第5周介绍如何由采样函数值近似导数。

Week 5 introduces derivative approximation from sampled function values.

学习目标

Learning Objectives

• 使用前向、后向和中心差分近似一阶导数。

• Use forward, backward, and central differences for first derivatives.

• 由二次插值推导三点公式。

• Derive the three-point formulas from quadratic interpolation.

• 在非均匀与均匀网格上进行导数估计。

• Handle both non-uniform and uniform grids in derivative estimation.

• 利用差商（Divided Difference）结构计算二阶导数近似。

• Compute second derivatives from divided-difference structure.

关键概念

Key Concepts

• 差分法通过邻近函数值近似导数。

• Finite differences approximate derivatives using nearby function values.

• 网格记号：步长 $h$，节点 $x_i=x_0+ih$（均匀网格）。

• Grid notation: step size $h$, nodes $x_i=x_0+ih$ (uniform case).

• 差商（Divided Difference）提供了从模型出发推导导数公式的路径。

• Divided differences provide a model-based path to derivative formulas.

• 在内点处，中心公式通常比单边公式更精确。

• Central formulas are often more accurate than one-sided formulas at interior points.

定义与公式

Definitions and Formulas

前向差分：

Forward difference:

$D^+f(x)=\dfrac{f(x+h)-f(x)}{h}$.

后向差分：

Backward difference:

$D^-f(x)=\dfrac{f(x)-f(x-h)}{h}$.

三点中心一阶导（均匀网格）：

Three-point central first derivative (uniform grid):

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

三点二阶导：

Three-point second derivative:

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

非均匀三点模型：若节点为 $p,q,r$，使用

Non-uniform three-point model: if nodes are $p,q,r$, use

$p_2(x)=f[p]+f[p,q](x-p)+f[p,q,r](x-p)(x-q)$，再计算 $p_2'(p),p_2'(q),p_2'(r)$。

$p_2(x)=f[p]+f[p,q](x-p)+f[p,q,r](x-p)(x-q)$ and evaluate $p_2'(p),p_2'(q),p_2'(r)$.

推导

Derivations

中心差分是前向与后向的平均

Central Difference as Average of Forward and Backward

$\dfrac{D^+f(x)+D^-f(x)}{2}=\dfrac{f(x+h)-f(x-h)}{2h}$.

该表达在泰勒展开中会抵消一阶偏差项。

This cancels first-order bias terms in Taylor expansion.

由牛顿二次插值推导三点公式

Three-Point Formula from Newton Quadratic

Fit $p_2(x)$ through $(p,P),(q,Q),(r,R)$.

通过 $(p,P),(q,Q),(r,R)$ 拟合 $p_2(x)$。

对 $p_2$ 求导并在各节点取值以近似导数。

Differentiate $p_2$ and evaluate at each node to approximate derivatives.

当间距均匀且 $p=x-h,q=x,r=x+h$ 时，公式化简为中心差分形式。

For uniform spacing $p=x-h,q=x,r=x+h$, formulas simplify to central differences.

例题精讲

Worked Examples

例题模块1：$f(x)=x^2$ 在 $x=1,h=0.1$ 的估计

Example Block 1: $f(x)=x^2$ at $x=1$, $h=0.1$

前向差分：$\dfrac{1.21-1}{0.1}=2.1$。

Forward difference: $\dfrac{1.21-1}{0.1}=2.1$.

后向差分：$\dfrac{1-0.81}{0.1}=1.9$。

Backward difference: $\dfrac{1-0.81}{0.1}=1.9$.

中心差分：$\dfrac{1.21-0.81}{0.2}=2.0$（本例恰为精确值）。

Central difference: $\dfrac{1.21-0.81}{0.2}=2.0$ (exact here).

例题模块2：$-0.1,0,0.1$ 的指数函数样本

Example Block 2: Exponential Samples at $-0.1,0,0.1$

使用三点框架估计 $f'(-0.1),f'(0),f'(0.1)$。

Use non-uniform/three-point framework to estimate $f'(-0.1),f'(0),f'(0.1)$.

与单边差分相比，中心估计在内点更接近真值 $e^x$。

Compared with one-sided differences, central estimates better match true value $e^x$ at interior point.

例题模块3：在 $0.9,1.0,1.1$ 的三点多项式恢复

Example Block 3: Three-Point Polynomial Recovery at $0.9,1.0,1.1$

对 $f(x)=x^2$，重构二次模型可得精确导数 $f'(0.9)=1.8$、$f'(1)=2$、$f'(1.1)=2.2$。

For $f(x)=x^2$, reconstructed quadratic gives exact derivatives $f'(0.9)=1.8$, $f'(1)=2$, $f'(1.1)=2.2$.

误差（Error）分析

Error Analysis

• 前向/后向差分是一阶精度：误差（Error） $O(h)$。

• Forward/backward differences are first-order accurate: error $O(h)$.

• 中心一阶导公式是二阶精度：误差（Error） $O(h^2)$。

• Central first-derivative formula is second-order accurate: error $O(h^2)$.

• 在均匀网格上，三点二阶导公式同样是二阶精度。

• Three-point second-derivative formula is also second-order accurate on uniform grids.

• 即使截断误差（Error）随 $h$ 减小，过小 $h$ 也会放大舍入误差（Error）。

• Too small $h$ can increase round-off error even if truncation error decreases.

常见错误

Common Mistakes

警示模块

Warning Block

• 在中心差分分母中误写 $2h$。

• Using $2h$ incorrectly in central denominator.

• 在非均匀节点上误用均匀网格公式。

• Applying uniform-grid formulas on non-uniform nodes.

• 在差商（Divided Difference）中混淆索引顺序。

• Mixing index order in divided differences.

• 解释导数量级时忽略 $h$ 的单位与尺度。

• Ignoring unit/scale of $h$ when interpreting derivative magnitude.

总结

Summary

总结模块

Summary Block

• 第5周建立了由离散数据进行导数估计的实用方法。

• Week 5 established practical derivative estimation from tabulated data.

• 三点框架统一了非均匀与均匀情形。

• The three-point framework unifies non-uniform and uniform formulas.

• 中心差分是相对于单边差分的重要精度提升。

• Central differences are a key accuracy upgrade over one-sided approximations.

练习题

Practice Questions

• 对 $f(x)=\ln x$，取 $h=0.1$，分别用前向、后向、中心差分估计 $f'(2)$。

• Estimate $f'(2)$ for $f(x)=\ln x$ using forward, backward, and central differences with $h=0.1$.

• 给定节点 $p=1,q=1.4,r=2$，用差商（Divided Difference）形式推导 $p_2'(q)$。

• Using nodes $p=1$, $q=1.4$, $r=2$, derive $p_2'(q)$ in divided-difference form.

• 在 $x=0$ 处比较 $f(x)=e^x$ 的数值二阶导与精确值，并测试多个 $h$。

• Compare numerical and exact second derivatives for $f(x)=e^x$ at $x=0$ for several $h$ values.