SciPy

Again I recommend this http://www.scipy.org/Wiki/Documentation?action=AttachFile&do=get&target=scipy_tutorial.pdf. There are many useful SciPy modules, in particular scipy.optimize, scipy.stats, scipy.linalg, scipy.linsolve, scipy.sparse, scipy.integrate, scipy.fftpack, scipy.signal, scipy.special. Most of these have relatively good documentation and often you can figure out what things do from the names of functions. I recommend exploring them. For example if you do

sage: import scipy
sage: from scipy import optimize

Then

sage: optimize.[tab]

will show a list of available functions. You should see a bunch of routines for finding minimum of functions. In particular if you do

sage: optimize.fmin_cg?

you find it is a routine that uses the conjugate gradient algorithm to find the minima of a function.

sage: scipy.special.[tab]

will show all the special functions that SciPy has. Spending a little bit of time looking around is a good way to familiarize yourself with SciPy. One thing that is sort of annoying, is that often if you do scipy.:math:\(\langle\) tab \(\rangle\). You won’t see a module that is importable. For example scipy.:math:\(\langle\) tab \(\rangle\) will not show a signal module but

sage: from scipy import signal

and then

signal.[tab]

will show you a large number of functions for signal processing and filter design. All the modules I listed above can be imported even if you can’t see them initially.

scipy.integrate

This module has routines related to numerically solving ODE’s and numerical integration. Lets give an example of using an ODE solver. Suppose you want to solve the ode

\(x''(t) + ux'(t)(x(t)^2-1)+x(t)=0\)

which as a system reads

\(x'=y\)

\(y'=-x+\mu y(1-x^2).\)

The module we want to use is odeint in scipy.integrate. We can solve this ode, computing the value of \((y,y')\), at 1000 points between \(0\), and \(100\) using the following code.

sage: import numpy
sage: from scipy import integrate
sage: def f_1(y,t):
....:    return[y[1],-y[0]-10*y[1]*(y[0]**2-1)]
sage: def j_1(y,t):
....:    return [ [0, 1.0],[-2.0*10*y[0]*y[1]-1.0,-10*(y[0]*y[0]-1.0)] ]
sage: x= numpy.arange(0,100,.1)
sage: y=integrate.odeint(f_1,[1,0],x,Dfun=j_1)

We could plot the solution if we wanted by doing

sage: pts = [(x[i],y[i][0]) for i in range(len(x))]
sage: point2d(pts).show()

Optimization

The Optimization module has routines related to finding roots, least squares fitting, and minimization. \(\langle\) To be Written \(\rangle\)