User:EGM6341.s11.team1.Chiu/Mtg2

Mtg 2: Fri, 7 Jan 11

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$$  \displaystyle I := \int_{a}^{b}f(x)dx \equiv I(f) $$     (1)
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$$ \equiv \ $$ means exact integration.

$$ := \ $$ means "equal by definition", non-symbolic (more useful).

$$ \overset{\underset{\mathrm{def}}{}}{=} \ $$ and $$ \overset{\underset{\mathrm{\Delta}}{}}{=} \ $$ are symbolic notation.

 One method  to approximate $$ I(f) \ $$: Approximate f by simpler functions $$ f_n \ $$, then integrate exactly $$ f_n \ $$.


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$$  \displaystyle f \cong \underbrace{f_n}_{ \color{blue} usually \ polynomial} \Rightarrow I(f) \cong I(f_n)\underset{ \color{red} \underset{ due \ to \ (1)}{\uparrow}}=\int_{a}^{b}f_n(x)dx $$
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Propagation of $$ f_n \ $$:


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$$  \displaystyle \left\| f-fn \right\| _ \underset{ \color{blue} \underset{infinity \ norm}{\uparrow}}{ \infty} \rightarrow \ as \ n \longrightarrow \infty $$
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$$  \displaystyle \left\| g \right\| _{ \infty}= \underset{x}{max} \left|  g(x) \right| $$
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 Note : $$ \mathbf{V} \in \underbrace{\mathbb{R}^{n}}_{ \color{blue} \mathbb{R} \times \cdots \times \mathbb{R} \ n \ times}$$

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Let $$ \left\{ \mathbf{e}_{i}, i=1, \ldots,n \right\} \ $$ at


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$$  \displaystyle \mathbf{e}_{i} \cdot \mathbf{e}_{j}= \underbrace{ \delta_{ij}}_{ \color{blue} Kronecker \ delta} = \begin{cases} & \text{ 1 } \ for \ i=j\\ & \text{ 0 } \ for \ i \neq j \end{cases} $$
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$$ \color{blue} \left\{ \mathbf{e}_{i} \right\} = \ $$ orthonormal basis for $$ \color{blue} \mathbb{R}^{n} \ $$


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$$  \displaystyle \Rightarrow \mathbf{V} = V_{i} \mathbf{e}_{i} \Rightarrow  \left\|  \mathbf{V} \right\|_{ \color{blue}2}= \left[  \sum_{i}  \left( V_{i} \right)^{2} \right]^{ \color{blue}1/2} $$
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Functions:


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$$  \displaystyle \left\| g \right\|_{2}= \left[ \int_{a}^{b} g^{2} \left( x \right) dx \right]^{ \color{blue}1/2} $$
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Similarly,


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$$  \displaystyle \left\| \mathbf{V} \right\|_{ \color{blue} \infty}= \underset{i}{max}  \left| V_{i} \right| $$ End method to approximate I(f)
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 Convergence of numerical integration :


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$$  \displaystyle E_{n}(f):=I(f)- \underbrace{I(f_{n})}_{ \color{blue} exact \ integration \ of \ \underbrace{f_n}_{ \color{blue} approximating \ function}}= \int_{a}^{b} \left[ f(x)-f_{n}(x) \right]dx $$
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$$  \displaystyle \left| E_{n}(f) \right| \leq \int_{a}^{b} \left| f(x)-f_{n}(x) \right|dx \leq  \left( b-a \right)  \left\| f-f_{n} \right\|_{ \color{blue} \infty} $$
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 Note:  ​


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$$  \displaystyle \left| \int_{a}^{b} g(x)dx \right| \leq \int_{a}^{b} \left| g(x) \right|dx $$
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 End Convergence of numerical integration 


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$$  \displaystyle \left\| g \right\|_{ \infty}= \underset{x}{max} \left| g \left( x \right) \right| \Rightarrow \left| g \left( x \right) \right| \leq  \left\| g \right\|_{ \infty}, \underset{ \color{blue} \underset{for \ all}{\uparrow}} \forall x \in  \left[ a,b \right] $$
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$$  \displaystyle \int_{a}^{b} \left| g(x) \right|dx \leq \int_{a}^{b} \underbrace{ \left\| g \right\|_{ \infty} }_{ \color{red} constant}dx = \left\| g \right\|_{ \infty} \left( b-a \right) $$
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2 ways to construct $$ \color{blue} f_n \ $$:

1) Taylor Series

2) Interpolating polynomial (FEM, FDM, FVM, Spectral)

 Taylor Series: 

Example: ​


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$$  \displaystyle I= \int_{0}^{1} \underbrace{ \frac{e^{x}-1}{x}}_{ \color{blue} f(x)}dx $$
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$$  \displaystyle f \left( x \right):= \frac{e^{x}-1}{x} $$ has a removable singularity at x=0
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End Example ​