Fear Mongering and Liberal Math

Take a look at this video clip of Nancy Pelosi fear-mongering. I don't want to make this a political blog site, but this is just the kind of idiotic fear-mongering that causes dangerous and expensive rushed decisions. If you want to give the Madame Speaker the benefit of the doubt, she made the same statement in a Chris Wallace interview on January 18th. Scare me once, shame on you, fool me twice shame on me...

"Every month that we do no have an economic recovery package, 500 million Americans lose their jobs" - Speaker of the House, Nancy Pelosi

I always thought there were only somewhere on the order of 300-305M Americans at the present time (at least according to the CIA and Census Bureau). So for her statement to be true, if we delay just one month, we would have to have 100% employment rate (that is, everyone has at least one job), and 83% of those people would have to have two jobs (again, I don't think this is true). Finally, in that first month, we would all have to lose our 1 or 2 jobs, that is 5/6ths of the jobs would have to be lost (I guess that includes her too, hopefully). Actually strictly speaking even that wouldn't work since she said 500M Americans would lose their jobs, not 500M American jobs would be lost. Semantics aside, so what would happen the next month? There would be no more American jobs to lose.... or is she assuming that we all have 4, or 6, or 8 jobs so we can stretch this out to 2, or 3 or 4 months?!?

Liberal math does not compute! Maybe that's why so many of the democrats in power have had recently-surfaced serious tax evasion problems. These are not little errors here and there, but often multi-year spanning, tens to hundreds of thousands of dollar "accidents" or "mistakes" made by the likes of Daschle, Geithner, Murphy, and Rangle. Tax evasion is another topic for another time though. I want to discuss something else while we are still on the sub-topic of unemployment.

Unemployment Timeline

Check out this unemployment timeline. Here are some interesting observations:

• The timeline does not have unemployment data points for the Great Depression years, but estimates were 23-25% for several years. So everyone comparing this, finally official, recession to the Great Depression is slapping the face those who survived through the 1930s. In fact in the '80s recession and many years before it had unemployment in excess of 10.0% (we are estimating that the current 2009 rate is ~7.5%), so what we are in is not even as bad as the '80s and other year s with regard to employment.
• For a lot of the 1990s and 2000s the unemployment rate was below 6.00. These years included three Bush terms (G.H.W. and G.W.) and two Clinton administrations, but I will let you decide who should get more acclaim for this.
• After the 2nd Gulf War began, you can see a prolonged decline in unemployment down to pre 9/11 rates. I am not promoting nor am I discounting the war, I am just saying that the statistics coincide and are probably correlated.
• It's interesting to see that the recent congress, the 110th Congress, had the Democrats as the majority party. It's since 100th's meeting in 2007 that there was a gradual rise in unemployment leading up to the current events. The majority lead has even increased in the succeeding congress, the 111th Congress.
  
• Just following a minimum wage increase, you can usually see an increase in unemployment, sometimes prolonged. Not being an economist, I can only speculate that if the two are correlated, then it is due to the sudden increase in operating costs that require spending cuts (in the form of decreased labor quantity).
  

Numerical Methods for Solving Initial Value Problems

Overview

I warned you that this blog would be random, and now we will discuss an undergraduate math topic. Recently, I was cataloging some of my old text books and I came across one on differential equations. The title is Differential Equations Computing and Modeling, by C. Henry Edwards and David E. Penney. It occurred to me that while I had done some small computation projects in this class using formulas in Excel and some MATLAB programs, most people do not have access to MATLAB (at least legally), though they should have access to a C or C++ compiler. So, without further ado, here is a small project (single source file) with multiple numerical methods for differential equation solution.

Introduction

First we need to define some terminology.

• ODE - Ordinary differential equation. This is an equation which contains functions of only one independent variable, and derivatives with respect to that variable.
• IVP - Initial value problem. This is an ODE, together with an initial condition. For example, dy/dx = y, y(0) = 1 is an IVP. (Coincidentally, if you solve this for y(1), you will determine Euler's costant e).
• Derivative - What is wrong with you? Go back to Calculus 1. Sorry, that was an inside joke.
  

Next, in order to solve your problem using these numeric methods (as implemented), you need to express your differential equation such that y'=dy/dx is on one side. For example:

y' - y = x

should be expressed as

dy/dx = y + x

Once this is done, you will be implementing dy/dx as a function ( that's a c++ function here) which takes two doubles and returns a double. Specifically, the signature is as below:

// This is your differential equation y'(x,y)... that is
// formulate the function dy/dx as a function of x,y
typedef double (*Y_primed)(double x, double y);

So, to implement the example function that was given, you could do something like the following

// This is the example y', following the signature above
double sample_y_primed(double x, double y) {
return x + y;
}

The Numerical Methods

Numerical methods are used for solving differential equations in an approximate way. There are three numerical methods used in this source. Space will not allow me to cover them thoroughly here, but here are some Wikipedia resources. You can find the algorithms there as well.

• Euler's Method - created by Leonhard Euler, mathematical genius. http://en.wikipedia.org/wiki/Euler_method
• Improved Euler's Method - also called Heun's Method, after Karl M. W. L. Heund. This method is an extension of the Euler Method. http://en.wikipedia.org/wiki/Heun%27s_method
  
• Runge-Kutta Method - A numerical method created by C. Runge and M.W. Kutta which has much lower error for a given number of steps, and thus has better accuracy for the same number of iterations as Euler's Method. Alternatively, a lot of computational efficiency can be gained at the expense of a little bit of accuracy when compared to Euler's Method. http://en.wikipedia.org/wiki/Runge-Kutta_methods

The implemented numerical methods all follow this prototype
double SomeNumericalMethod(Y_primed f, double initial_x, double initial_y, double final_x, unsigned int steps)

... where f, is the y' function covered earlier, initial_x is the x-value at which the initial condition is true, initial_y is the y value at the initial condition, final_x is the x-value at which to calculate the new y value, and steps is the number of iterations. The approximate new y-value, that is the approximate y(final_x), is returned.

Here is a small example:

dy/dx = x + y, y(0) = 1

We shall call ImprovedEulersMethod() function

double y_of_1 = ImprovedEulersMethod(sample_y_primed, 0, 1, 1, 10);

You should find that y_of_1 is near 3.4282, which is approximately equal to the real y-value y(1) = 3.4366. Increase the number of steps (20, 40, 80, ...) to see how the accuracy increases.

Downloading

Here's a link to the C++ source code.

Unexplored Ideas

1. What is the error of each method as a function of the number of iterations used?
2. How many iterations does it take to have n-decimal places accurate?
3. Derive and prove why the special IVPs and final x-values for e, ln(2), and pi arrive at those values.