A quadrature formula based on Chebyshev rational functions

Abstract

Several generalisations to the classical Gauss quadrature formulas have been made over the last few years. When the integrand has singularities near the interval of integration, formulas based on rational functions give more accurate results than the classical quadrature rules based on polynomials. In this paper we present one such generalisation which uses results from the theory of orthogonal rational functions. Compared to similar existing formulas, it has the advantage of improved stability and smaller quadrature weights.

Rational interpolation: II. Quadrature and convergence

“…The main advantage of the upper bound given in the previous theorem is that the integral (19) is always computable by the Residue Theorem; see [16,Lemma 2]. We can also obtain an error bound for the RIQs on the unit circle, considered in the previous section.…”

“…Since the calculation of the projection coefficients J σ (ϕ k ) could take a lot of time (especially for higher degrees and/or different poles 5 ), we considered the auxiliary functions (see also [19,Section 3])…”

Positive rational interpolatory quadrature formulas on the unit circle and the interval

“…All the computations were done with 30 digits in MAPLE 9.5. , with α = J(β) ∈ C I . In [22,Thm. 3.2(2)] it is proved that…”

A generalized eigenvalue problem for quasi-orthogonal rational functions

“…11]. For the special case in which this subset is a real half-line or an interval, we refer to [7,8] and [16,33] respectively, while some computational aspects have been dealt with in e.g., [17,[34][35][36]38].…”

“…Similar as in the polynomial case, interpolating in the nodes of a rational Gaussian quadrature rule has nice properties regarding convergence and numerical stability. See e.g., [22,31,33,37,39,43]) and [1][2][3]30,42], for their usefulness in numerical integration and in numerically solving differential equations using rational spectral methods, respectively.…”