Karl Deckers

Rational Gauss-Chebyshev quadrature formulas for complex poles outside [-1,1]

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside [-1,1] to arbitrary complex poles outside [-1,1]. The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [-1,1].

Algorithm 882: Near-Best Fixed Pole Rational Interpolation with Applications in Spectral Methods

We present a numerical procedure to compute the nodes and weights in rational Gauss-Chebyshev quadrature formulas. Under certain conditions on the poles, these nodes are near best for rational interpolation with prescribed poles (in the same sense that Chebyshev points are near best for polynomial interpolation). As an illustration, we use these interpolation points to solve a differential equation with an interior boundary layer using a rational spectral method. The algorithm to compute the interpolation points (and, if required, the quadrature weights) is implemented as a Matlab program.

Rational Szego quadratures associated with Chebyshev weight functions

In this paper we characterize rational Szegýo quadrature formulas associated with Chebyshev weight functions, by giving explicit expressions for the corresponding para-orthogonal rational functions and weights in the quadratures. As an application, we give characterizations for Szegýo quadrature formulas associated with rational modifications of Chebyshev weight functions. Some numerical experiments are finally presented.

Computing rational Gauss-Chebyshev quadrature formulas with complex poles: The algorithm

We provide an algorithm to compute arbitrarily many nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with complex poles outside [-1, 1]. Contrary to existing rational quadrature formulas, the computational effort is very low, even for extremely high degrees, and under certain conditions on the poles it can be shown that the complexity is of order O(n). This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions on [-1, 1] with arbitrary complex poles outside this interval.

A generalized eigenvalue problem for quasi-orthogonal rational functions

In general, the zeros of an orthogonal rational function (ORF) on a subset of the real line, with poles among

A numerical solution of the constrained weighted energy problem

{\{\alpha_1,\ldots,\alpha_n\}\subset(\mathbb{C}_0\cup\{\infty\})}

Full length article: An extension of the associated rational functions on the unit circle

, are not all real (unless