Fast Generation of Quadrature Rules with Some Special Properties

Abstract

We consider fast methods, based on FFT techniques, for obtaining families of quadrature rules approximating

$$\smallint\!^{{\text{nh}}}_{0}({\text{nh}}-{\text{s}})^{-\frac{1}{2}}\,\,\varphi({\text{s}}){\text{ds}}\quad\quad\quad\quad ({\text{n}}\,=\,1,\,2,\,3,\,\,\,....,{\text{N}};\,\,\,{\text{N}}=\,2^{{\text{R}}};\,{\text{R}}\,\,\epsilon\,\,{\text{Z}}_{+}\!)$$

. The rules are fractional quadrature rules derived from approximations to $\smallint\!^{nh}_{0}\,\,\varphi({\text{s}}){\text{ds}}\,({\text{n}}\,\,\epsilon\,\,{\text{Z}}_{+}\!)$ generated by implicit linear multistep formulae. Suitable “starting” weights, computed using the Björck-Pereyra algorithm and FFT techniques, produce formulae with good order accuracy for functions φ(s) of the form $\varphi({\text{s}})\,=\,\varphi_{0}\,+\varphi_{1}{\text{s}}^{1/2}\,+\,\varphi_{2}{\text{s}}\,+\,\varphi_{3}{\text{s}}^{3/2}\,+\,\varphi_{4}{\text{s}}^{2}\,+\,\,\ldots\,{\text{as\,\,\,s}}\,\to\,0$. The discussion is associated with FORTRAN 77 code given in [2].