L3_Approximation - MA2213 Lecture 3 Approximation Piecewise...

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MA2213 Lecture 3 Approximation
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Piecewise Linear Interpolation p. 147 ] , [ , ) ( 2 1 1 2 1 2 x x x x x x x x = l 1 x ] , [ , ) ( 2 1 1 2 2 1 x x x x x x x x = l 3 x 3 l 2 x 1 l 2 l 1 1 1 1 x 1 x 2 x 2 x 3 x 3 x ] , [ , 0 ) ( 3 2 1 x x x x = l ] , [ , ) ( 3 2 2 3 3 2 x x x x x x x x = l can use Nodal Basis Functions ] , [ , ) ( 3 2 2 3 2 3 x x x x x x x x = l ] , [ , 0 ) ( 2 1 3 x x x x = l
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Introduction Problem : Find / evaluate a function P that (i) P belongs to a specified set S of functions , and (ii) P best approximates a function f among the functions in the set S Approximates = match, fit, resemble If S were the set of ALL functions the choice P = f solves the problem – “you can’t get any closer to somewhere than by being there”. If S is not the set of all functions then S must be defined carefully Furthermore, we must define the approximation criteria used to compare two approximations
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Set of Functions In practice S is closed under sums and multiplication by numbers – this means that it is a vector space Example: Bases for S = { Polynomials of degree < n } : Furthermore, S is usually finite dimensional } ,..., : ) ( ) ( { 1 1 R c c x b c x P S n j n j j = = = and then S admits a basis } ) ( ),. .., ( { 1 x b x b n Monomial Basis Lagrange’s Basis Newton’s Basis (used with divided differences) ) ( ),. .., ( 1 x L x L n 1 2 ,..., , , 1 n x x x ) ( ) ( ),. .., )( ( ), ( , 1 1 1 2 1 1 n x x x x x x x x x x L For distinct nodes n x x < < L 1 For possibly repeated nodes n x x L 1
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Approximation Criteria Least Squares p. 178-187 In many engineering and scientific problems, data is acquired from measurements http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss observed that measurement errors usually have “Gaussian Statistics” and he invented an optimum method to deal with such errors. Minimax or Best Approximation p. 159-165 Arises in optimal design, game theory as well as in mathematics of uniform convergence
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Least Squares Criteria Least Squares (over an finite set) = m k k k x P y 1 2 )) ( ( minimize Least Squares (over an interval) b a dx x P x 2 ) ) ( ) ( f ( minimize
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Least Squares Over a Finite Set p. 319-333 If = = m k k k x P y 1 2 )) ( ( then we minimize ) ( ) ( 1 x b c x P j n j j = = [] 2 11 ) ( ∑∑ == m k n j k j j k x b c y by choosing coefficients n c c ,..., 1 to satisfy n i x b c y c m k n j k j j k i ,..., 1 , 0 ) ( 2 = =
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Least Squares Over a Finite Set Remark: these are n equations in n variables Since ( ) = ∑∑ == m k n j k j j k i x b c y c 11 2 ) ( ( ) n i x b x b c y k i n j k j j k m k ,..., 1 ), ( ) ( 2 1 1 = = = n c c ,..., 1 the coefficients satisfy the equations ( ) n i y x b c x b x b m k k k i j n j m k k j k i ,..., 1 , ) ( ) ( ) ( 1 = = =
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Least Squares Equations [ ] n i y x b c x b x b m k k k i j n j m k k j k i ,..., 1 , ) ( ) ( ) ( 1 11 = = = == Construct matrices = ) ( ) ( ) ( ) ( 1 1 1 1 m n m n x b x b x b x b B L M O M L = n c c c M 1 = m y y y M 1 The least squares equations are y B Bc B T T = The interpolation equations y Bc = hold if and only if m k y x P
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This note was uploaded on 01/06/2012 for the course MA 2213 taught by Professor Michael during the Fall '07 term at National University of Singapore.

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L3_Approximation - MA2213 Lecture 3 Approximation Piecewise...

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