romberg_gauss_intro_2010

romberg_gauss_intro_2010 - Revisiting Numerical...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Revisiting Numerical Integration: Getting More from Fewer Points Part II March 2010 Bruce Cohen Lowell High School, SFUSD bic@cgl.ucsf.edu http://www.cgl.ucsf.edu/home/bic David Sklar San Francisco State University dsklar@sfsu.edu Ver. 5.00
Background image of page 1

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

View Full DocumentRight Arrow Icon
Plan Romberg integration: getting more with fewer points Gaussian quadrature: getting even more with even fewer points Bibliography
Background image of page 2
Romberg Integration The method arises from a technique called Richardson extrapolation which can be used whenever the error E ( h ) can be expanded in a series of the form ( 29 2 3 4 1 2 3 4 E h c h c h c h c h = + + + + L ( 29 ( 29 ( 29 , b a I f x dx T h E h = = + ( 29 2 4 6 8 2 4 6 8 E h c h c h c h c h = + + + + L We’ll illustrate Richardson’s technique by applying it to the trapezoidal rule. ( 29 ( 29 2 with E h O h = In fact we can show that if f can be expanded in a Taylor series on each subinterval then E can be expanded in a series of the form
Background image of page 3

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

View Full DocumentRight Arrow Icon
To implement the method we don’t need to know the coefficients, we need only know that they exist. Assume that we have computed trapezoidal estimates for I for h, h/2, h/4, h/8, so we have Just looking at the first two we have ( 29 ( 29 ( 29 ( 29 , 2 , 4 , 8 . T h T h T h T h ( 29 ( 29 ( 29 2 4 6 8 2 4 6 8 I T h E h T h c h c h c h c h = + = + + + + + L 2 4 6 8 2 4 6 8 2 4 6 8 and 2 2 2 2 2 2 2 h h h h h h h I T E T c c c c = + = + + + + + L ( 29 ( 29 4 6 8 2 2 4 6 8 4 6 8 2 1 2 2 2 2 2 h h h h I T T h c c c - = - + + + + L Which we can rewrite as ( 29 ( 29 ( 29 4 6 8 2 2 4 6 8 4 6 8 2 1 2 1 2 2 2 2 2 h h h h h I T T T h c c c - = - + - + + + + L We can eliminate the h 2 term by multiplying the second by 2 2 and subtracting the first giving
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 17

romberg_gauss_intro_2010 - Revisiting Numerical...

This preview shows document pages 1 - 5. Sign up to view the full document.

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