Numerical Python

My experiments with Python continues on. This time it’s playing with the numerical and plotting capabilities of Python.

The shortest path between two points on a plane is the straight line connecting the two points. There’s a way to derive the equation of the shortest path using variational calculus. This experiment derives such an equation by doing a high dimensional gradient descent on the space of equations that satisfy the Euler-Lagrange equation.

We are looking for a curve that minimizes the arc length between the two points. We define y(x) as the curve, y_x as the x differential of y.

$I[y] = \int_a^b \sqrt{1 +(y_x(x))^2} dx$

The corresponding Euler-Lagrange equation is

$\frac{d}{dx} \frac{\partial \sqrt{1 +(y_x)^2}}{\partial y_x} = 0$

The plot below was generated by matplotlib, a plotting library that is almost exactly similar to the Matlab plotting interface. I was surprised at how close Python code can be to Matlab code. Nowhere in my code do I have to loop over the dimensions of the vectors. The numerical calls are passed over to high performance math routines written in Fortran. The initial condition was a sine wave and you can see that it eventually converges to the true solution.

Matplotlib

Get the code here.

This entry was posted on Friday, July 13th, 2007 at 1:34 am and is filed under Computing. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

One Response to “Numerical Python”

Fitting data with Python » Ganesh Swami says:

February 3, 2008 at 7:36 am

[...] a heavy user of the numerical capabilities of Python. I’ve written about my experiments before, but now I’m writing production quality code with numpy and [...]