User:EGM6341.s11.TEAM1.WILKS/Mtg45

=EGM6321 - Principles of Engineering Analysis 1, Fall 2010=

[[media: 2010_12_01_16_31_38.djvu | Mtg 45:]] Fri, 3 Dec 10

= Derive Legendre equation (cont'd) =

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Differentiate [[media: 2010_11_30_19_50_24.djvu | Eq.(1)p.44-5]] : $$ \left [ (1- \mu^2 )P_n' \right ]'+n \left [ \mu\ P_n'+P_n \right ]-n P_{n-1}' =0 \ $$ Where: $$ \left [ (1- \mu^2)P_n' \right ] \rightarrow \ (1- \mu^2)P_n''-2 \mu\ P_n' \ $$ and $$ P_{n-1}' = \mu\ P_n'-n P_n \ $$ Obtain Legendre differential equation [[media: 2010_09_02_13_55_50.djvu | Eq.(1)p.5-4]] and [[media: 2010_10_14_14_02_27.djvu | Eq.(1)p.24-1]].

= Application: Gauss-Legendre (GL) quadrature =

Important application, e.g. Finite Element

Origin of "quadrature"
The word quadrature comes from quadrilateral (note use of quad in both words).



Greeks: measure areas, where Area equals sum of quadrilaterals

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The word cubature comes from the word cube: Volume is equal to the sum of cubes.

Gauss-Legendre quadrature formula
with $$ \left \{ x_j \right \} \ $$ being the roots of $$ P_n(x) \Rightarrow P_n(x_j)=0 \ $$ for $$ j=1,...,n \ $$, and $$\displaystyle n$$ the degree of the polynomial $$\displaystyle P_n$$.

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

Weights and error
Theorem :

End Theorem

[[media: 2010_12_01_16_31_38.djvu | Page 45-3]]
Example: $$ n=2 \ $$ (2 integration points)

[[media: 2010_11_09_15_00_14.djvu | Eq.(4)p.36-2]] : $$ P_2(x)=\frac{1}{2}(3x^2-1) \ $$

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



[[media: 2010_11_09_15_00_14.djvu | Eq.(5)p.36-2]] : $$ P_2'(x)=3x, P_3(x)=\frac{1}{2}(5x^3-3x) \ $$

$$ w_1=\frac{-2} { (2+1) (3) ( -\frac{1} { \sqrt{3} } ) \frac{1}{2} \left [ 5( -\frac{1}{ \sqrt{3} } )^3 -3 ( -\frac{1}{ \sqrt{3} } ) \right ] } = 1 \ $$

HW8.2

$$ w_2=1 \ $$. More generally, verify the table for Gauss Legendre quadrature in wikipedia, which gives the analytical expressions for $$ \left \{ x_j \right \} \ $$ and $$ \left \{ w_j \right \} \ $$ , $$ j=1,...,5 \ $$, after verifying the expressions for $$ P_n(x) \ $$, with $$ n=1,...,5 \ $$.

[[media: 2010_12_01_16_31_38.djvu | Page 45-4]]
Gaussian quadrature

Extend this table to $$\displaystyle n=6$$.

See [[media: Egm6321.f09.mtg35.djvu | Fall 2009 HW. pg.35-2]] and [[media: 2010_11_30_14_54_10.djvu | HW7.9 P41-2]].

Evaluate $$ \left \{ x_j \right \} \ $$ and $$ \left \{ w_j \right \} \ $$ and compare results to

NIST Handbook

END HW8.2

HW8.4

Back to [[media: 2010_11_11_08_24_37.djvu | HW7.4 P38-5]]. Complete the first three non-zero coefficients using Gauss Legendre quadrature up to within 5% accuracy. see [[media: Egm6321.f09.mtg34.djvu | Fall 2009 P34-2]]

END HW8.4

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Question: How does Gauss Legendre quadrature compare to other quadratures, e.g. trapezoidal rule?

Answer: Look at $$ E_n(f) \ $$ [[media: 2010_12_01_16_31_38.djvu | Eq.(5) P45-2]].

Consider $$ f \in \mathcal P_{2n-1} \ $$ (set of polynomials of degree less than or equal $$ 2n-1 \ $$ )

$$ \Rightarrow \ f^{(2n)}(x)=0 \Rightarrow \ E_n(f)=0 \ $$ i.e., the GL quadrature can integrate exactly any polynomial of degree less than or equal to $$ 2n-1 \ $$ using only n integration points (almost half).

Example: $$ 2n-1 = 3 \Rightarrow \ n=2 \Rightarrow \ 2n=4 \ $$

$$ f \in \mathcal P_3 \Rightarrow \ f(x) = \sum_{j=0}^3 c_j x^j \Rightarrow \ f^{(4)}(x)=0 \ $$

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Trapezoidal Rule (more details in EGM6341) $$ I(f) = \int_{a}^{b} f(x)\, dx \ $$ $$ h=\frac{b-a}{n} \ $$ where n=number of trapezoidal panels



$$ E_n(f)= -\frac{(b-a)h^2}{12} f^{(2)}(\eta), \eta \in [ a,b ] \ $$ Trapezoidal rule can only integrate exactly a straight line. Also $$ E_n(f) \rightarrow \ 0 \ $$ as $$ h \rightarrow \ 0 \ $$ ($$ \Leftrightarrow n \rightarrow \ \infty \ $$ ), even for a simple polynomial of degree 2 (parabola)