60 lines
1.5 KiB
Markdown
60 lines
1.5 KiB
Markdown
# Math4501 Lecture 1
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In many practical problems (ODEs (ordinary differential equations), PdEs (partial differential equations), System of equations)
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closed-form analytical solutions are unknown.
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-> resort ot computational algorithms (approximation)
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For example,
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Deep learning classifiers
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**Root finding**
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$$
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f(x)=\sum_{i=1}^n a_i x^i
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$$
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for $n\geq 5$.
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find all roots $x\in \mathbb{R}$ of $f(x)=0$.
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**Investment**
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Invest a dollars every month return with the rate $r$.
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$g(r)=a\sum_{i=1}^n (1+r)^i=a\left[\frac{(1+r)^{n+1}-(1+r)}{r}\right]$
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Say want $g(r)=b$ for some $b$.
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$f(r)=a(1+n)^{n+1}-a(1+n)-br=0$
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use Newton's method to find $r$ such that $f(r)=0$.
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Since $f$ is non-linear, that is $f(x+y)\neq f(x)+f(y)$.
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Let
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$$
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f_1(x_1,\dots, x_m)=0\\
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\vdots\\
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f_m(x_1,\dots, x_m)=0
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$$
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be a system of $m$ equations $\vec{f} \mathbb{R}^m \to \mathbb{R}^m$. and $f_1(\vec{x})=\vec{0}$.
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If $\vec{f}$ is linear, note that
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$$
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\begin{aligned}
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\vec{f}(\vec{x})&=\vec{f}(\begin{bmatrix}x_1\\ \vdots\\ x_m\end{bmatrix})\\
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&=\vec{f}(x_1\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix}+x_2\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix}+\cdots+x_m\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
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&=x_1\vec{f}(\begin{bmatrix}1\\ 0\\ \vdots\\ 0\end{bmatrix})+x_2\vec{f}(\begin{bmatrix}0\\ 1\\ \vdots\\ 0\end{bmatrix})+\cdots+x_m\vec{f}(\begin{bmatrix}0\\ 0\\ \vdots\\ 1\end{bmatrix})\\
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&=A\vec{x}
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\end{aligned}
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$$
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where $\vec{e}_i$ is the $i$-th standard basis vector.
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Gaussian elimination (LU factorization) |