题号:14630    题型:解答题    来源:2024年丘成桐大学生数学竞赛(计算与应用数学)-无答案
Consider the integral
$$
\int_0^{\infty} f(x) \mathrm{d} x
$$
where $f$ is continuous, $f^{\prime}(0) \neq 0$, and $f(x)$ decays like $x^{-1-\alpha}$ with $\alpha>0$ in the limit $x \rightarrow \infty$.
(a) Suppose you apply the equispaced composite trapezoid $\mathrm{r}$ ule with $n$ subintervals to approximate
$$
\int_0^L f(x) \mathrm{d} x
$$

What is the asymptotic error formula for the error in the limit $n \rightarrow \infty$ with $L$ fixed?
(b) Suppose you consider the quadrature from (a) to be an ap proximation to the full integral from 0 to $\infty$. How should $L$ in crease with $n$ to optimize the asymptotic rate of total error $d$ ecay? What is the rate of error decrease with this choice of $L$ ?
(c) Make the following change of variable $x=\frac{L(1+y)}{1-y}$, $y=\frac{x-L}{x+L}$ in the original integral to obtain
$$
\int_{-1}^1 F_L(y) \mathrm{d} y
$$

Suppose you apply the equispaced composite trapezoid rule; what is the asymptotic error formula for fixed $L$ ?
(d) Depending on $\alpha$, which method - domain truncation of ch ange-of-variable - is preferable?
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