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Unformatted text preview: 07.05.1 Chapter 07.05 Gauss Quadrature Rule of Integration After reading this chapter, you should be able to: 1. derive the Gauss quadrature method for integration and be able to use it to solve problems, and 2. use Gauss quadrature method to solve examples of approximate integrals. 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 07.00A07.00G. 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 Gauss quadrature rule of approximating integrals of the form ( ) ∫ = b a dx x f I where ) ( x f is called the integrand, = a lower limit of integration = b upper limit of integration 07.05.2 Chapter 07.05 Figure 1 Integration of a function. Gauss Quadrature Rule To derive the trapezoidal rule from the method of undetermined coefficients, we approximated Background: ) ( ) ( ) ( 2 1 b f c a f c dx x f b a + ≈ ∫ (1) Let the right hand side be exact for integrals of a straight line, that is, for an integrated form of ( ) ∫ + b a dx x a a 1 So ( ) b a b a x a x a dx x a a + = + ∫ 2 2 1 1 ( ) − + − = 2 2 2 1 a b a a b a (2) But from Equation (1), we want ( ) ) ( ) ( 2 1 1 b f c a f c dx x a a b a + = + ∫ (3) to give the same result as Equation (2) for x a a x f 1 ) ( + = . ( ) ( ) ( ) b a a c a a a c dx x a a b a 1 2 1 1 1 + + + = + ∫ ( ) ( ) b c a c a c c a 2 1 1 2 1 + + + = (4) Hence from Equations (2) and (4), ( ) ( ) ( ) b c a c a c c a a b a a b a 2 1 1 2 1 2 2 1 2 + + + = − + − Gauss Quadrature Rule 07.05.3 Since a and 1 a are arbitrary constants for a general straight line a b c c − = + 2 1 (5a) 2 2 2 2 1 a b b c a c − = + (5b) Multiplying Equation (5a) by a and subtracting from Equation (5b) gives 2 2 a b c − = (6a) Substituting the above found value of 2 c in Equation (5a) gives 2 1 a b c − = (6b) Therefore ∫ + ≈ b a b f c a f c dx x f ) ( ) ( ) ( 2 1 ) ( 2 ) ( 2 b f a b a f a b − + − = (7) Derivation of twopoint Gauss quadrature rule The twopoint Gauss quadrature rule is an extension of the trapezoidal rule approximation where the arguments of the function are not predetermined as Method 1: a and b , but as unknowns 1 x and 2 x . So in the twopoint Gauss quadrature rule, the integral is approximated as ∫ = b a dx x f I ) ( ) ( ) ( 2 2 1 1 x f c x f c + ≈ There are four unknowns 1 x , 2 x , 1 c and 2 c . These are found by assuming that the formula...
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 Spring '08
 Kaw,A
 xdx, Gauss Quadrature Rule

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