Four Techniques for One Dimensional Riemann Definite Integration by James Pate Williams, Jr. BA, BS, MSwE, PhD

The four methods considered in this study are as follows:

  1. Trapezoidal Rule
  2. Simpson’s Rule
  3. Gauss-Legendre Quadrature
  4. Monte Carlo Method

The trapezoidal rule requires (n  + 2) function evaluations, n real number increments, and six additional real number arithmetic operations. Simpson’s rule involves (n + 2) function evaluations, n real number increments, and ten additional real number arithmetic operations. Gauss-Legendre quadrature uses n function evaluations, 3 * n real number arithmetic operations, 2 * n index operations, and five additional arithmetic operations. Finally, the Monte Carlo Method requires n function evaluations, n random number generations, 2 * n + 3 additional real number arithmetic operations. The Gauss-Legendre quadrature also involves some complicated orthogonal polynomial operations to determine the abscissas and weights. Below are some results from our test C# application.

ResultsForm 10_25_2018 5_20_34 PMResultsForm 10_25_2018 5_21_14 PMResultsForm 10_25_2018 5_21_28 PMResultsForm 10_25_2018 5_22_26 PMResultsForm 10_25_2018 5_22_47 PMResultsForm 10_25_2018 5_23_05 PMResultsForm 10_25_2018 5_23_52 PMResultsForm 10_25_2018 5_24_19 PMResultsForm 10_25_2018 5_24_39 PM

We conclude from the preceding dearth of tests that for given n the order of accuracy is generally Gauss-Legendre, Simpson’s, Trapezoidal, and finally Monte Carlo.

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Author: jamespatewilliamsjr

My whole legal name is James Pate Williams, Jr. I was born in LaGrange, Georgia approximately 70 years ago. I barely graduated from LaGrange High School with low marks in June 1971. Later in June 1979, I graduated from LaGrange College with a Bachelor of Arts in Chemistry with a little over a 3 out 4 Grade Point Average (GPA). In the Spring Quarter of 1978, I taught myself how to program a Texas Instruments desktop programmable calculator and in the Summer Quarter of 1978 I taught myself Dayton BASIC (Beginner's All-purpose Symbolic Instruction Code) on LaGrange College's Data General Eclipse minicomputer. I took courses in BASIC in the Fall Quarter of 1978 and FORTRAN IV (Formula Translator IV) in the Winter Quarter of 1979. Professor Kenneth Cooper, a genius poly-scientist taught me a course in the Intel 8085 microprocessor architecture and assembly and machine language. We would hand assemble our programs and insert the resulting machine code into our crude wooden box computer which was designed and built by Professor Cooper. From 1990 to 1994 I earned a Bachelor of Science in Computer Science from LaGrange College. I had a 4 out of 4 GPA in the period 1990 to 1994. I took courses in C, COBOL, and Pascal during my BS work. After graduating from LaGrange College a second time in May 1994, I taught myself C++. In December 1995, I started using the Internet and taught myself client-server programming. I created a website in 1997 which had C and C# implementations of algorithms from the "Handbook of Applied Cryptography" by Alfred J. Menezes, et. al., and some other cryptography and number theory textbooks and treatises.

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