# Math4501 Lecture 1

In many practical problems (ODEs (ordinary differential equations), PdEs (partial differential equations), System of equations)

closed-form analytical solutions are unknown.

-> resort ot computational algorithms (approximation)

For example,

Deep learning classifiers

**Root finding**

$$
f(x)=\sum_{i=1}^n a_i x^i
$$

for $n\geq 5$.

find all roots $x\in \mathbb{R}$ of $f(x)=0$.

**Investment**

Invest a dollars every month return with the rate $r$.

$g(r)=a\sum_{i=1}^n (1+r)^i=a\left[\frac{(1+r)^{n+1}-(1+r)}{r}\right]$

Say want $g(r)=b$ for some $b$.

$f(r)=a(1+n)^{n+1}-a(1+n)-br=0$

use Newton's method to find $r$ such that $f(r)=0$.

Since $f$ is non-linear, that is $f(x+y)\neq f(x)+f(y)$.

Let

$$
f_1(x_1,\dots, x_m)=0\\
\vdots\\
f_m(x_1,\dots, x_m)=0
$$

be a system of $m$ equations $\vec{f} \mathbb{R}^m \to \mathbb{R}^m$. and $f_1(\vec{x})=\vec{0}$.

If $\vec{f}$ is linear, note that

$$
\begin{aligned}
\vec{f}(\vec{x})&=\vec{f}(\begin{bmatrix}x_1\\ \vdots\\ x_m\end{bmatrix})\\
&=\vec{f}(x_1\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix}+x_2\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix}+\cdots+x_m\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
&=x_1\vec{f}(\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix})+x_2\vec{f}(\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix})+\cdots+x_m\vec{f}(\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
&=A\vec{x}
\end{aligned}
$$

where $\vec{e}_i$ is the $i$-th standard basis vector.

Gaussian elimination (LU factorization)