences of Simultaneous Padé–Faber Approximants

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