Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

Abstract

Abstract.We present an elegant algorithm for stably and quickly generating the weights of Fejér's quadrature rules and of the Clenshaw-Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions.

Pricing of early-exercise Asian options under Lévy processes based on Fourier cosine expansions

“…,n q + 2) can be pre-computed as the Discrete Cosine Transform (DCT) of a vector of real numbers, in O (n q log n q ) computational cost by an FFT-based algorithm for the DCT. We refer the reader to [15] for more details regarding the fast calculation of Clenshaw-Curtis quadrature rules.…”

“…Furthermore, the derivative of the continuation value as well as of the payoff function, with respect to y m , can be easily computed by (19) and (15). By prescribing the initial values for the Newton iterations as y 0…”

“…with g( y m+1 , t m+1 ) defined in (15). The error in the continuation valueĉ( y m , x m , t m ) is composed of two parts.…”

Matrix Based Operatorial Approach to Differential and Integral Problems

“…A Chebyshev series is in fact a Fourier cosine series so that the FFT and iFFT may be used to transform the spectral representation of y into the physical one and conversely, the physical representation into the spectral representation. The quadrature weights w could also be calculated by a fast algorithm given in (Waldvogel, 2006). The first code of level0, inspired from chebpts.m of chebfun (Trefethen et al, 2011) is …”

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

“…An FFT-based implementation is then feasible, in analogy to what happens in the univariate case with the Clenshaw-Curtis formula (cf. [15]). The algorithm is quite similar to that in Table 1, and we do not describe it for brevity.…”