Linear Regression

Hi there, Long time no see, sorry I have been rather quite taken with various business ventures and so I have not been working on my projects as much as I should have. As I have had a lot of downtime recently as the semester winds down I have been taking my free time at work to learn some celestial mechanics.

The notes I am using are great and so I will link to them here.  It starts off with a primer on numerical methods, one of the things he asks you to do is create a program which will compute the  least squares solution to an overdetermined system, then apply that program to create a least squares fit to a set of data. This was the method Gauss used to create tables of ephermrides for asteroids, but it presents and interesting set of computational problems,

1) how does one represent a matrix in memory
2) how does one solve a system of linear equations in a finite number of steps
3) how does one then use that solution as a polynomial to evaluate a coefficent of determination.

This has occupied my time on and off for the past couple of weeks. It suffices to say I learned a lot about representing mathematical objects in a computer. Polynomials are really quite facinating actually. For example it is possible to compute a polynomial at any unknown value in Ω(n) time where n is the degree of the polynomial.

Another interesting question is how does one test a linear least squares code once its been written. The answers are found in my github repository. Here I will only present my results:

The test data and fit

Polynomial model: 
Degree: 1
Coefficients:
a_1: 7.939715
a_0: 1438.362123
r: 0.997310
r²: 0.994628