University of Florida/Egm6321/F10.TEAM1.WILKS/Mtg35

EGM6321 - Principles of Engineering Analysis 1, Fall 2009

Mtg 35: Thurs, 12Nov09

Page 35-1

Gauss Legendre quadrature (numerical integration}

Quadrature; QUAD-->quadrilateral-->Greek: measuring areas

File:EGM6321 F10 TEAM1 WILKS xc mtg35 1.svg

Area $=\sum$ Quadrilaterals

Cubature; CUBE; Volume $=\sum$ cubes

$I(f):=\underset{-1}{\overset{1}{\int }}f(x)dx$

$I_n(f):=\sum _{j=1}^n{w}_{j}f(x_j)dx$ 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

$-1<x_1<x_2<...<x_n<1$

Page 35-2

$\begin{array}{c}I(f)=I_n(f)+E_n(f)\end{array}$

$\begin{array}{c}{w}_j=\frac{-2}{(n+1){P_n}^{\prime }(x_j)P_{n+1}(x_j)}\end{array}$

where j=1,2,...,n

$\begin{array}{c}{E}_n(f)=\frac{2^{2n+1}(n!{)}^4}{(2n+1){[(2n)!]}^2}\frac{f^{(2n)}(\eta )}{(2n)!}\end{array}$

for $\eta \in [-1,+1]$

Ex: $n=2$ (2 point interpolation)

Eq.(3) P.31-3 $P_2(x)=\frac{1}{2}(3x^2-1)$

$\Rightarrow x_{1,2}=\pm \frac{1}{\sqrt{3}}$

Eq.(4) P.31-3 ${P_2}^{\prime }(x)=3x,P_3(x)=\frac{1}{2}(5x^3-3x)$

$W_1=\frac{2}{(2+1)(3)(\frac{-1}{\sqrt{3}})\frac{1}{2}[5(\frac{-1}{\sqrt{3}}{)}^3-3(\frac{-1}{\sqrt{3}})]}=1$

Page 35-3

$W_2=1$

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 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 $E_n(f)$, Eq.(3) P.35-2. Consider $f\in \mathrm{P}{}_{2n-1}$...

References

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