Scipy -- Library of scientific algorithms for Python

Based on lecture at http://github.com/jrjohansson/scientific-python-lectures.

The SciPy framework builds on top of the low-level NumPy framework for multidimensional arrays, and provides a large number of higher-level scientific algorithms. Some of the topics that SciPy covers are:

Each of these submodules provides a number of functions and classes that can be used to solve problems in their respective topics.

In this lecture we will look at how to use some of these subpackages.

To access the SciPy package in a Python program, we start by importing everything from the scipy module.

WARNING: In the new version of python many functionalities are now moved from scipy to numpy, but they are still available in scipy and a deprecated warning is displayed. The work-around is to first import functions from scipy and after that from numpy, to overwrite scipy functions with the same name.

If we only need to use part of the SciPy framework we can selectively include only those modules we are interested in. For example, to include the linear algebra package under the name la, we can do:

Special functions

A large number of mathematical special functions are important for many computional physics problems. SciPy provides implementations of a very extensive set of special functions. For details, see the list of functions in the reference documention at http://docs.scipy.org/doc/scipy/reference/special.html#module-scipy.special.

To demonstrate the typical usage of special functions we will look in more detail at the Bessel functions:

Integration

Numeric integration with general purpose quadrature

Numerical evaluation of a function of the type

$\displaystyle \int_a^b f(x) dx$

is called numerical quadrature, or simply quadature. SciPy provides a series of functions for different kind of quadrature, for example the quad, dblquad and tplquad for single, double and triple integrals, respectively.

The quad function takes a large number of optional arguments, which can be used to fine-tune the behaviour of the function (try help(quad) for details).

The basic usage is as follows:

If we need to pass extra arguments to integrand function we can use the args keyword argument. Let's say we want to evaluate

$f(x) = \displaystyle \int_0^x \frac{j_n(t)}{t} dt$

Higher-dimensional integration works in the same way:

Note that we pass lambda functions for the limits for the y integration, since these in general can be functions of x.

\begin{equation} \int_{x_a}^{x_b} \int_{y_a(x)}^{y_b(x)} f(x,y) dy dx \end{equation}

Ordinary differential equations (ODEs)

SciPy provides two different ways to solve ODEs: An API based on the function odeint, and object-oriented API based on the class ode. Usually odeint is easier to get started with, but the ode class offers some finer level of control.

Here we will use the odeint functions. For more information about the class ode, try help(ode). It does pretty much the same thing as odeint, but in an object-oriented fashion.

To use odeint, first import it from the scipy.integrate module

A system of ODEs should be formulated in standard form, which is:

$y' = f(y, t)$

where

$y = [y_1(t), y_2(t), ..., y_n(t)]$

and $f$ is some function that gives the derivatives of the function $y_i(t)$. To solve an ODE we need to know the function $f$ and an initial condition, $y(0)$.

Note that higher-order ODEs can always be written in this form by introducing new variables for the intermediate derivatives.

Once we have defined the Python function f and array y_0 (that is $f$ and $y(0)$ in the mathematical formulation), we can use the odeint function as:

y_t = odeint(f, y_0, t)

where t is and array with time-coordinates for which to solve the ODE problem. y_t is an array with one row for each point in time in t, where each column corresponds to a solution y_i(t) at that point in time.

We will see how we can implement f and y_0 in Python code in the examples below.

Example: double pendulum

Let's consider a physical example: The double compound pendulum, described in some detail here: http://en.wikipedia.org/wiki/Double_pendulum

The equations of motion of the pendulum are given on the wiki page:

${\dot \theta_1} = \frac{6}{m\ell^2} \frac{ 2 p_{\theta_1} - 3 \cos(\theta_1-\theta_2) p_{\theta_2}}{16 - 9 \cos^2(\theta_1-\theta_2)}$

${\dot \theta_2} = \frac{6}{m\ell^2} \frac{ 8 p_{\theta_2} - 3 \cos(\theta_1-\theta_2) p_{\theta_1}}{16 - 9 \cos^2(\theta_1-\theta_2)}.$

${\dot p_{\theta_1}} = -\frac{1}{2} m \ell^2 \left [ {\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + 3 \frac{g}{\ell} \sin \theta_1 \right ]$

${\dot p_{\theta_2}} = -\frac{1}{2} m \ell^2 \left [ -{\dot \theta_1} {\dot \theta_2} \sin (\theta_1-\theta_2) + \frac{g}{\ell} \sin \theta_2 \right]$

To make the Python code simpler to follow, let's introduce new variable names and the vector notation: $x = [\theta_1, \theta_2, p_{\theta_1}, p_{\theta_2}]$

${\dot x_1} = \frac{6}{m\ell^2} \frac{ 2 x_3 - 3 \cos(x_1-x_2) x_4}{16 - 9 \cos^2(x_1-x_2)}$

${\dot x_2} = \frac{6}{m\ell^2} \frac{ 8 x_4 - 3 \cos(x_1-x_2) x_3}{16 - 9 \cos^2(x_1-x_2)}$

${\dot x_3} = -\frac{1}{2} m \ell^2 \left [ {\dot x_1} {\dot x_2} \sin (x_1-x_2) + 3 \frac{g}{\ell} \sin x_1 \right ]$

${\dot x_4} = -\frac{1}{2} m \ell^2 \left [ -{\dot x_1} {\dot x_2} \sin (x_1-x_2) + \frac{g}{\ell} \sin x_2 \right]$

examples for subplot creation: https://towardsdatascience.com/the-many-ways-to-call-axes-in-matplotlib-2667a7b06e06

matplotlib examples: https://matplotlib.org/3.1.1/gallery/index.html

See animation in pendulum.py.

(To get it work within jupyter seems a bit challenging at the moment.)

Example: Damped harmonic oscillator

ODE problems are important in computational physics, so we will look at one more example: the damped harmonic oscillation. This problem is well described on the wiki page: http://en.wikipedia.org/wiki/Damping

The equation of motion for the damped oscillator is:

$\displaystyle \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega^2_0 x = 0$

where $x$ is the position of the oscillator, $\omega_0$ is the frequency, and $\zeta$ is the damping ratio. To write this second-order ODE on standard form we introduce $p = \frac{\mathrm{d}x}{\mathrm{d}t}$:

$\displaystyle \frac{\mathrm{d}p}{\mathrm{d}t} = - 2\zeta\omega_0 p - \omega^2_0 x$

$\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t} = p$

In the implementation of this example we will add extra arguments to the RHS function for the ODE, rather than using global variables as we did in the previous example. As a consequence of the extra arguments to the RHS, we need to pass an keyword argument args to the odeint function:

Fourier transform

Fourier transforms are one of the universal tools in computational physics, which appear over and over again in different contexts. SciPy provides functions for accessing the classic FFTPACK library from NetLib, which is an efficient and well tested FFT library written in FORTRAN. The SciPy API has a few additional convenience functions, but overall the API is closely related to the original FORTRAN library.

To use the fftpack module in a python program, include it using:

To demonstrate how to do a fast Fourier transform with SciPy, let's look at the FFT of the solution to the damped oscillator from the previous section:

Properties of Fourier transform of a real signal: \begin{eqnarray} && F(\omega) = \int e^{i\omega t} x(t) dt\\ && F^*(\omega) = F(-\omega)\\ && Re(F(\omega)) = Re(F(-\omega))\\ && Im(F(\omega)) = -Im(F(-\omega)) \end{eqnarray}

Since the signal is real, the spectrum is symmetric. We therefore only need to plot the part that corresponds to the postive frequencies. To extract that part of the w and F we can use some of the indexing tricks for NumPy arrays that we saw in Lecture 2:

As expected, we now see a peak in the spectrum that is centered around 1, which is the frequency we used in the damped oscillator example.

Optimization

Optimization (finding minima or maxima of a function) is a large field in mathematics, and optimization of complicated functions or in many variables can be rather involved. Here we will only look at a few very simple cases. For a more detailed introduction to optimization with SciPy see: http://scipy-lectures.github.com/advanced/mathematical_optimization/index.html

To use the optimization module in scipy first include the optimize module:

Finding a minima

Let's first look at how to find the minima of a simple function of a single variable:

We can use the fmin_bfgs function to find the minima of a function:

We can also use the brent or fminbound functions. They have a bit different syntax and use different algorithms.

Finding a solution to a function, i.e., zeros

To find the root for a function of the form $f(x) = 0$ we can use the fsolve function.

It is based on Powell's hybrid method as implemented in MINPACK’s library (hybrd): https://www.extremeoptimization.com/Documentation/Mathematics/Solving-Equations/Solving-Systems-of-Non-Linear-Equations.aspx

Powell's dogleg method, also called Powell's hybrid method, attempts to minimize the sum of the squares of the function values. It does this using a combination of Newton's method and the steepest descent method. This is a so-called trust region method. This means that every step moves the current point to within a finite region. This makes the method more stable than Newton's method.

On the other hand, the fact that the method is, in essence, a specialized minimizer means that the algorithm can get stuck in a local minimum that does not correspond to a solution of the system of equations.