Fast and accurate computation of Chebyshev coefficients in the complex plane

Abstract

Chebyshev expansion coefficients can be computed efficiently by using the FFT, and for smooth functions the resulting approximation is close to optimal, with computations that are numerically stable. Given sufficiently accurate function samples, the Chebyshev expansion coefficients can be computed to machine precision accuracy. However, the accuracy is only with respect to absolute error, and this implies that very small expansion coefficients typically have very large relative error. Upon differentiating a Chebyshev expansion, this relative error in the small coefficients is magnified and accuracy may be lost, especially after repeated differentiation. At first sight, this seems unavoidable. Yet, in this paper, we focus on an alternative computation of Chebyshev expansion coefficients using contour integrals in the complex plane. The main result is that the coefficients can be computed with machine precision relative error, rather than absolute error. This implies that even very small coefficients can be computed with full floating point accuracy, even when they are themselves much smaller than machine precision. As a result, no accuracy is lost after differentiating the expansion, and even the 100th derivative of an analytic function can be computed with near machine precision accuracy using standard floating point arithmetic. In some cases, the contour integrals can be evaluated using the FFT, making the approach both highly accurate and fast. Accurate computation of high-order derivativesComputing derivatives of a function numerically is a notoriously ill-conditioned problem, especially for high-order derivatives [21]. It was shown by Bornemann in [4] that computation of high-order derivatives through Cauchy integrals in the complex plane is, in fact, stable. To be precise, consider the power series of a function analytic at the

Numerical initial data deformation exploiting a gluing construction: I. Exterior asymptotic Schwarzschild

“…Optimal here entails selection of r * (n) such that round-off error is minimised during numerical work. The recent work of [85] extends this analysis to the case of Chebyshev expansion coefficients which may be considered to be embedded as the Taylor coefficients of a particular integral transformation there described. As a preliminary, define the Bernstein ellipse Γ E with foci at ±1 and major and minor semi-axis lengths summing to the "radius" parameter r B :…”

“…In complete analogy to [9] the absolute and relative stability of the above approximation are then considered by [85]. It is shown that evaluation of the Chebyshev coefficients is absolutely stable.…”

“…, 1}; N−k+1 = 0. Unfortunately, the search for r * (n) requires extensive use of asymptotic approximations to infer the condition number κ directly [9,85]. In order to provide a practical, numerical method for the approximate determination of κ we propose to instead approximate κ on some Γ E (r B ) via the m-point trapezoidal rule of Eq.(4.16).…”

“…In order to provide a practical, numerical method for the approximate determination of κ we propose to instead approximate κ on some Γ E (r B ) via the m-point trapezoidal rule of Eq.(4.16). The convexity of log(κ) together with monotonicity of r * (n) [9,85] allows for us to employ the downhill simplex minimisation algorithm [75]. We initialise the search at n = 0 with r B = 1 constructing r * (0).…”

Quantifying the ill-conditioning of analytic continuation

“…Analytic continuation in a region bounded by an ellipse is mentioned as Example 3 of [12], and a detailed analysis of algorithms in this geometry is presented in [7]. Results for ellipses analogous to those of [5] for disks can be found in [28].…”