romberg

Romberg - Chapter 07.04 Romberg Rule of Integration After reading this chapter you should be able to 1 derive the Romberg rule of integration and 2

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Chapter 07.04 Romberg Rule of Integration After reading this chapter, you should be able to: 1. derive the Romberg rule of integration, and 2. use the Romberg rule of integration to solve problems. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems like) countless applications for integral calculus. You can read about some of these applications in Chapters 07.00A-07.00G. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods has been developed to simplify the integral. Here, we will discuss the Romberg rule of approximating integrals of the form () = b a dx x f I ( 1 ) where ) ( x f is called the integrand = a lower limit of integration = b upper limit of integration 07.04.1
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07.04.2 Chapter 07.04 Figure 1 Integration of a function. Error in Multiple-Segment Trapezoidal Rule The true error obtained when using the multiple segment trapezoidal rule with segments to approximate an integral n () b a dx x f is given by () () n f n a b E n i i t = = 1 2 3 12 ξ ( 2 ) where for each i , i is a point somewhere in the domain ( ) [ ] ih a h i a + + , 1 , and the term () n f n i i = 1 can be viewed as an approximate average value of ( ) x f in . This leads us to say that the true error in Equation (2) is approximately proportional to [ b a , ] t E 2 1 n E t α ( 3 ) for the estimate of using the -segment trapezoidal rule. () b a dx x f n Table 1 shows the results obtained for dt t t 30 8 8 . 9 2100 140000 140000 ln 2000 using the multiple-segment trapezoidal rule.
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Romberg rule of Integration 07.04.3 Table 1 Values obtained using multiple segment trapezoidal rule for = 30 8 8 . 9 2100 140000 140000 ln 2000 dt t t x . n
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This note was uploaded on 04/30/2010 for the course MAE 107 taught by Professor Rottman during the Fall '08 term at UCSD.

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Romberg - Chapter 07.04 Romberg Rule of Integration After reading this chapter you should be able to 1 derive the Romberg rule of integration and 2

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