User:EGM6321.F10.TEAM1.WILKS/Mtg35

=EGM6321 - Principles of Engineering Analysis 1, Fall 2009= Mtg 35: Thurs, 12Nov09

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Gauss Legendre quadrature (numerical integration} Quadrature; QUAD-->quadrilateral-->Greek: measuring areas Area $$ = \sum $$ Quadrilaterals Cubature; CUBE; Volume $$ = \sum  $$ cubes $$ I(f):=\int_{-1}^{1} f(x)\, dx \ $$ $$ I_n(f):=\sum_{j=1}^n w_jf(x_j)\, dx \ $$ with $$ \left \{ x_j \right \} \ $$ the roots for $$ P_n(x)=0 \ $$, where n is the degree of $$ P_n(x) \ $$ and $$ w_j \ $$ being the weight $$ -1[[media: Egm6321.f09.mtg35.djvu | Page 35-2]]
where j=1,2,...,n for $$ \eta\ \in \left [ -1,+1 \right ] \ $$ Ex: $$ n=2 \ $$ (2 point interpolation) [[media: Egm6321.f09.mtg31.djvu | Eq.(3) P.31-3]]  $$ P_2(x)=\frac{1}{2}(3x^2-1) \ $$ $$ \Rightarrow \ x_{1,2}= \pm \ \frac{1}{ \sqrt{3} } \ $$ [[media: Egm6321.f09.mtg31.djvu | Eq.(4) P.31-3]] $$ P_2'(x)=3x, P_3(x)=\frac{1}{2}(5x^3-3x) \ $$ $$ W_1=\frac{2}{(2+1)(3)(\frac{-1}{ \sqrt{3} }) \frac{1}{2} \left [ 5 (\frac{-1}{\sqrt{3}}) ^3-3(\frac{-1}{\sqrt{3}}) \right ] }=1 \ $$

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$$ W_2=1 \ $$ HW verify table for Gauss Legendre quadrature in wikipedia, analytical expression of $$ \left \{ x_j \right \} \ $$ and $$ \left \{ w_j \right \}, j=1,...,n \ $$ and  $$ n=1,...,5 \ $$ (n=integration points) after verifying the expression for  $$ P_n(x) \ $$ with $$ n=1,...,6 \ $$; (see [[media: Egm6321.f09.mtg31.djvu  | HW p31-3]] ) Evaluate numerically $$ \left \{ x_j \right \} \ $$ and $$ \left \{ w_j \right \} \ $$ and compute results with Abram & Stegum (see lecture plan) Question: How does Gauss Legendre quadrature compare to other quadrature methods, e.g. trapezoidal rule? Answer: Look at $$ E_n(f) \ $$, [[media:Egm6321.f09.mtg35.djvu | Eq.(3) P.35-2]]. Consider  $$ f \in \Rho\ _{2n-1} \ $$... END HW