On numerical integration with high-order quadratures: with application to the Rayleigh–Sommerfeld integral

The paper focusses on the advantages of using high-order Gauss–Legendre quadratures for the precise evaluation of integrals with both smooth and rapidly changing integrands. Aspects of their precision are analysed in the light of Gauss’ error formula. Some “test examples” are considered and evaluated in multiple precision to 200 significant decimal digits with David Bailey’s multiprecision package to eliminate truncation/rounding errors. The increase of precision on doubling the number of subintervals is analysed, the relevant quadrature attribute being the precision increment. In order to exemplify the advantages that high-order quadrature afford, the technique is then used to evaluate several plots of the Rayleigh–Sommerfeld diffraction integral for axi-symmetric source fields defined on a planar aperture. A comparison of the high-order quadrature method against various FFT-based methods is finally given.