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# 17 - 17 Numerical Integration 17 of Functions Chapter...

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17: Numerical Integration 17: Numerical Integration of Functions of Functions

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Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 2 Chapter Objectives Objectives : Know to use techniques that provide acceptable accuracy with minimum effort. Understand use of Richardson extrapolation to improve an estimate. Apply Richardson extrapolation to get Romberg Integration Understand how Gauss Quadrature provides better estimates than Newton- Cotes. Know how to apply MATLAB quad() and quad1() functions to integrate functions.
Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 3 17.1 Introduction Chapter 16 dealt with integration of tabulated data: small, fixed, number of data points. This chapter deals with integration of functions: can compute as many data points as needed to get the desired accuracy. base techniques are often the same (Trapezoidal rule, Simpson's 1/3 rule, etc.) We desire methods that: achieve a desired accuracy require minimum effort to do so this is usually characterized by trying to limit the number of function evaluations. E.G.: In section 16.3.3, 10 001 function evaluations were used to get an accurate estimate of an integral can an acceptable result be obtained with fewer function evaluations?

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Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 4 17.2.1 Richardson's Extrapolation If error behaviour is known, two estimates can be improved to form a better estimate. Notation: I(h) – estimate of integral using a value of h . E(h) – truncation error associated with that estimate. If we know that: that means: the ratio of errors: E = O h p E C h p where C is a constant E h 1  ≈ C h 1 p E h 2  ≈ C h 2 p E h 1 E h 2 C h 1 p C h 2 p h 1 h 2 p The following material is presented differently than in the text, p. 306.
Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 5 Richardson's Extrapolation (2) or: Now, given two estimates using h 1 and h 2 The exact value is: Unfortunately, neither I , nor E(h 1 ) , nor E(h 2 ) are knowable. Substituting (a) into (b): This can be solved for E(h 2 ) , to give an estimate of the truncation error: E h 1  = E h 2 h 1 h 2 p (a) I = I h 1   E h 1  = I h 2   E h 2 (b) I h 1   E h 2 h 1 h 2 p = I h 2   E h 2 E h 2  ≈ I h 1 − I h 2 1 h 1 / h 2 p

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Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 6 Richardson's Extrapolation (3) Now use this in the r.h.s of (b) to get an estimate of the true value: This can be re-arranged to: Two less-accurate results are extrapolated to form a more accurate estimate. It can be shown that the error associated with the improved estimate is: I = I h 2   E h 2  ≈ I h 2   I h 1 − I h 2 1 h 1 / h 2 p I h 1 / h 2 p I h 2  − I h 1 h 1 / h 2 p 1 E = O h p 2
Nov 21, 2006 08:54 ECOR2606 -- Hassan & Holtz 7 Richardson's Extrapolation (4) Example: using 1 and 3 panels of Simpson's 1/3 rule to compute: (see notes, chapter 16).

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17 - 17 Numerical Integration 17 of Functions Chapter...

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