IS CS-2020S2-05

题目来源：Problem 5 日期：2024-08-11 题目主题：Math-数值分析-复合梯形法

解题思路

这道题目涉及数值积分的复合梯形法，以及数值微分和误差分析。首先需要利用数值微分方法来近似函数的二阶导数，然后通过分析 IEEE 754 双精度浮点运算的特性来讨论数值误差的问题。接着，我们需要表达复合梯形法的积分近似公式，最后推导出利用两种不同分割方法得到的积分结果之间的关系。

Solution

Question 1

To approximate the second derivative $f_k^{\prime \prime }$ at $x_k$ using the values $f_{k-1}$, $f_k$, and $f_{k+1}$, we can use the central difference formula:

$$f_k^{\prime \prime }\approx \frac{f_{k-1}-2f_k+f_{k+1}}{h^2}$$

The error in this approximation is $O(h^2)$. This formula is derived from the Taylor series expansion of $f(x)$ around $x_k$.

Question 2

The approximation for $f_k^{\prime \prime }$ becomes more accurate as $h$ approaches zero theoretically. However, in the context of IEEE 754 double precision floating point operations, as $h$ decreases, the difference between $f_{k+1}$ and $f_{k-1}$ also becomes very small. This can lead to a significant loss of precision due to rounding errors, which is known as catastrophic cancellation. Therefore, in practice, there is a limit to the accuracy of this approximation when $h$ becomes too small.

Question 3

The composite trapezoidal rule $J_n$ for approximating the integral $J$ is given by:

$$J_n=\frac{h}{2}\left(f_0+2\sum _{i=1}^{n-1}{f}_i+f_n\right)$$

where $h=\frac{b-a}{n}$ and $f_i=f(x_i)$.

Question 4

When $f(x)$ is approximated by a quadratic function on each subinterval, the error $E_n=J_n-J$ can be related to the integral approximations $J_n$ and $J_{2n}$ as follows:

$$E_n=\frac{J_{2n}-J_n}{3}$$

This formula arises from the fact that the error in the trapezoidal rule is proportional to $h^2$, and doubling the number of intervals reduces the error by a factor of 4. By subtracting the two approximations, we can eliminate the leading order error term, leaving a smaller error proportional to $h^4$.

知识点

数值分析数值积分复合梯形法数值微分误差分析浮点运算

难点思路

在第二问中，理解浮点数运算中的精度问题是关键。尤其是当差值变小时，浮点运算的误差可能会导致结果不准确。

解题技巧和信息

1. 数值微分：当计算导数时，中央差分法通常比前向或后向差分法具有更高的精度，但需要注意数值稳定性。
2. 数值积分：复合梯形法的误差分析依赖于对被积函数的光滑性假设，常见的误差公式与分割数 $n$ 的关系密切。
3. 误差消除：使用不同分割方式计算积分时，考虑两次计算的结果，可以有效减少误差，这种方法类似于龙贝格积分法。

重点词汇

• Trapezoidal rule: 梯形法
• Central difference: 中央差分
• IEEE 754 double precision: IEEE 754 双精度浮点数
• Catastrophic cancellation: 灾难性消减

参考资料

1. Burden, R. L., & Faires, J. D. (2011). Numerical Analysis (9th ed.). Brooks Cole. Chapter 4: Numerical Differentiation and Integration.
2. Heath, M. T. (2002). Scientific Computing: An Introductory Survey (2nd ed.). McGraw-Hill. Chapter 8: Numerical Integration.