Category: Numerical Analysis

My Near-Term Agenda by James Pate Williams, Jr. BA, BS, MSwE, PhD

Merry Christmas to all you devout Christians. I am not one of you. I am about to embark on a mission to carefully annotate with open source C# computer code my copy of “Exterior Ballistics, 1935” by Professor Ernest Edward Herrmann of then the United States Naval Academy at Annapolis, Maryland. This book set the standard for naval gunnery in World War II. Of course, our navy and especially our battle-wagons had the largest rifled artillery of any United States service. The 105 mm = 105 mm / 25.4 mm / inch = 4.13 inches, 120 mm / 25.4 mm / inch = 4.72 inches, and 155 mm / 25.4 mm / inch = 6.10 inch of our excellent United States Army and United States Marine Corps (semper fidelis) are puny in comparison to the mighty 8, 10, 12, 14, and finally 16 inch mostly rifled artillery of our incredible navy’s cruisers, dreadnoughts, and battleships of the World War I and World War II era ships. Even a destroyer of the USN Fletcher class had 5-inch (127 mm) / 38 caliber rifled artillery which had a 5 inch * 38 = 190-inch barrel length. Our mightiest naval artillery was, of course, my favorite the mighty 16 inch (406.4 mm) / 50 caliber rifles that had a barrel length of 16 * 50 inches = 800 inches = 66.6 feet!

Thanks,

James Pate Williams, Jr.

Bachelor of Arts Chemistry LaGrange College 1979

Bachelor of Science Computer Science LaGrange College 1994

Master of Software Engineering Auburn University 2000

Doctor of Philosophy Computer Science Auburn University 2005

Gratis Open Source Computer Software Developer Since Summer 1978

1980 – 1983 Graduate Work in Chemistry and Mathematics at Georgia Tech

A Current Website I developed for my friends Wesley “Wes” and Missy Cochran:

http://thecochrancollection.com/Home

Powers of Two – Excel by James Pate Williams, Jr. BA, BS, MSwE, PhD

First Function in Excel (Assumes that You Have Access to an Office 365 Subscription)

Please attempt the following procedure:

  1. Type Excel in the Windows 10 Search Box
  2. Select the Excel App
  3. Select Blank workbook
  4. Maximize the Excel Window
  5. Type x in Cell A1
  6. Tab to Cell B1
  7. Type y in Cell B1
  8. Type 0 in Cell A2
  9. Type 1 in Cell B2
  10. Type 1 in Cell A3
  11. Type 2 in Cell B3
  12. Type 2 in Cell A4
  13. Type 4 in Cell B4
  14. Type 3 in Cell A5
  15. Type 8 in Cell B5
  16. Type 4 in Cell A6
  17. Type 16 in Cell B6
  18. Type 5 in Cell A7
  19. Type 32 in Cell A8
  20. Highlight Cells A1 and B1
  21. From the Toolbar Select Alignment and Right Alignment
  22. Select File Save As
  23. Select “Documents” and “Powers of Two” as the filename
  24. Highlight Cells A1 to B7
  25. Select Insert from Toolbar
  26. Select Charts Scatter
  27. Select the Chart
  28. Select the Big + Sign on the Right
  29. Label the y-axis “y = 2 ^ n”
  30. Label the x-axis “n”
  31. Relabel the Title of the Chart as “Powers of Two”

Note that x is in the finite set { 0, 1, 2, 3, 4, 5 }

Note that y is the function y = 2 ^ x where ^ is the exponentiation operator

Powers of Two TablePowers of Two Chart

Global Primary Greenhouse Gas Concentrations by James Pate Williams Jr BA, BS, MSwE, PhD

I designed and implemented a C# computer language application to model the global greenhouse gas concentrations data found on the NOAA website:

https://www.esrl.noaa.gov/gmd/aggi/aggi.html

I used the latest recommended data for time period 1979 to 2017. The concentrations of three greenhouse gases were modeled: carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O).

The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:

                                Simple Linear Regression Parameters

See the website:

https://en.wikipedia.org/wiki/Simple_linear_regression

Some plots of the concentrations in parts per million (PPM) and parts per billion (PPB) are given below.

                                                Carbon Dioxide Concentration in Parts Per Million
                                                        Methane Concentration in Parts Per Billion
                                                 Nitrous Oxide Concentration in Parts Per Billion
                                             Carbon Dioxide Concentration in Parts Per Million
                                                      Methane Concentration in Parts Per Billion
                                                   Nitrous Oxide Concentration in Parts Per Billion
              Simple Linear Regression Parameters
              Simple Linear Regression Parameters
             Simple Linear Regression Parameters
                                                         Greenhouse Gas Concentrations Table

NOAA Contiguous United States of America Precipitation by James Pate Williams Jr BA, BS, MSwE, PhD

I designed and implemented a C# computer language application to model the precipitation data found on the NOAA website:

https://www.ncdc.noaa.gov

I used the latest recommended data for time period 1895 to 2017. The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:

Simple  Linear Regression Equations

See the website:

https://en.wikipedia.org/wiki/Simple_linear_regression

Some plots of the contiguous U.S. precipitation are shown below. The climate is getting wetter thus some parts of the U.S.maybe more prone to floods.

Precipitation Plot
Precipitation Plot
Precipitation Plot
Precipitation Plot
Precipitation Plot
Precipitation Plot
Precipitation Plot
Simple Linear Regression Parameters
Precipitation Table Experimental and Theoretical Values

NOAA Contiguous United States of America Temperature Anomaly by James Pate Williams Jr BA, BS, MSwE, PhD

I designed and implemented a C# computer language application to model the temperature anomaly data found on the NOAA website:

https://www.ncdc.noaa.gov

I used the latest recommended data for time period 1895 to 2017. The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:

Simple Linear Regression Equations

See the website:

https://en.wikipedia.org/wiki/Simple_linear_regression

Below are some plots of the temperature anomaly.

Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
Temperature Anomaly Actual Anomaly in Red Theoretical in Blue
C# Application Main Form
Simple Linear Regression Parameters
Table Function Form with Experimental and Theoretical Anomalies

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.

Calculating a Few Digits of the Transcendental Number Pi by Throwing Darts by James Pate Williams, BA, BS, MSwE, PhD

Suppose you have a unit square with a circle of unit diameter inscribed . You can compute a few digits of the transcendental number, pi, 3.1415926535897932384626433832795…, by using the algorithm described as follows. Let n be the number of darts to throw and h be the number of darts that land within the inscribed circle.

h = 0

for i = 1 to n do

Choose two random numbers x and y such that x and y are contained in the interval 0 to 1 inclusive that is x and y contained in [0, 1]

Let u = x – 1 / 2 and v = y – 1 / 2

if u * u + v * v <= 0.25 = 1 / 4 then h = h + 1

next i

pi = 4 * h / n

Below are the results of a C# Microsoft Visual Studio simulation project. In the first case we throw 100,000 darts and get two significant digits of pi and then we throw a 1,000,000 darts and five significant digits of pi are computed. Of course, in a previous entry by this author we can calculate hundreds or thousands of digits of pi in relatively little time:

https://jamespatewilliamsjr.wordpress.com/2018/07/01/the-bailey-borwein-plouffe-formula-for-calculating-the-first-n-digits-of-pi/

MainForm 10_16_2018 3_04_04 AMMainForm 10_16_2018 3_05_45 AMMainForm 10_16_2018 3_09_36 AMMainForm 10_16_2018 3_09_54 AM

MonteCarloPi Source Code