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N32 - ME 510 NUMERICAL METHODS FOR ME II Richardsons...

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ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Richardson’s Extrapolation “Richardson’s Extrapolation to the Limit or “Deferred Approach to the Limit”: Finding a more accurate answer using two inaccurate ones. Applicable to evaluation of functional values, derivatives, integrals, and solution of differential equations Extrapolating Polynomials 1 0 1 ( ) = a + a x + a x + .... p p p p f x g14 g14 0 (0) a first-order approximation to f(0) f g124 How do you find a better approximation to f(0)? Remember Taylor series expansion ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Compute f(qx) where 0 > q > 1: g11 g12 g11 g12 1 0 1 ( ) = a + a q x + a q x + .... p p p p f qx g14 g14 Solve for a p x p : 1 0 1 ( ) = a + a x + a x + .... p p p p f x g14 g14 g11 g12 1 ( ) ( ) a x = + O x 1 g14 g16 g16 p p p p f x f qx q Substitute: g11 g12 1 0 ( ) ( ) (0) = a - + O x 1 higher-order approximation to f(0) p p f x f qx f q g14 g16 g16 Example: 1 ( ) = , (0) ? x e f x f x g16 g0
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ME – 510 NUMERICAL METHODS FOR ME II g51g85g82g73g17g3g39g85g17g3g41g68g85g88g78g3g36g85g213g81g111 Fall 2007 Richardson’s Extrapolation for Derivatives : 2 ( ) = ( ) + '( ) + ''( ) 2! h f x h f x hf x f g91 g14 ( ) ( ) '( ) = ''( ) , 2 f x h f x h f x f x x h h g91 g91 g14 g16 g16 g31 g31 g14 First-order approximation O(h) Find a better approximation to the first derivative, f(x): 2 ( ) ( ) '( ) = '''( ) , 2 3!
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