数值分析⚓︎
课程信息
- 学分:2.5
- 教师:黄劲
- 教材:Numeric Analysis, 7th Edition
目录
上课顺序为:1 -> 2 -> 6 -> 7 -> 9 -> 3 -> 8 -> 4 -> 5,建议复习的时候也按这个顺序来。
- Chap 1: Mathematical Premiminaries
- Chap 2: Solutions of Equations in One Variable
- Chap 3: Interpolation and Polynomial Approximation
- Chap 4: Numerical Differentiation and Integration
- Chap 5: Initial-Value Problems for Ordinary Differential Equations
- Chap 6: Direct Methods for Solving Linear Systems
- Chap 7: Iterative Techniques in Matrix Algebra
- Chap 8: Approximation Theory
- Chap 9: Approximating Eigenvalues
剩下三章不会介绍,但这里还是列一下标题:
- Chap 10: Numerical Solutions of Nonlinear Systems for Equations
- Chap 11: Boundary-Value Problems for Ordinary Differential Equations
- Chap 12: Numerical Solutions to Partial Differential Equations
可能有用的东西
- 常用的泰勒展开式:
- 指数函数:\(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)(收敛半径:\(\infty\))
- 正弦函数:\(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\)(收敛半径:\(\infty\))
- 余弦函数:\(\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\)(收敛半径:\(\infty\))
- 自然对数函数:\(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}\)(收敛区间:\((-1, 1]\))
- 几何级数(或二项式展开的特例
) :\(\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots = \sum_{n=0}^{\infty} x^n\)(收敛区间:\((-1, 1)\)) - 推广的二项式定理:\((1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!} x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!} x^3 + \dots = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^n\),其中 \(\binom{\alpha}{n} = \frac{\alpha(\alpha-1)\dots(\alpha-n+1)}{n!}\)(收敛区间:\((-1, 1)\),当 \(\alpha\) 为非负整数时,级数是有限项的)
参考资料
- PPT、教材
- Jiepeng 学长的笔记
- CrazySpottedDove 的笔记:记录了很多 hj 老师课上提到的一些很有价值的观点(
但笔者没好好听课,所以就参考这位大佬的笔记了 hh)
复习提纲 by hj 老师
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