(4.2) -Richardson Extrapolation
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1 (.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as approaces 0 wit rate of convergence O G, orf L O G if tere exists a positive constant K suc tat lim F L G K. Tis means F L as fast as G 0as 0. Wen K 0, F L faster ten G 0as 0, and we denote tat F L o G. Example Sow tat sin Since sin 5 5!..., sin and o and sin lim sin lim sin O. 5!... lim lim 5! ! Example Sow tat f x 0 f x 0 f x 0 O and f x 0 f x 0 f x 0 f x 0 o f x 0 f x 0 f x 0! f c were c is in x 0,x 0 f x 0 f x 0 f x 0! f c lim f x 0 f x 0 f x 0 f x 0 f x 0 lim f c f x 0 K f x 0 f x 0 f x 0! f c f x 0 f x 0 f c f x 0 lim f x 0 f x 0 f x 0 f x 0 lim f c f x 0 0. Example Sow tat f x 0 f x 0 f x 0 O and f x 0 f x 0 6 f x 0 f x 0 o
2 f x 0 f x 0 f x 0 f x 0! f c f x 0 f x 0 f x 0 f x 0! f c f x 0 f x 0 f x 0 f c f c f x 0 f c f x 0 f x 0 f x 0 6 f c lim f x 0 f x 0 f x 0 lim f c 6 6 f x 0 K f x 0 f x 0 6 f x 0 f x 0 f x 0 6 f c 6 f x 0 f x 0 6 f c f x 0 lim f x 0 f x 0 6 f x 0 f x 0 lim 6 f c f x Ricardson Extrapolation Ricardson Extrapolation is a metod to generate ig-accuracy O k approximation formulas using lower-accuracy O m, m k formulas. Here is te idea. Let D be approximated by a formula D wit approximation error A A A... Ten D D O or D D A o. For example, approximate f x 0 by te forward difference formula: f x 0 f x 0 f x 0 D f x 0, f x 0! f x 0 D f x 0 f x 0..., A f x 0! D D O or D D f x 0 o! Back to te formula: (*) D D A A..., replace by in (*), we ave (**) D D / A A A... Multiplying to bot sides of (**), we ave (***) D D / A A A... Compute (***) (*): Since (***)-(*) is also D D D, (***)-(*) D / D A A..., A f x 0...
3 D D / D A A... D / D B B... Hence, D D / D Ten Replace by O D / D D / D ( ) D D B B B... A A... B o.defined D / D. in ( ): ( ) D D/ B B B... Multiplying to bot sides of ( ): ( ) D D / B B B... Observe tat ( ) ( ) D D D. So, D ( ) ( ) D / D B B... D / D C C... Hence, D D / D O D / D C o. Continue tis process and define D k k D k k / D, for k. k We ave D D k O k D k k o k for k. Example Te distance of a car at te time t is given below. t in ours s in miles Estimate te speed wen t by te forward difference formula and Ricardson extrapolation metod. By te forward difference formula: D v x 0 s x 0 s x 0 O, were x 0 Let D s s.
4 D D D / D / D 0 D 0 D / D 0 D Similarly, consider (*) D D A A... D O D A o. For example, approximate f x 0 by te central difference formula: f x 0 f x 0 f x 0 f x 0 f 5 x 0 5!... (**) D D/ A A... Multiplying to bot sides of (**): (***) D D / A A... (***) (*) D D / D A... were For k, D (***) (*) D / D B B 6... D B B 6... D D / D. D k k D k k / D, D D k k O k D k k o k. Example Te same example above, approximate v using te central difference formula and te Ricardson extrapolation metod. t in ours s in miles Use te central difference formula: D v s s O D O s s D
5 D 75 0 D / D / D 0 D / D Example Complete te following tables using Ricardson extrapolation formulas basing on giving orders. O O O O O O O O O O O 6 O Summary: a. For a sequence of increments:,,,... D D A A A... D k k D k k / D k D D A A A 6... D k k D k k / D k D D A A 6 A 9... D k 8k D k k / D 8 k b. For a sequence of increments:,,,... D D k O k D D k O k D D k O k 5
6 D D A A A... D k k D k k / D k D D A A A 6... D k 9k D k k / D 9 k D D A A 6 A 9... D k 7k D k k / D 7 k D D k O k D D k O k D D k O k Exercises:. Te distance of a car at te time t is given below. t in ours s in miles Estimate te speed wen t using te backward difference formula and Ricardson extrapolation metod.. Complete te following table. O O O Te distance of a car at te time t is given below. t in ours distance in miles a. Estimate v using te central difference formula wit 9 b. Let te central difference formula wit be D.DefineD k 9k D k k / D, k,,,. 9 k k Use te results in a. to find all possible D starting wit. c. Suppose we know v D A A A 6... were A i s are constants. Ten v D B B 6... Find B (in term of A ). 6
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