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Unformatted text preview: MA2213 Lecture 7 Optimization Topics The Best Approximation Problem pages 159165 Chebyshev Polynomials pages 165171 Finding the Minimum of a Function Method of Steepest Descent Constrained Minimization Gradient of a Function http://en.wikipedia.org/wiki/Optimization_(mathematics) http://www.mat.univie.ac.at/~neum/glopt/applications.html What is argmin ? 4 1 ] 1 , [ x 1) (x x min = 2 1 ] 1 , [ x 1) (x x argmin = } 1 , { x) (1 x argmin ] 1 , [ x = Optimization Problems 2 j n 1 j V f ) ) f(x ( argmin j y = Least Squares : given or dx x y 2 b a V f )) ( ) f(x ( argmin ]) , ([ b a C V compute Spline Interpolation : } ,..., 1 , ) ( f ]), , ([ f { 2 n i y x b a C V i i = = b x x x x a n n = < < < < = 1 2 1 given where compute dx x 2 b a ' ' V s ] ) ( [s argmin ) , ( ),..., ( 1 , 1 n n y x y x or ]) , ([ b a C y and a subspace LS equations page 179 are derived using differentiation. Spline equations pages 149151 are derived similarly. The Best Approximation Problem p.159 ]) , ([ f b a C Definition For and integer ]  ) ( ) ( f  max [ min ) f ( x p x b x a P p n n } degree of s polynomial { n P n where Definition The best approximation problem is to compute ]  ) ( ) ( f  max [ min arg x p x b x a P p n n Best approximation pages 159165 is more complicated Best Approximation Examples ]  ) (  max [ min arg 1 1 x p e m x x P p n n  5431 . 1 2 / ) ( 1 = + = e e m 1752 . 1   max 1 1 =  m e x x 2643 . 1 1752 . 1 ) ( 1 + = x x m 279 .  ) (  max 1 1 1 =  x m e x x Best Approximation Degree 0 Best Approx. Error Degree 0 Best Approximation Degree 1 Best Approx. Error Degree 1 Properties of Best Approximation 1.Best approximation gives much smaller error than Taylor approximation. 2. Best approximation error tends to be dispersed over the interval rather that at the end. Figures 4.43 and 4.14 on page 162 display the error for the degree 3 Taylor Approximation (at x = 0) and the error for the Best Approximation of degree 3 over the interval [1,1] for exp(x), together with the figures in the preceding slides, support assertions on pages 162163: 3. Best approximation error is oscillatory, it changes sign at least n+1 times in the interval and the sizes of the oscillations will be equal. Theoretical Foundations ]) , ([ f b a C Theorem 1. (Weierstrass Approximation Theorem 1885). If then there exists a and polynomial p such that ]. , [ ,  ) ( ) ( f  b a x x p x  Proof Weierstrasss original proof used properties of solutions of a partial differential equation called the heat equation. A modern, more constructive proof based on Bernstein polynomials is given on pages 320323 of Kincaid and Cheneys Numerical Analysis: Mathematics of Scientific Computing, Brooks Cole, 2002....
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 Fall '07
 Michael
 Polynomials, Numerical Analysis, Approximation

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