I : Question one : What is nonlinearity?
非线性是指两个变量间的数学关系，不是直线，而是曲线、曲面、或不确定的属性，是不成简单比例（即线性）关系的。
1: Prototype scalar equation: Find x∗ ∈ R such that
F(x∗) = 0
for a given function F that is differentiable with
derivative F′(x) ̸= constant (non-constant derivative)2: Prototype system of equations: Find ⃗x∗ ∈ Rn such that
F⃗ ( ⃗x ∗ ) = ⃗0
with Jacobian matrix F⃗′(⃗x) ̸= constant (non-constant Jacobian)
II: Question two : Why do we need numerical methods to solve nonlinear equations?
非线性方程，就是因变量与自变量之间的关系不是线性的关系，这类方程很多，例如平方关系、对数关系、指数关系、三角函数关系等等。
现实生活中大部分事情都是nonlinearity, nonlinearity可以很好的描述的生活中大部分事物。
III: Example of Nonlinear Equation:
-
Polynomial(多项式) equations :
截屏2021-12-29 10.52.34.png
(common approximations(近似值) for other nonlinear equations)
Trigonometric equations
sin(x)+2cos(x)=0etc.Many physics equations:
n1 sin θ1 = n2 sin θ2, E = mc2 etc.-
Neural Network
Neural Network.png
IV: Newton Method
It is rare for nonlinear equations to have a closed form solution that can be found through algebraic manipulation so people must require (approximate) solution.
People have the differentiable function F(x)
and its derivative F′(x).People have one initial point x0 that is close to the
unknown zero x∗ such that F(x∗) = 0.-
People look for a sequence of iterates x0,x1,...such that
iterates method.png -
牛顿法具体思路：
牛顿法具体思路 -
Derivation(推导) of Newton’ Method
牛顿法的推演 -
Pros and cons（优点和缺点）
Performance
Fast (quadratic(二次方) convergence rate) (二次收敛率)
Not robust（不稳健性）Other issues
Requires the derivative function
Requires a “good” initial guess
