20 - Richardson extrapolation If the composite trapezoidal...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
4/7/2010 1 Richardson extrapolation: If the composite trapezoidal rule is used to evaluate an integral twice, once using a step size of h 1 and once using a step size of h 2 , we have: h E h I h E h I I ) ( ) ( ) ( ) ( 2 2 1 1 The errors are roughly proportional to the squares of the the step sizes: i i i i h h E h h I I using error when ) ( using obtained value ) ( integral of value true where 2 2 2 3 ) ( 12 ) ( ) ( 12 ) ( ) ( f h a b f n a b h E 2 1 2 1 2 2 2 1 2 1 ) ( ) ( ) ( ) ( h h h E h E h h h E h E Eliminating E ( h 1 ) from the original equation and solving for E ( h 2 ) gives: 2 2 1 2 1 2 2 2 2 2 1 2 1 ) / ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( h h h I h I h E h E h I h h h E h I This result leads to a useful expression for the true value of the integral: The equation takes two estimates with errors of order h 2 and produces an estimate 2 2 1 2 1 2 2 2 ) / ( 1 ) ( ) ( ) ( ) ( ) ( h h h I h I h I h E h I I with an error of order h 4 . When h 2 = h 1 /2 (when the step size is halved) the equation becomes: ) ( 3 1 ) ( 3 4 1 2 h I h I I
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4/7/2010 2 Example: Suppose we want to evaluate Solving analytically (or using quad ) gives I = 12.9910004 Composite trapezoidal integration with h = 8 gives I (8) = 9.2256830 (29% error) 9 1 2 . 0 dx xe x Composite trapezoidal integration with h = 4 gives I (4) = 11.9704303 (7.9% error) Combining the two estimates gives an improved estimate: ) error % 8 . 0 ( 8853461 . 12 ) 8 ( 3 1 ) 4 ( 3 4 I I I The two initial integrations involve a grand total of three function evaluations (the two points used to find I (8) can be reused in finding I (4) ) Interesting point: Applying Simpson’s 1/3 rule to the three points gives exactly the same result. This will always be the case when h 1 = b a (try proving this). Romberg integration: The improved estimates obtained by combining initial estimates can be combined to produce still better estimates and so on. k=1 k=2 k=3 k=4 j=1 h 1 I 11 I 12 I 13 I 14 j=2 h 2 I 21 I 22 I 23 Romberg Pyramid j=3 h 3 I 31 I 32 j=4 h 4 I 41 Initial estimates (obtained using ever smaller step sizes) Best estimate 1 1 1 1 , 1 , 1 1 , 2 1 4 4 j j k k j k j k k j h h I I I Example (same problem as last slide, analytical result = 12.991000416828392): 9.225682988289043 12.885346094001475 12.990342105576776 12.990999499741445 11.970430317573367 12.983779854853319 12.990989227957623 12.730442470533331 12.990538642138604 12.925514599237287 This pyramid required a total of just nine function evaluations.
Background image of page 2
4/7/2010 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 04/14/2010.

Page1 / 10

20 - Richardson extrapolation If the composite trapezoidal...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online