Lecture 8

Lecture 8 - Introduction to Numerical Integration, as an...

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EE103 Slides 8A (SEJ) 1 Introduction to Numerical Integration, as an Application of Polynomial Interpolation 1 1 1 2 () _ [,] , , . bb m aa m m f x dx P x dx error term P x interpolates f at m equally spaced points in a b m P x a polynomial =+ ∫∫ Newton-Cotes Rules 1 m f xdx P xdx EE103 Slides 8A (SEJ) 2 ab 0 a zero degree approximation, P "1 " The m case = Rectangular Approximation 0 P xf a 0 P b
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EE103 Slides 8A (SEJ) 3 a b 1 2 b m a mP x = = :( ) Trapezoid Rule (m = 2) 1 1 2 b a Px f a f bb a = +− () ( () ) ( ) 1 = EE103 Slides 8A (SEJ) 4 a b c=(a+b)/2 2 2 aa f xdx P xdx e r ro r te rm P x interpolates afa cfc bfb =+ ∫∫ _ ( , ( )),( , ( )) Simpson’s Rule (m=3) 1 m P xdx
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EE103 Slides 8A (SEJ) 5 2 4 3 2 () [() ] b a h Px d x fa fc fb ba h =+ + = Exercise: Derive the above. Simpson’s Rule (m=3) EE103 Slides 8A (SEJ) 6 Third degree polynomial (red), interpolating at the m=4 points, 0,1,2,3 3 rd Degree Polynomial
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EE103 Slides 8A (SEJ) 7 ab Trapezoid Approximation Simpson’s Approximation Trapezoid, Simpson EE103 Slides 8A (SEJ) 8 01 ,, , n xx x [] 0 interpolant nn xa b x x P x ∈= ,[ , ] , ( ) () () () ( ) 1 1 ! n fx f xP x x x x x x x n ξ + =+ −− + " ( ) ( ) ( ) ( ) ( ) ( ) 0011 xf f f x , , Polynomial Approx Theorem () ( ) ( ) ( ) 1 0 1 n n bb b ni aa a i f x dx P x dx x x dx n + = + ∫∫ !
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EE103 Slides 8A (SEJ) 9 () () () () bb aa Then c a b so that fugud u f c gud u ∃∈ = ∫∫ [,] Mean Value Theorem for Integrals ,,' [ , ] IF f g continuous IF g doesn t change sign on a b ( ) () () ( ) b a xa b b f xg u d u f u g u d u f u d u ≤≤ min ( ) ( ) max ( ) ( ) 1 () hx 2 b a u b a f x gudu = ( ) 12 :( ) ( ) ( ) note h x h x x h x α = b a f u g udu f x ( ) EE103 Slides 8A (SEJ) 10 b a
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This note was uploaded on 03/30/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Winter '08 term at UCLA.

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Lecture 8 - Introduction to Numerical Integration, as an...

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