CSC_349_HANDOUT#31

CSC_349_HANDOUT#31 - COMPUTER SCIENCE 349A Handout Number...

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1 COMPUTER SCIENCE 349A Handout Number 31 RICHARDSON'S EXTRAPOLATION (Section 22.2.1) -- the technique of combining two different numerical approximations that depend on a parameter (usually a stepsize h ) in order to obtain a new approximation having a smaller truncation error. Let M denote some value to be computed, for example, f ( x 0 ) or ′ ′ f ( x 0 or f ( x ) dx . a b Let N 1 ( h ) denote a formula (that depends on a parameter h that can take on different values) for computing an approximation to M , and suppose that the form of the truncation error of this formula is a known infinite series in powers of h . For example, the most common case is that the truncation error is O ( h 2 ) and is an infinite series with only even powers of h , that is, (1) { 4 4 4 43 4 4 4 42 1 L 3 2 1 ) ( is error truncation 6 3 4 2 2 1 stepsize using ion approximat computed 1 value exact 2 ) ( h O h h K h K h K h N M + + + + = where the values K i are some (possibly unknown) constants. The parameter h can be any positive value, but as h 0, the truncation error 0 ; that is, N 1 ( h ) M . If (1) holds, then using a stepsize of h /2 , (2) L + + + + = 64 16 4 2 6 3 4 2 2 1 1 h K h K h K h N M . In order to obtain an O ( h 4 ) approximation to M , we need to determine a linear combination of equations (1) and (2) in which the O ( h 2 ) terms cancel out; this will occur if we compute ) 1 ( ) 2 ( 4 × , which gives L = 16 15 4 3 ) ( 2 4 4 6 3 4 2 1 1 h K h K h N h N M M ,
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2 or (by solving for M ) (3) 4 4 4 43 4 4 4 42 1 L 4 4 4 4 4 4 1 ) ( is ) ( of error truncaton the 6 3 4 2 ) ( called is to ion approximat new this 1 1 1 4 2 2 16 5 4 3 ) ( 2 2 h O h N h N M h K h K h N h N h N M +
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CSC_349_HANDOUT#31 - COMPUTER SCIENCE 349A Handout Number...

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