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content/Math4501/Math4501_L1.md

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 # Math4501 Lecture 1
 
-In many practical problems (ODEs, PdEs, Sys of eqn)
+In many practical problems (ODEs (ordinary differential equations), PdEs (partial differential equations), System of equations)
 
 closed-form analytical solutions are unknown.
 

content/Math4501/Math4501_L2.md

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+# Math4501 Lecture 2
+
+Solving non-linear equations
+
+Let $\vec{f}:\mathbb{R}^n\to\mathbb{R}^n$ we want to solve $\vec{f}(\vec{x})=\vec{0}$. ($m$ equations, $m$ variables)
+
+In case if $\vec{f}$ is linear, we can solve it by Gaussian elimination.
+
+Closely related to the problem: eigenvalue problem.
+
+related to root finding problem for polynomial.
+
+## Polynomial approximations
+
+Let $f:[0,1]\to\mathbb{R}$ be a continuous function.
+
+Find polynomial $p_n$ of degree $n$ such that $p_n(x_i)=f(x_i)$ for $i=0,1,\cdots,n$.
+
+Then, some key questions are involved:
+
+1. How to compute $c_0,c_1,\cdots,c_n$?
+2. If $f$ is continuously differentiable, does $p_n'$ approximate $f'$?
+3. If $f$ is integrable, does $\int_0^1 p_n(x)dx$ approximate $\int_0^1 f(x)dx$?
+
+Deeper questions:
+
+Is the approximation **efficient**?
+
+## Scalar problem
+
+Problem 1: Let $f:[a,b]\to\mathbb{R}$ be a continuous function. Find $\xi\in[a,b]$ such that $f(\xi)=0$.
+
+Problem 2: Let $f:[a,b]\to\mathbb{R}$ be a continuous function. Find $\xi\in[a,b]$ such that $f(\xi)=\xi$.
+
+P1, P2 are equivalent. $f(x)\coloneqq f(x)-x$ is a continuous function.
+
+[Intermediate value theorem](https://notenextra.trance-0.com/Math4121/Math4121_L3#definition-5121-intermediate-value)
+
+> Some advantage in solving P1 as P2
+
+### When does a solution exists
+
+Trivial case: $f(x)=0$ for some $x\in[a,b]$.
+
+Without loss of generality, assume $f(a)f(b)<0$, Then there exists $\xi\in(a,b)$ such that $f(\xi)=0$.
+
+Bisection algorithm:
+
+```python
+def bisection(f, a, b, tol=1e-6, max_iter=100):
+    # first we setup two sequences $a_n$ and $b_n$
+    # require:
+    # |a_n - b_n| \leq 2^{-n} (b-a)
+    for i in range(max_iter):
+        c = (a + b) / 2
+        if c < tol or f(c) == 0:
+            return c
+        elif f(a) * f(c) < 0:
+            b = c
+        else:
+            a = c
+    return None
+```
+
+Let $f(a_n)<0$ for all $n$ and $f(b_n)>0$ for all $n$.
+
+$\lim_{n\to\infty} f(a_n)\leq 0$ and $\lim_{n\to\infty} f(b_n)\geq 0$.
+
+If limit exists, then $\lim_{n\to\infty} f(a_n)=\lim_{n\to\infty} f(b_n)=0$.
+
+Such limit exists by the sequence $a_n$ and $b_n$ is Cauchy and we are in real number field.
+
+This can be used to solve P2:
+
+Recall that if we define $f(x)\coloneqq g(x)-x$, then $f(x)=0$ if and only if $f(a)f(b)<0$. That is $(g(a)-a)(g(b)-b)\leq 0$.
+

content/Math4501/_meta.js

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+export default {
+    index: "Course Description",
+    "---":{
+        type: 'separator'
+    },
+    Math4501_L1: "Numerical Applied Mathematics (Lecture 1)",
+    Math4501_L2: "Numerical Applied Mathematics (Lecture 2)",
+}

content/Math4501/index.md

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+# Math4501
+
+Numerical Applied Mathematics