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

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Gauss Legendre quadrature (numerical integration}

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



Area ${\displaystyle =\sum }$ Quadrilaterals

Cubature; CUBE; Volume ${\displaystyle =\sum }$ cubes

${\displaystyle I(f):=\int _{-1}^{1}f(x)\,dx\ }$

${\displaystyle I_{n}(f):=\sum _{j=1}^{n}w_{j}f(x_{j})\,dx\ }$ with ${\displaystyle \left\{x_{j}\right\}\ }$ the roots for ${\displaystyle P_{n}(x)=0\ }$ , where n is the degree of ${\displaystyle P_{n}(x)\ }$ and ${\displaystyle w_{j}\ }$ being the weight

${\displaystyle -1<x_{1}<x_{2}<...<x_{n}<1\ }$

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${\displaystyle \displaystyle {\begin{aligned}I(f)=I_{n}(f)+E_{n}(f)\end{aligned}}}$	(1)

${\displaystyle \displaystyle {\begin{aligned}w_{j}={\frac {-2}{(n+1)P_{n}'(x_{j})P_{n+1}(x_{j})}}\end{aligned}}}$	(2)

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

${\displaystyle \displaystyle {\begin{aligned}E_{n}(f)={\frac {2^{2n+1}(n!)^{4}}{(2n+1)\left[(2n)!\right]^{2}}}{\frac {f^{(2n)}(\eta \ )}{(2n)!}}\end{aligned}}}$	(3)

for ${\displaystyle \eta \ \in \left[-1,+1\right]\ }$

Ex: ${\displaystyle n=2\ }$ (2 point interpolation)

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

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

Eq.(4) P.31-3 ${\displaystyle P_{2}'(x)=3x,P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)\ }$

${\displaystyle W_{1}={\frac {2}{(2+1)(3)({\frac {-1}{\sqrt {3}}}){\frac {1}{2}}\left[5({\frac {-1}{\sqrt {3}}})^{3}-3({\frac {-1}{\sqrt {3}}})\right]}}=1\ }$

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${\displaystyle W_{2}=1\ }$

HW: verify table for Gauss Legendre quadrature in wikipedia, analytical expression of ${\displaystyle \left\{x_{j}\right\}\ }$ and ${\displaystyle \left\{w_{j}\right\},j=1,...,n\ }$ and ${\displaystyle n=1,...,5\ }$ (n=integration points) after verifying the expression for ${\displaystyle P_{n}(x)\ }$ with ${\displaystyle n=1,...,6\ }$; (see HW p31-3 )

Evaluate numerically ${\displaystyle \left\{x_{j}\right\}\ }$ and ${\displaystyle \left\{w_{j}\right\}\ }$ 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 ${\displaystyle E_{n}(f)\ }$, Eq.(3) P.35-2. Consider ${\displaystyle f\in \mathrm {P} \ _{2n-1}\ }$...

References