Numerical Methods/Numerical Integration

<< Numerical Methods Often, we need to find the integral of a function that may be difficult to integrate analytically (ie, as a definite integral) or impossible (the function only existing as a table of values). Some methods of approximating said integral are listed below. Trapezoidal Rule Consider some function, possibly unknown, , with known values over the interval [a,b] at n+1 evenly spaced points xi of spacing , and . Further, denote the function value at the ith mesh point as . Using the notion of integration as "finding the area under the function curve", we can denote the integral over the ith segment of the interval, from to as: = (1) Since we may not know the antiderivative of , we must approximate it. Such approximation in the Trapezoidal Rule, unsurprisingly, involves approximating (1) with a trapezoid of width h, left height , right height . Thus, (1) = (2) (2) gives us an approximation to the area under one interval of the curve, and must be repeated to cover the entire interval. For the case where n = 2, = (3) Collecting like terms on the right hand side of (3) gives us: or Now, substituting in for h and cleaning up, To motivate the general version of the trapezoidal rule, now consider n = 4, Following a similar process as for the case when n=2, we obtain Proceeding to the general case where n = N, Image:Trapizoidruleexample.jpg This is an example of what the trapezoidal rule would represent graphicly, here . Example Approximate to within 5%. First, since the function can be exactly integrated, let us do so, to provide a check on our answer. = (4) We will start with an interval size of 1, only considering the end points. (4) Relative error = Hmm, a little high for our purposes. So, we halve the interval size to 0.5 and add to the list (4) Relative error = Still above 0.01, but vastly improved from the initial step. We continue in the same fashion, calculating and , rounding off to four decimal places. (4) Relative error = We are well on our way. Continuing, with interval size 0.125 and rounding as before, (4) Relative error = Since our relative error is less than 5%, we stop. Error Analysis Let y=f(x) be continuous, well-behaved and have continuous derivatives in [x0,xn]. We expand y in a Taylor series about x=x0,thus- Simpson's Rule Consider some function possibily unknown with known values over the interval [a,b] at n+1 evently spaced points then it defined as where and and . Example Evaluate by taking ( must be even) Solution: Here Since & so Now when then And since , therefore for , , , , , the corresponding values are , , , , , Incomplete ... Completed soon Error Analysis Simpson's 3/8 The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. His also worked in the areas of numerical interpolation and probability theory. Theorem (Simpson's 3/8 Rule) Consider over , where , , and . Simpson's 3/8 rule is . This is an numerical approximation to the integral of over and we have the expression . The remainder term for Simpson's 3/8 rule is , where lies somewhere between , and have the equality . Proof Simpson's 3/8 Rule Simpson's 3/8 Rule Composite Simpson's 3/8 Rule Our next method of finding the area under a curve is by approximating that curve with a series of cubic segments that lie above the intervals . When several cubics are used, we call it the composite Simpson's 3/8 rule. Theorem (Composite Simpson's 3/8 Rule) Consider over . Suppose that the interval is subdivided into subintervals of equal width by using the equally spaced sample points for . The composite Simpson's 3/8 rule for subintervals is . This is an numerical approximation to the integral of over and we write . Proof Simpson's 3/8 Rule Simpson's 3/8 Rule Remainder term for the Composite Simpson's 3/8 Rule Corollary (Simpson's 3/8 Rule: Remainder term) Suppose that is subdivided into subintervals of width . The composite Simpson's 3/8 rule . is an numerical approximation to the integral, and . Furthermore, if , then there exists a value with so that the error term has the form . This is expressed using the "big " notation . Remark. When the step size is reduced by a factor of the remainder term should be reduced by approximately . Algorithm Composite Simpson's 3/8 Rule. To approximate the integral , by sampling at the equally spaced sample points for , where . Notice that and . Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations. Computer Programs Simpson's 3/8 Rule Simpson's 3/8 Rule Mathematica Subroutine (Simpson's 3/8 Rule). Object oriented programming. Example 1. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4. Solution 1. Example 2. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160. Solution 2. Example 3. Find the analytic value of the integral (i.e. find the "true value"). Solution 3. Example 4. Use the "true value" in example 3 and find the error for the Simpson' 3/8 rule approximations in example 2. Solution 4. Example 5. When the step size is reduced by a factor of the error term should be reduced by approximately . Explore this phenomenon. Solution 5. Example 6. Numerically approximate the integral by using Simpson's 3/8 rule with m = 1, 2, 4. Solution 6. Example 7. Numerically approximate the integral by using Simpson's 3/8 rule with m = 10, 20, 40, 80, and 160. Solution 7. Example 8. Find the analytic value of the integral (i.e. find the "true value"). Solution 8. Example 9. Use the "true value" in example 8 and find the error for the Simpson's 3/8 rule approximations in example 7. Solution 9. Example 10. When the step size is reduced by a factor of the error term should be reduced by approximately . Explore this phenomenon. Solution 10. Various Scenarios and Animations for Simpson's 3/8 Rule. Example 11. Let over . Use Simpson's 3/8 rule to approximate the value of the integral. Solution 11. Animations (Simpson's 3/8 Rule Simpson's 3/8 Rule). Internet hyperlinks to animations. Research Experience for Undergraduates Simpson's Rule for Numerical Integration Simpson's Rule for Numerical Integration Internet hyperlinks to web sites and a bibliography of articles. Headline text Example Error Analysis References and further reading Eric W. Weisstein. "Trapezoidal Rule." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TrapezoidalRule.html Davis, P. J., & Rabinowitz, P. (2007). Methods of numerical integration. Courier Corporation. Main Page - Mathematics bookshelf - Numerical Methods