Dynamical Control of Computations Using the Trapezoidal and Simpson's Rules

Abstract

If $I_n$ is the approximation of a definite integral $\int_{a}^{b}f(x)dx$ with step $\frac{b-a}{2^n}$ using the trapezoidal rule (respectively Simpson's rule), if $C_{a,b}$ denotes the number of significant digits common to $a$ and $b$, we show, in this paper, that $C_{I_{n},I_{n+1}} = C_{I_ {n},I}+\log_{10}(\frac{4}{3})+\mathcal{O}(\frac{1}{4^n})$ (respectively $C_{I_{n},I_{n+1}} = C_{I_ {n},I}+\log_{10}(\frac{16}{15})+\mathcal{O}(\frac{1}{16^n})$). According to the previous theorems, using the CADNA library which allows on computers to estimate the round-off error effect on any computed result, we can compute dynamically the optimal value of $n$ to approximate $I$ and we are sure that the exact significant digits of $I_n$ are in common with the significant digits of $I$.