EE103 Section8

EE103 Section8 - EE103 Winter 2009 Lecture Notes(SEJ Section 8 SECTION 8 INTRODUCTION TO NUMERICAL INTEGRATION(An Application of Polynomial

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EE103 Winter 2009 Lecture Notes (SEJ) Section 8 SECTION 8: INTRODUCTION TO NUMERICAL INTEGRATION (An Application of Polynomial Interpolation) ............................................................................................................................................ 143 Background Information: .................................................................................................................... 143 Newton-Cotes Quadrature: ................................................................................................................. 144 Basic Idea: ............................................................................................................................................. 145 Rectangular Rule ( 1 m = ): .............................................................................................................. 145 Trapezoid Rule ( 2 m = ): ................................................................................................................ 145 Simpson’s Rule ( 3 m = ): ................................................................................................................. 147 Simpson’s is Actually () 5 h Ο : (This subsection is a digression and may be omitted) ........... 148 Basic Newton-Cotes Error Bounds on [,] ab : ................................................................................... 150 Composite Integration .......................................................................................................................... 151 Composite Trapezoid Rule .............................................................................................................. 151 Composite Simpson’s Rule .............................................................................................................. 153 Composite Newton-Cotes Error Bounds : .......................................................................................... 155 Usage of Error Bounds ..................................................................................................................... 155 Trapezoid Rule: ............................................................................................................................ 156 Simpson’s Rule: ............................................................................................................................ 157
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EE 103 Lecture Notes Winter 2009 (SEJ) Section 8 143 SECTION 8: INTRODUCTION TO NUMERICAL INTEGRATION (An Application of Polynomial Interpolation) Numerical integration is an important topic for engineers and computer scientists to become familiar with. The introductory methods and analyses to follow are meant only to introduce you to this important area; the analyses are based upon polynomial interpolation and the methods we'll discuss are among the simplest. Background Information: There is an important theorem, similar to Taylor’s Theorem, that relates a function ( ) f x and an appropriate interpolating polynomial. This result will be useful for the analysis of the methods to be introduced. Theorem: Let 01 ,, , n x xx be n+1 distinct points, and let f have n+1 continuous derivatives on the interval [ ] , ab . Then, for each [ ] , x , () () () ( ) ( ) 1 1! n nn fx f xP x x x x x n ξ + =+ −− + " where () ( ) , x and n Px is the unique interpolating polynomial, of degree n , which interpolates the n+1 points ( ) ( ) ( ) ( ) 0011 ,, , , xfx There is one more result which we’ll need. Fact: (Weighted Mean Value Theorem for Integrals) If f is continuous on [ ] , and g is integrable on [ ] , and ( ) gx does not change sign on [ ] , , then there is a point c , acb << , such that () () () () bb aa f xgxd x f c gxd x = ∫∫ (Note: If 1 , then this says 1 b a f xd ba = )
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EE 103 Lecture Notes Winter 2009 (SEJ) Section 8 144 Proof: In the first place, we’ll assume the intuitively clear result: If h is continuous on [ ] , ab and if () ( ) 12 hx α ≤≤ , [ ] ,, x xa b then, there exists [ ] , ca b such that ( ) hc = . Now, let ( ) 1 2 argmin argmax axb x fx x = = Then ( ) () () () ( ) () bb b aa a f x gzd z f zgzd z f x z ∫∫ Therefore, let () () b a z = Then, taking () () () b a z = We have that there exists c such that b a f cg z d z = Newton-Cotes Quadrature: We’ll discuss the approximation 1 b b m a a f xd x P xd x where 1 m P is the interpolating polynomial for m equally spaced points in the interval [,] Therefore, by the theorem above: () ( ) ( ) ( ) 1 1 0 01 ! , m m b mi a i m x x x x d x m x b ξ = =+ ==
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EE 103 Lecture Notes Winter 2009 (SEJ) Section 8 145 Basic Idea: We are interested in evaluating, numerically () b a f xd x , a<b There are two parts of such an investigation: 1. The statement of the method 2. The derivation of an error bound Rectangular Rule ( 1 m = ): Given any arbitrary curve. We could, for instance, approximate the integral by () () ( ) () 0 bb aa x f b b a P xd x ≈− = ∫∫ How good is this approximation? (Clearly, not so good) By Taylor’s Theorem () ( ) ( ) ( )( ) f xf bf x x b ξ = +− This gives us ( ) ()( ) ( ) 2 1 2 b a f
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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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EE103 Section8 - EE103 Winter 2009 Lecture Notes(SEJ Section 8 SECTION 8 INTRODUCTION TO NUMERICAL INTEGRATION(An Application of Polynomial

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