Appendix C - Teach Yourself Scheme in Fixnum Days[Go to...

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Teach Yourself Scheme in Fixnum Days [Go to first , previous , next page; contents ; index ] Appendix C Numerical techniques Recursion (including iteration) combines well with Scheme's mathematical primitive procedures to implement various numerical techniques. As an example, let's implement Simpson's rule, a procedure for finding an approximation for a definite integral. C.1 Simpson's rule The definite integral of a function f ( x ) within an interval of integration [ a , b ] can be viewed as the area under the curve representing f ( x ) from the lower limit x = a to the upper limit x = b . In other words, we consider the graph of the curve for f ( x ) on the x , y -plane, and find the area enclosed between that curve, the x -axis, and the ordinates of f ( x ) at x = a and x = b . [t-y-scheme-Z-G-1.gif] According to Simpson's rule, we divide the interval of integration [ a , b ] into n evenly spaced intervals, where n is even. (The larger n is, the better the approximation.) The interval boundaries constitute n + 1 points on the x -axis, viz, x 0 , x 1 , ... , x i , x i +1 , ... , x n , where x 0 = a and x n = b . The length of each interval is h = ( b - a )/ n , so each x i = a + ih . We then calculate the ordinates of f ( x ) at the interval boundaries. There are n + 1 such ordinates, viz, y 0 , ... , y i , ... , y n , where y i = f ( x i ) = f ( a + ih ). Simpson's rule approximates the definite integral of f ( x ) between a and b with the value 12 : [t-y-scheme-Z-G-2.gif] We define the procedure integrate-simpson to take four arguments: the integrand f ; the x -values at the limits a and b ; and the number of intervals n . (define integrate-simpson (lambda (f a b n) ;... (1 of 9) [2/6/2008 11:43:05 AM]
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