The choice - 1 (Intro)

2011 Jul 22

[ Get the code at github ]

A programming language is low level when its programs require attention to the irrelevant. Quote: – Alan J. Perlis

It’s not only that physicists can’t program (we rarely use version control, we don’t plan, we don’t test… I suppose most labs score zero on The Joel Test). It is that we program, we program a lot and we still mostly use Fortran!

Now I’m not implying that Fortran is not a good language. Fortran is just fine. What I’m saying is that Fortran is a very bad choice for non-programmers who do a lot of programming.

Over the years some have turned to other languages like C, C++ and Matlab. Some, perhaps more enlightened, have even turned to Python. Naturally, I, too, turned to other languages and soon began wondering about what else might be out there (language-wise) to help a poor physicist do his job.

Could there be a language that won’t break everything apart just because a number is consider “special”? (What’s a float or a double? Why do I care?) Could there be one that implements a few-lines-of-equations-model in a few lines of code? One that will help me in my work not make the problem just … different?

I’ve been googling and poking around on stackoverflow and programmers.SE for some answers but nothing particularly interesting came up. It, thus, seems that I have to try them all.

“Rules”

no extra packages

idiomatic code for each language

efficiency is not so much of an issue

extendable code

The system

I’ll be solving, as an example, the following 2D system of ordinary differential equations:

$$ \begin{aligned} \frac{df_1(x)}{dx} =& 1 - f_2(x)\\ \frac{df_2(x)}{dx} =& f_1(x) - x \end{aligned} $$

with initial conditions:

$$\begin{aligned} x_0 &= 0\\ f_1(x_0) &= 1\\ f_2(x_0) &= 0 \end{aligned}$$

in the range $$[0, 3]$$

The exact solution is $$\begin{aligned} f_1(x)&=\cos{x}+x\\ f_2(x)&=\sin{x}\end{aligned}$$

solution

To numerically solve the system we treat the function values $$f_i(x)$$ as parameters of the differential functions:

$$\begin{aligned} F_1(x,y_1,y_2) &= \frac{df_1(x,y_1,y_2)}{dx} =& 1 - y_2\\ F_2(x,y_1,y_2) &= \frac{df_2(x,y_1,y_2)}{dx} =& y_1 - x \end{aligned}$$

We will end up with a list of \(y_{i,0},\ldots,y_{i,n}\) values that will represent an approximation of the values \(f_i(x_k),\, x_k = x + k\cdot\delta x\).

The method: Runge Kutta-4

Runge Kutta method iterates from a current known value of $$f(x)$$ to the “next” value at $$x+\delta x$$. So, knowing $$f(x_0)$$ and choosing a small step $$\delta x$$ the method gives us $$f(x_0 + \delta x)$$ by calculating a “correction” to $$f(x_0)$$.

We thus have: $$ f(x_0 + \delta x) = f(x_0) + \Delta()$$ where the correction $$\Delta()$$ depends on $$\delta x, x_0, f(x_0), f\prime(x,y)$$ and is given by:

$$ \Delta(\delta x, x_0, f(x_0), f\prime(x,y)) = \frac{1}{6}\delta x (a + 2b + 2c + d) $$

where

$$\begin{aligned} a =& f\prime(x_0, f(x_0))\
b = & f\prime(x_0 + \frac{1}{2}\delta x, f(x_0) + \frac{1}{2}a\delta x)\
c = & f\prime(x_0 + \frac{1}{2}\delta x, f(x_0) + \frac{1}{2}b\delta x)\
d = & f\prime(x_0 + \delta x, f(x_0) + c\delta x)\
\end{aligned}$$

We feed $$\Delta()$$ with the initial known values.

In many dimensions

To apply RK4 to a system of ODEs

for each set of equations $$F_i = f_i\prime(x_0, y_1, \ldots, y_n)$$ we have a different set of parameters $$a_i, b_i, c_i, d_i$$ and a correction function $$f_i(x_0 + \delta x) = f_i(x_0) + \Delta(\delta x, a_i,b_i,c_i,d_i) $$

$$\begin{aligned} a_i &= F_i(x_0 ,& y_1,& \ldots ,& y_n) \\ b_i &= F_i(x_0+\frac{1}{2}\delta x ,& y_1 + \frac{1}{2}a_1\delta x ,& \ldots ,& y_n + \frac{1}{2}a_n\delta x)\\ c_i &= F_i(x_0+\frac{1}{2}\delta x ,& y_1 + \frac{1}{2}b_1\delta x ,& \ldots ,& y_n + \frac{1}{2}b_n\delta x)\\ d_i &= F_i(x_0+\delta x ,& y_1 + c_1\delta x ,& \ldots ,& y_n + c_n\delta x)\\ \end{aligned}$$

$$ \Delta() = \frac{1}{6}\delta x (a_i + 2 b_i + 2 c_i + d) $$

$$ f_i(x_0 + \delta x) = f_i(x_0) + \Delta(\delta x, a_i,b_i,c_i,d_i) $$

The Code

I’ll use a C implementation as a reference.

C (reference)

Adapted from numerical recipes in C:

<{{static/src/the-choice-1-intro/rk4.c}}

for example: <{{static/src/the-choice-1-intro/rk4_ex.c}}

Coming up: J, LISP, R, Fortran, Python, Haskell, Pari/GP, maxima.

Any other languages that you would like to see?

See Also