User:EGM6341.S11.team5.cavalcanti/Mtg9

Mtg 9: Fri, 21 Jan 11 [[media: Nm1.s11.mtg9.djvu | Page 9-1]]

_____________________________________________________________________ Newton-Cotes cont'd p8-3



[[media: Nm1.s11.mtg9.djvu | Page 9-2]]  Lagrange (n intervals) :

Note:    $$\begin{array}{c}{l}_{i,n}\left(x\right)\in {P}_{n}\\ i=\mathrm{0,1,.}\mathrm{..},n\end{array}$$(set of poly. of deg. $$\leqslant n$$)  "End Note" 

Trapezoidal Rule: (simple)

$$\left\lbrack a,b\right\rbrack , \left.\begin{matrix} x_{0}=a\\ x_{1}=b \end{matrix}\right\} $$ 2 nodes

$$f_{1}(x)=p_{1}(x)=\sum_{i=0}^{1}l_{i}(x)f(x_{i})=l_{0,1}(x)f(x_{0})+l_{1,1}(x)f(x_{1})$$ [[media: Nm1.s11.mtg9.djvu | Page 9-3]] 



Note: Used extensively in 1-D Finite Element Methods.  "End Note" 

(1)&(2) → $${l}_{i,\color{red}{n}}\left({x}_{j}\right)={\delta }_{\mathit{ij}}$$ Kronecker δ p.2-2

$${l}_{\mathrm{0,\color{red}{1}}}\left(x\right),{l}_{\mathrm{1,\color{red}{1}}}\left(x\right)$$: linear funcs → f1(x) is also linear since f1(x) is a linear combination of l0, 1(x) and l1, 1(x). (i.e., f1(x) is a straight line)

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