Category: Numerical Analysis

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