L3_Approximation

# 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 - - = 1 x ] , [ , ) ( 3 2 2 3 2 3 x x x x x x x x - - = ] , [ , ) ( 2 1 1 2 2 1 x x x x x x x x - - = ] , [ , 0 ) ( 2 1 3 x x x x = 3 x 3 2 x 1 2 1 1 1 1 x 1 x 2 x 2 x 3 x 3 x ] , [ , 0 ) ( 3 2 1 x x x x = ] , [ , ) ( 3 2 2 3 3 2 x x x x x x x x - - = can use Nodal Basis Functions
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 For distinct nodes n x x < < 1 For possibly repeated nodes n x x 1
Approximation Criteria Least Squares p. 178-187 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. In many engineering and scientific problems, data is acquired from measurements 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
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 1 1 ) ( = = - 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 1 1 = = - = =

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Least Squares Over a Finite Set Remark: these are n equations in n variables Since n c c ,..., 1 the coefficients ( 29 = - = = m k n j k j j k i x b c y c 1 1 2 ) ( ( 29 n i x b x b c y k i n j k j j k m k ,..., 1 ), ( ) ( 2 1 1 = - - = = satisfy the equations ( 29 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 1 1 = = ∑ ∑ = = =
Least Squares Equations Construct matrices [ ] 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 1 1 = = ∑ ∑ = = = = ) ( ) ( ) ( ) ( 1 1 1 1 m n m n x b x b x b x b B = n c c c 1 = m y y y 1 The least squares equations are y B Bc B T T = The interpolation equations y Bc =

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L3_Approximation - MA2213 Lecture 3 Approximation Piecewise...

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