第7周先回顾数值微分，再重点推导积分误差（Error）并讨论外推改进。

Week 7 reviews numerical differentiation and then emphasizes integration error derivation and extrapolation.

学习目标

Learning Objectives

• 在均匀网格上重新推导三点微分公式。

• Re-derive three-point differentiation formulas on uniform grids.

• Derive Simpson error order from Taylor expansion.

• 由泰勒展开推导 Simpson 误差（Error）阶。

• 比较梯形法与中点法误差（Error）结构。

• Compare trapezoid and midpoint error structures.

• Understand Romberg-style extrapolation as weighted error cancellation.

• 理解 Romberg 型外推本质是加权误差（Error）消去。

关键概念

Key Concepts

• 对光滑函数，求积误差（Error）可按 $h$ 的幂展开。

• For smooth $f$, quadrature error can be expanded in powers of $h$.

• Simpson can be interpreted as a weighted combination of trapezoid and midpoint rules.

• Simpson 可解释为梯形法与中点法的加权组合。

• 通过选取权重消去主导误差（Error）项。

• Error cancellation is achieved by choosing weights that eliminate leading terms.

• 外推通过组合粗网格与细网格结果提升精度。

• Extrapolation combines coarse and fine-grid estimates to improve accuracy.

定义与公式

Definitions and Formulas

均匀三点导数回顾：

Uniform three-point derivative review:

$f'(x)\approx\dfrac{f(x+h)-f(x-h)}{2h},\quad f''(x)\approx\dfrac{f(x+h)-2f(x)+f(x-h)}{h^2}$.

Single-panel Simpson on $[-h,h]$:

区间 $[-h,h]$ 的单面板 Simpson：

$S(h)=2h\cdot\dfrac{f(-h)+4f(0)+f(h)}{6}$.

Classical local Simpson error form:

经典 Simpson 局部误差（Error）形式：

$S(h)-\int_{-h}^{h}f(x)\,dx = C h^5,\quad C\propto f^{(4)}(\xi)$.

梯形法与中点法主导误差（Error）符号相反，可用于加权抵消。

Trapezoid and midpoint leading errors have opposite signs, enabling weighted cancellation.

推导

Derivations

Simpson Error via Taylor Series

用泰勒级数推导 Simpson 误差（Error）

Expand $f(x)=a+bx+cx^2+dx^3+ex^4+\cdots$ around $0$.

1. 在 $0$ 点附近展开 $f(x)=a+bx+cx^2+dx^3+ex^4+\cdots$。

2. 在对称区间 $[-h,h]$ 上逐项积分。

3. Integrate term-by-term over symmetric interval $[-h,h]$.

4. Compare with Simpson weighted average of $f(-h),f(0),f(h)$.

5. 与 Simpson 对 $f(-h),f(0),f(h)$ 的加权平均比较。

6. 首个未被消去的项与 $h^5$ 成正比，因此局部误差（Error）为 $O(h^5)$。

7. The first non-canceled term is proportional to $h^5$, so the local error is $O(h^5)$.

梯形-中点混合

Trapezoid-Midpoint Mixing

设 $E_T,E_M$ 分别表示梯形法与中点法误差（Error）。

Let $E_T$ and $E_M$ denote trapezoid and midpoint errors.

选择混合公式 $\lambda T+(1-\lambda)M$ 使主导 $O(h^3)$ 面积误差（Error）为零。

Choose mixture $\lambda T+(1-\lambda)M$ so leading $O(h^3)$ area error vanishes.

The resulting weights recover Simpson's $(1,4,1)/6$ pattern.

由此得到 Simpson 的 $(1,4,1)/6$ 权重结构。

Coarse-Fine Extrapolation (Romberg Idea)

粗细网格外推（Romberg 思想）

Suppose $I_h=I+Kh^p+\cdots$ and $I_{h/2}=I+K(h/2)^p+\cdots$.

设 $I_h=I+Kh^p+\cdots$，$I_{h/2}=I+K(h/2)^p+\cdots$。

通过线性组合可消去 $Kh^p$，得到更高阶估计。

A linear combination can eliminate $Kh^p$ and produce higher-order estimates.

This is the central mechanism of Romberg extrapolation.

这就是 Romberg 外推的核心机制。

例题精讲

Worked Examples

例题模块1：三点微分回顾

Example Block 1: Review of Three-Point Differentiation

给定 $x-h,x,x+h$ 处样本，使用中心公式计算 $f'(x)$ 与 $f''(x)$。

Given samples at $x-h,x,x+h$, use central formulas for $f'(x)$ and $f''(x)$.

这与本周复习表格结构一致。

This matches the week review table structure.

Example Block 2: Simpson on $\int_0^1\dfrac{1}{1+x^2}dx$ 例题模块2：Simpson 计算 $\int_0^1\dfrac{1}{1+x^2}dx$

Using points $0,0.5,1$, Simpson gives $\dfrac{1}{6}(1+4\cdot0.8+0.5)=0.78333\ldots$.

使用点 $0,0.5,1$，Simpson 给出 $\dfrac{1}{6}(1+4\cdot0.8+0.5)=0.78333\ldots$。

真值为 $\pi/4\approx0.785398\ldots$，误差（Error）较小。

True value is $\pi/4\approx0.785398\ldots$, so error is small.

例题模块3：误差（Error）-权重图解释

Example Block 3: Error-Weight Plot Interpretation

课堂比较显示梯形与中点法主导误差（Error）项具有比例关系且趋势相反。

Lecture comparison indicates trapezoid and midpoint leading terms have ratio and opposite tendency.

Weighted mixing suppresses leading error and motivates Simpson-type coefficients.

加权混合可抑制主误差（Error），从而得到 Simpson 型系数。

Example Block 4: Five-Point Newton-Cotes Pattern (Lecture Extension) 例题模块4：五点 Newton-Cotes 结构（课堂扩展）

组合粗细网格估计可得到与 $(7,32,12,32,7)$ 成比例的五点权重。

Combining coarse and fine estimates yields a five-point weight vector proportional to $(7,32,12,32,7)$.

This is consistent with higher-order closed Newton-Cotes construction.

这与高阶闭型 Newton-Cotes 构造一致。

误差（Error）分析

Error Analysis

• Simpson local panel error is $O(h^5)$, much smaller than trapezoid/midpoint local errors.

• Simpson 单面板局部误差（Error）为 $O(h^5)$，显著小于梯形/中点法。

• 复化方法会累积各面板误差（Error），从而形成不同全局阶次。

• Composite methods accumulate local errors across panels, giving different global orders.

• 外推通过消去已知渐近误差（Error）项来提高阶次。

• Extrapolation improves order by canceling known asymptotic terms.

常见错误

Common Mistakes

警示模块

Warning Block

• 混淆局部误差（Error）阶与全局误差（Error）阶。

• Mixing local-error and global-error orders.

• 在泰勒误差（Error）常数中遗漏阶乘因子。

• Dropping factorial factors in Taylor-based error constants.

• Using Simpson combination weights incorrectly when panel widths differ.

• 在面板宽度不一致时错误套用 Simpson 组合权重。

• 在缺少光滑性条件时误以为外推必然提高精度。

• Assuming extrapolation always improves accuracy without smoothness conditions.

总结

Summary

总结模块

Summary Block

• 第7周贯通了微分回顾、积分误差（Error）理论与外推设计。

• Week 7 connected differentiation review, integration error theory, and extrapolation design.

• 核心原则是结构化的：先识别主导误差（Error）项，再通过加权组合将其消去。

• The main principle is structural: identify dominant error terms, then cancel them by weighted combinations.

• This principle underlies Simpson refinement and Romberg-style acceleration.

• 这一原则支撑了 Simpson 改进与 Romberg 型加速。

练习题

Practice Questions

• Derive Simpson local error on $[-h,h]$ up to the $h^5$ term using Taylor expansion.

• 用泰勒展开推导 $[-h,h]$ 上 Simpson 的局部误差（Error）到 $h^5$ 项。

• Show algebraically how combining trapezoid and midpoint rules yields Simpson weights.

• 用代数推导说明如何由梯形法与中点法组合得到 Simpson 权重。

• Given coarse and fine Simpson estimates, design an extrapolated estimate that cancels the leading error term.

• 给定粗网格与细网格 Simpson 结果，构造能消去主导误差（Error）项的外推估计。