Entries by Mark44

Why Can’t My Computer Do Simple Arithmetic?

The first computer I owned was an Apple IIe computer, with a CPU that ran at slightly over 1 Megahertz (MHz), and with 64 Kilobytes (KB) of RAM, together with two 5 1/4″ floppy drives that could each store 140 Kilobytes (KB) of data. My current computer (which cost about the same as the Apple computer) is vastly more […]

What Are Eigenvectors and Eigenvalues?

Two important concepts in Linear Algebra are eigenvectors and eigenvalues for a linear transformation that is represented by a square matrix. Besides being useful in mathematics for solving systems of linear differential equations, diagonalizing matrices, and other applications, eigenvectors and eigenvalues are used in quantum mechanics and molecular physics, chemistry, geology, and many other scientific disciplines. Some definitions: An eigenvector for […]

Partial Fractions Decomposition

Partial fractions decomposition is an algebraic technique that can be used to decompose (break down) a product of rational expressions into a sum of simpler rational expressions. A rational expression is one in which both the numerator and denominator are polynomials. A proper rational expression is one in which the degree of the numerator is strictly less than the degree […]

Solving Nonhomogeneous Linear ODEs using Annihilators

My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations of the form F(y, y’, y”, …) = 0. In this Insights article we will look at equations of the form F(y, y’, y”, …) = g(t), for certain functions g. Ex. 1: y” – 4y’ + 3y […]

Solving Homogeneous Linear ODEs using Annihilators

In this Insights article we’ll look at a limited class of ordinary differential equations — homogeneous linear ODES with constant coefficients. Although there are many differential equations that are outside the scope of this article, there are many applications in mechanics and electronics of the types of differential equations we’ll be looking at. Homogeneous Equations A […]

Simple Python Debugging with Pdb: Part 2

This Insight article is the continuation of the first article, Simple Python Debugging with Pdb: Part 1. In this article, let’s look at another important capability of debuggers: breakpoints. When you set a breakpoint in a program, the debugger executes all of the code up to the breakpoint, and then halts. This allows you to inspect variables […]

Simple Python Debugging with Pdb: Part 1

I’m pretty new with Python, so I was looking around for some debugging tools. At first I dismissed Pdb (Python debugger) as being too primitive, but after seeing a blog post about using Pdb, I decided that I should give it a try. To get started with Pdb import the pdb module insert a call […]