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.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence known to converge to 0. Te sequence x n n is said to converge to x wit rate of convergence O n or x n x O n if tere exists a positive constant K suc tat Note tat te above inequality implies tat Note tat usually we coose n K 0. For example, x n x K n for all sufficiently large n. n p lim n x n x n K. for some p 0or n n were if it works and n n, n n. Now 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 (a small-o). Example Sow tat sin Since sin 5 5!..., sin and o and sin lim sin lim sin O. 5!... 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, we ave
2 Ten lim f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0! f c. 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 we ave we ave we ave f x 0 f x 0 f x 0 f x 0! f x 0 f x 0 f x 0 f x 0! f c f c,and f x 0 f x 0 f x 0 f c f c f x 0 f c lim f x 0 f x 0 lim f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 f x 0 6 f c, 6 f c 6 f x 0 K. 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,. 6 f x 0 f x 0 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.
3 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..., A f x 0 Compute (***) (*): (***)-(*) D / D A A... Since (***)-(*) is also D D D, D D / D A A... D / D A A... D / D B B... Hence, D D / D O D / D B o.defined D / D. Ten ( ) D D B B B... Replace by 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...
4 We ave D k k D k k / D, for k. k 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 0 5 s in miles Example 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. 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 D (***) (*) D / D B B 6... D B B 6...
5 For k, 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 0 5 s in miles Use te central difference formula: D 5 0 D / D / D v.5 0 D 0 D / s s O D O s s D D Example Complete te following tables using Ricardson extrapolation formulas basing on giving orders. O O O O O O O O
6 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 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 k D k k / D k D D k O k D D k O k D D k O k D D k O k D D k O k D D k O k Exercises:. Sow tat cos O and cos O.. Te distance of a car at te time t is given below. t in ours 0 5 s in miles Estimate te speed wen t using te backward difference formula and Ricardson extrapolation metod. Wat is your coice of? 6
7 . Complete te following table (computer A, B and C) using Ricardson s extrapolation formulas. O (D ) O (D ) O ( D D D / D / D 8 / A. 68 ) O (D ) B C Te distance of a car at te time t is given below. t in ours distance in miles a. Estimate v, te velocity wen t, 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 ).
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