# 20 - Richardson extrapolation If the composite trapezoidal...

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

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

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

View Full Document
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.
4/7/2010 3

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

View Full Document
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
Ask a homework question - tutors are online