The Golden Number and Numerical Integration

Abstract

The golden number $\omega =\frac{\sqrt{5}-1}{2}$ is not only useful for the golden section method, but it also plays an important role in the theory of diophantine approximation. This inspired us to think about the connection between numerical integration and the golden number. In 1960 we found the very efficient quadrature formula

$$\int\limits_{0}^{1}{{}}\int\limits_{0}^{1}{{}}f\left( x,y \right)dxdy\approx \frac{1}{{{F}_{m}}}\sum\limits_{k=1}^{{{F}_{m}}}{{}}f\left( \frac{k}{{{F}_{m}}}\left\{ \frac{{{F}_{m}}-{{1}^{k}}}{{{F}_{m}}}\left. {} \right\} \right. \right)$$

(5.1)

that is based on the rational approximation of ω

$$\left| \frac{{{F}_{m}}-1}{{{F}_{m}}}-\omega \right|<\frac{1}{\sqrt{5}F_{m}^{2}}$$

and where {ع} denotes the fractional part of ع.