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(Thomas) Simpson’s rule A great mathematician Riley Wang Mr. Sidanycz Block 2nd Thomas Simpson • 1. Thomas Simpson, born on August 20, 1710, in Market Bosworth, Leicestershire, England, was the son of a self-taught weaver. • 2. Simpson’s father naturally expected his son to take up the same profession as his “ol’ man”. However, with the occurrence of a solar eclipse in 1724, Thomas Simpson turned to “mathematical interests”, changing his life forever. Thomas Simpson • 3. By 1735, he was able to solve puzzles concerning infinitesimal calculus. • 4. Not only did Simpson work on mathematics, but he also delved heavily into probability theory and the concept of approximation and error. • 5. Simpson died on May 14, 1761, he is usually remembered for his contribution to numerical integration: “Simpson’s Rule”. . Simpson’s rule • 1.Discussed here at the Holistic Numerical Methods Institute, Simpson’s Rule “has become a popular and useful special case of the Newton-Cotes formula for approximating an integral”. • Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation: Simpson’s rule Simpson’s Quadrati interpolation Error of Simpson's rule Simpson's rule Example Example 1: Approximating the graph of y = f(x) with parabolic arcs across successive pairs of intervals to obtain Simpson's Rule Since only one quadratic function can interpolate any three (non-colinear) points, we see that the approximating function must be unique for each interval . Note that the following quadratic function interpolates the three points Example of Simpson's rule Then you get • Finally we get To nine decimal places, we get better approximation Probability Theory This part of mathematics is concerned with the analysis of random phenomena that much of Simpson’s life was dedicated to. Simpson found this to be true by deriving several equations with an end result of p =(1 + i)ax/ax + 1 He used several proofs and De Moivre’s work to compile to this answer. This ’branch’ of Simpson’s work in mathematics deals with things such as mortality rates and life insurance. How much it is better than trapezoid rule Table 7.2 Composite Trapezoidal Rule for f (x) = 2 + sin(2√x) over [1, 6] M h T ( f, h) ET ( f, h) = O(h2) 10 0.5 8.19385457 −0.01037540 20 0.25 8.18604926 −0.00257006 40 0.125 8.18412019 −0.00064098 80 0.0625 8.18363936 −0.00016015 160 0.03125 8.18351924 −0.00004003 Table 7.3 Composite Simpson Rule for f (x) = 2 + sin(2√x) over [1, 6] M h S( f, h) ES( f, h) = O(h4) 5 0.5 8.18301549 0.00046371 10 0.25 8.18344750 0.00003171 20 0.125 8.18347717 0.00000204 40 0.0625 8.18347908 0.00000013 80 0.03125 8.18347920 0.00000001 Now we get how Simpson's rule works and why it has better approximation