## Archive for the ‘python’ Category

If you **really** want to learn the differences between Python 2 and Python 3, I suggest you try converting a non-trivial software project. I’m in the middle of doing one now and am learning all kinds of little gotchas over and above the standard stuff that everyone knows such as changes to print, integer division and removal of xrange.

The most recent one I learned about (about 10 minutes ago) amounted to this

#Python 2.7.6 (default, Mar 22 2014, 22:59:56) [GCC 4.8.2] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> [x for x in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> x 9

compared to

Python 3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2] on linux Type "help", "copyright", "credits" or "license" for more information. >>> [x for x in range(10)] [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] >>> x Traceback (most recent call last): File "", line 1, in NameError: name 'x' is not defined

This is well documented (This StackOverflow Q+A is great!) but I didn’t know about it and, in the code I was looking at, there was a heck of a lot of complication between the list comprehension and when ‘x’ was used. As such, it took me a while to figure out!

Another change that had me scratching my head for a while is the fact that Python 3 ignores the __metaclass__ hook. I didn’t know this little fact but discovered it while debugging failing tests!

Of course, once you know these little gotchas, you’ll probably not be caught out by them again in your next Python 2->Python 3 porting project but they got me wondering…..

**What changes from Python 2->Python 3 have really caught you out at some point?**

Linear Algebra – Foundations to Frontiers (or LAFF to its friends) is a popular, high quality and free MOOC that, as the title suggests, teaches aspects of linear algebra in a way that takes the student from the very basics through to some cutting edge techniques. I worked through much of it last year and thoroughly enjoyed the approach it took — focusing on programming aspects from the very beginning. The course authors are also among the developers of the FLAME project, a high performance linear algebra library, and one of the interesting aspects of the LAFF course (for me at least) was that it taught linear algebra in a way that also allowed you to understand the approaches used in the algorithms behind FLAME.

Last year, all of the programming assignments in LAFF were done in Python, making use of the IPython notebook. This year, the software stack will be different and will be based on MATLAB. I understand that everyone who signs up to LAFF will be **able to get a free MATLAB license from Mathworks for the duration of the course**. Understandably, this caused quite a bit of discussion between the LAFF team and software/language geeks like me. In a recent Facebook thread, I asked about the switch and received the reply

*‘MATLAB will be free during the course. There are open source equivalents, but Mathworks staff is supporting the use of MATLAB (staff for us). There were some who never got the IPython notebooks to work properly. We are really excited at the opportunity to innovate again and perhaps clear up snags in the programming issues we had. It was complicated to support IPython on all of the operating systems and machines that participants use. MATLAB promises to be easier and will allow us again to concentrate on the Linear Algebra’ – LAFF UTx*

I’m sufficiently interested in this change from IPython to MATLAB that I’ll be signing up for the course again this year and I encourage you to do the same — I believe that the programming-centric teaching approach taken by LAFF is extremely well done and your time would be well-spent working through the course.

The course starts on 28th January 2015 so sign up now!

Here’s the trailer for last year’s course.

Given a symmetric matrix such as

What’s the nearest correlation matrix? A 2002 paper by Manchester University’s Nick Higham which answered this question has turned out to be rather popular! At the time of writing, Google tells me that it’s been cited 394 times.

Last year, Nick wrote a blog post about the algorithm he used and included some MATLAB code. He also included links to applications of this algorithm and implementations of various NCM algorithms in languages such as MATLAB, R and SAS as well as details of the superb commercial implementation by The Numerical algorithms group.

I noticed that there was no Python implementation of Nick’s code so I ported it myself.

Here’s an example IPython session using the module

In [1]: from nearest_correlation import nearcorr In [2]: import numpy as np In [3]: A = np.array([[2, -1, 0, 0], ...: [-1, 2, -1, 0], ...: [0, -1, 2, -1], ...: [0, 0, -1, 2]]) In [4]: X = nearcorr(A) In [5]: X Out[5]: array([[ 1. , -0.8084125 , 0.1915875 , 0.10677505], [-0.8084125 , 1. , -0.65623269, 0.1915875 ], [ 0.1915875 , -0.65623269, 1. , -0.8084125 ], [ 0.10677505, 0.1915875 , -0.8084125 , 1. ]])

This module is in the early stages and there is a lot of work to be done. For example, I’d like to include a lot more examples in the test suite, add support for the commercial routines from NAG and implement other algorithms such as the one by Qi and Sun among other things.

Hopefully, however, it is just good enough to be useful to someone. Help yourself and let me know if there are any problems. Thanks to Vedran Sego for many useful comments and suggestions.

- github repository for the Python NCM module, nearest_correlation
- Nick Higham’s original MATLAB code.
- NAG’s commercial implementation – callable from C, Fortran, MATLAB, Python and more. A superb implementation that is significantly faster and more robust than this one!

The Mathematics department at The University of Manchester runs a third year undergraduate module called ‘Problem solving by computer’ which invites students to solve complex mathematical problems by doing a little programming. Along with some interesting mathematics, the course exposes students to a wide variety of languages and numerical libraries including MATLAB, Octave, NAG, Mathematica and, most recently, Python.

Earlier this year, Python was introduced as an option for students who wanted to use it for a project in this course and, despite only being given two lectures in the language, quite a few people chose to use it. Much of this success must be attributed to the Python for MATLABers notes written by Manchester PhD student, Sophia Coban which is why I’m providing links to them here.

When porting code between MATLAB and Python, it is sometimes useful to produce the exact same set of random numbers for testing purposes. Both Python and MATLAB currently use the Mersenne Twister generator by default so one assumes this should be easy…and it is…provided you use the generator in Numpy and avoid the seed 0.

**Generate some random numbers in MATLAB**

Here, we generate the first 5 numbers for 3 different seeds in MATLAB. Our aim is to reproduce these in Python.

>> format long >> rng(0) >> rand(1,5)' ans = 0.814723686393179 0.905791937075619 0.126986816293506 0.913375856139019 0.632359246225410 >> rng(1) >> rand(1,5)' ans = 0.417022004702574 0.720324493442158 0.000114374817345 0.302332572631840 0.146755890817113 >> rng(2) >> rand(1,5)' ans = 0.435994902142004 0.025926231827891 0.549662477878709 0.435322392618277 0.420367802087489

**Python’s default random module**

According to the documentation,Python’s random module uses the Mersenne Twister algorithm but the implementation seems to be different from MATLAB’s since the results are different. Here’s the output from a fresh ipython session:

In [1]: import random In [2]: random.seed(0) In [3]: [random.random() for _ in range(5)] Out[3]: [0.8444218515250481, 0.7579544029403025, 0.420571580830845, 0.25891675029296335, 0.5112747213686085] In [4]: random.seed(1) In [5]: [random.random() for _ in range(5)] Out[5]: [0.13436424411240122, 0.8474337369372327, 0.763774618976614, 0.2550690257394217, 0.49543508709194095] In [6]: random.seed(2) In [7]: [random.random() for _ in range(5)] Out[7]: [0.9560342718892494, 0.9478274870593494, 0.05655136772680869, 0.08487199515892163, 0.8354988781294496]

**The Numpy random module**

Numpy’s random module, on the other hand, seems to use an identical implementation to MATLAB for seeds other than 0. In the below, notice that for seeds 1 and 2, the results are identical to MATLAB’s. For a seed of zero, they are different.

In [1]: import numpy as np In [2]: np.set_printoptions(suppress=True) In [3]: np.set_printoptions(precision=15) In [4]: np.random.seed(0) In [5]: np.random.random((5,1)) Out[5]: array([[ 0.548813503927325], [ 0.715189366372419], [ 0.602763376071644], [ 0.544883182996897], [ 0.423654799338905]]) In [6]: np.random.seed(1) In [7]: np.random.random((5,1)) Out[7]: array([[ 0.417022004702574], [ 0.720324493442158], [ 0.000114374817345], [ 0.30233257263184 ], [ 0.146755890817113]]) In [8]: np.random.seed(2) In [9]: np.random.random((5,1)) Out[9]: array([[ 0.435994902142004], [ 0.025926231827891], [ 0.549662477878709], [ 0.435322392618277], [ 0.420367802087489]])

**Checking a lot more seeds**

Although the above interactive experiments look convincing, I wanted to check a few more seeds. All seeds from 0 to 1 million would be a good start so I wrote a MATLAB script that generated 10 random numbers for each seed from 0 to 1 million and saved the results as a .mat file.

A subsequent Python script loads the .mat file and ensures that numpy generates the same set of numbers for each seed. It outputs every seed for which Python and MATLAB differ.

On my mac, I opened a bash prompt and ran the two scripts as follows

matlab -nodisplay -nodesktop -r "generate_matlab_randoms" python python_randoms.py

The output was

MATLAB file contains 1000001 seeds and 10 samples per seed Random numbers for seed 0 differ between MATLAB and Numpy

**System details**

- Late 2013 Macbook Air
- MATLAB 2014a
- Python 2.7.7
- Numpy 1.8.1

One of my favourite MATLAB books is The MATLAB Guide by Desmond and Nicholas Higham. The first chapter, called ‘A Brief Tutorial’ shows how various mathematical problems can be easily explored with MATLAB; things like continued fractions, random fibonacci sequences, fractals and collatz iterations.

Over at the SIAM blog, Don MacMillen, demonstrates how its now possible, trivial even, to rewrite the entire chapter as an IPython notebook with all MATLAB code replaced with Python.

The notebook is available as a gist and can be viewed statically on nbviewer.

What other examples of successful MATLAB->Python conversions have you found?

The IPython project started as a procrastination task for Fernando Perez during his PhD and is currently one of the most exciting and important pieces of software in computational science today. Last month, Fernando joined us at The University of Manchester after being invited by Nick Higham of the department of Mathematics under the auspices of the EPSRC Network Numerical Algorithms and High Performance Computing

While at Manchester, Fernando gave a couple of talks and we captured one of them using the University of Manchester Lecture Podcasting Service (itself based on a Python project). Check it out below.

A lot of people don’t seem to know this….and they should. When working with floating point arithmetic, it is not necessarily true that a+(b+c) = (a+b)+c. Here is a demo using MATLAB

>> x=0.1+(0.2+0.3); >> y=(0.1+0.2)+0.3; >> % are they equal? >> x==y ans = 0 >> % lets look >> sprintf('%.17f',x) ans = 0.59999999999999998 >> sprintf('%.17f',y) ans = 0.60000000000000009

These results have nothing to do with the fact that I am using MATLAB. Here’s the same thing in Python

>>> x=(0.1+0.2)+0.3 >>> y=0.1+(0.2+0.3) >>> x==y False >>> print('%.17f' %x) 0.60000000000000009 >>> print('%.17f' %y) 0.59999999999999998

If this upsets you, or if you don’t understand why, I suggest you read the following

Does anyone else out there have suggestions for similar resources on this topic?

In a recent Stack Overflow query, someone asked if you could switch off the balancing step when calculating eigenvalues in Python. In the document A case where balancing is harmful, David S. Watkins describes the balancing step as* ‘the input matrix A is replaced by a rescaled matrix A* = D ^{-1}AD, where D is a diagonal matrix chosen so that, for each i, the ith row and the ith column of A* have roughly the same norm.’ *

Such balancing is usually very useful and so is performed by default by software such as MATLAB or Numpy. There are times, however, when one would like to switch it off.

In MATLAB, this is easy and the following is taken from the online MATLAB documentation

A = [ 3.0 -2.0 -0.9 2*eps; -2.0 4.0 1.0 -eps; -eps/4 eps/2 -1.0 0; -0.5 -0.5 0.1 1.0]; [VN,DN] = eig(A,'nobalance') VN = 0.6153 -0.4176 -0.0000 -0.1528 -0.7881 -0.3261 0 0.1345 -0.0000 -0.0000 -0.0000 -0.9781 0.0189 0.8481 -1.0000 0.0443 DN = 5.5616 0 0 0 0 1.4384 0 0 0 0 1.0000 0 0 0 0 -1.0000

At the time of writing, it is not possible to directly do this in Numpy (as far as I know at least). Numpy’s eig command currently uses the LAPACK routine DGEEV to do the heavy lifting for double precision matrices. We can see this by looking at the source code of numpy.linalg.eig where the relevant subsection is

lapack_routine = lapack_lite.dgeev wr = zeros((n,), t) wi = zeros((n,), t) vr = zeros((n, n), t) lwork = 1 work = zeros((lwork,), t) results = lapack_routine(_N, _V, n, a, n, wr, wi, dummy, 1, vr, n, work, -1, 0) lwork = int(work[0]) work = zeros((lwork,), t) results = lapack_routine(_N, _V, n, a, n, wr, wi, dummy, 1, vr, n, work, lwork, 0)

My plan was to figure out how to tell DGEEV not to perform the balancing step and I’d be done. Sadly, however, it turns out that this is not possible. Taking a look at the reference implementation of DGEEV, we can see that the balancing step is **always performed **and is not user controllable–here’s the relevant bit of Fortran

* Balance the matrix * (Workspace: need N) * IBAL = 1 CALL DGEBAL( 'B', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )

So, using DGEEV is a dead-end unless we are willing to modifiy and recompile the lapack source — something that’s rarely a good idea in my experience. There is another LAPACK routine that is of use, however, in the form of DGEEVX that allows us to control balancing. Unfortunately, this routine is not part of the **numpy.linalg.lapack_lite** interface provided by Numpy and I’ve yet to figure out how to add extra routines to it.

I’ve also discovered that this functionality is an open feature request in Numpy.

**Enter the NAG Library**

My University has a site license for the commercial Numerical Algorithms Group (NAG) library. Among other things, NAG offers an interface to all of LAPACK along with an interface to Python. So, I go through the installation and do

import numpy as np from ctypes import * from nag4py.util import Nag_RowMajor,Nag_NoBalancing,Nag_NotLeftVecs,Nag_RightVecs,Nag_RCondEigVecs,Integer,NagError,INIT_FAIL from nag4py.f08 import nag_dgeevx eps = np.spacing(1) np.set_printoptions(precision=4,suppress=True) def unbalanced_eig(A): """ Compute the eigenvalues and right eigenvectors of a square array using DGEEVX via the NAG library. Requires the NAG C library and NAG's Python wrappers http://www.nag.co.uk/python.asp The balancing step that's performed in DGEEV is not performed here. As such, this function is the same as the MATLAB command eig(A,'nobalance') Parameters ---------- A : (M, M) Numpy array A square array of real elements. On exit: A is overwritten and contains the real Schur form of the balanced version of the input matrix . Returns ------- w : (M,) ndarray The eigenvalues v : (M, M) ndarray The eigenvectors Author: Mike Croucher (www.walkingrandomly.com) Testing has been mimimal """ order = Nag_RowMajor balanc = Nag_NoBalancing jobvl = Nag_NotLeftVecs jobvr = Nag_RightVecs sense = Nag_RCondEigVecs n = A.shape[0] pda = n pdvl = 1 wr = np.zeros(n) wi = np.zeros(n) vl=np.zeros(1); pdvr = n vr = np.zeros(pdvr*n) ilo=c_long(0) ihi=c_long(0) scale = np.zeros(n) abnrm = c_double(0) rconde = np.zeros(n) rcondv = np.zeros(n) fail = NagError() INIT_FAIL(fail) nag_dgeevx(order,balanc,jobvl,jobvr,sense, n, A.ctypes.data_as(POINTER(c_double)), pda, wr.ctypes.data_as(POINTER(c_double)), wi.ctypes.data_as(POINTER(c_double)),vl.ctypes.data_as(POINTER(c_double)),pdvl, vr.ctypes.data_as(POINTER(c_double)),pdvr,ilo,ihi, scale.ctypes.data_as(POINTER(c_double)), abnrm, rconde.ctypes.data_as(POINTER(c_double)),rcondv.ctypes.data_as(POINTER(c_double)),fail) if all(wi == 0.0): w = wr v = vr.reshape(n,n) else: w = wr+1j*wi v = array(vr, w.dtype).reshape(n,n) return(w,v)

Define a test matrix:

A = np.array([[3.0,-2.0,-0.9,2*eps], [-2.0,4.0,1.0,-eps], [-eps/4,eps/2,-1.0,0], [-0.5,-0.5,0.1,1.0]])

Do the calculation

(w,v) = unbalanced_eig(A)

which gives

(array([ 5.5616, 1.4384, 1. , -1. ]), array([[ 0.6153, -0.4176, -0. , -0.1528], [-0.7881, -0.3261, 0. , 0.1345], [-0. , -0. , -0. , -0.9781], [ 0.0189, 0.8481, -1. , 0.0443]]))

This is exactly what you get by running the MATLAB command **eig(A,’nobalance’).**

Note that unbalanced_eig(A) **changes the input matri**x A to

array([[ 5.5616, -0.0662, 0.0571, 1.3399], [ 0. , 1.4384, 0.7017, -0.1561], [ 0. , 0. , 1. , -0.0132], [ 0. , 0. , 0. , -1. ]])

According to the NAG documentation, this is the real Schur form of the balanced version of the input matrix. I can’t see how to ask NAG to not do this. I guess that if it’s not what you want unbalanced_eig() to do, you’ll need to pass a copy of the input matrix to NAG.

**The IPython notebook**

The code for this article is available as an IPython Notebook

**The future**

This blog post was written using Numpy version 1.7.1. There is an enhancement request for the functionality discussed in this article open in Numpy’s git repo and so I expect this article to become redundant pretty soon.

A question I get asked a lot is ‘How can I do nonlinear least squares curve fitting in X?’ where X might be MATLAB, Mathematica or a whole host of alternatives. Since this is such a common query, I thought I’d write up how to do it for a very simple problem in several systems that I’m interested in

This is the Python version. For other versions,see the list below

- Simple nonlinear least squares curve fitting in Julia
- Simple nonlinear least squares curve fitting in Maple
- Simple nonlinear least squares curve fitting in Mathematica
- Simple nonlinear least squares curve fitting in MATLAB
- Simple nonlinear least squares curve fitting in R

**The problem**

xdata = -2,-1.64,-1.33,-0.7,0,0.45,1.2,1.64,2.32,2.9 ydata = 0.699369,0.700462,0.695354,1.03905,1.97389,2.41143,1.91091,0.919576,-0.730975,-1.42001

and you’d like to fit the function

using nonlinear least squares. You’re starting guesses for the parameters are p1=1 and P2=0.2

For now, we are primarily interested in the following results:

- The fit parameters
- Sum of squared residuals

Future updates of these posts will show how to get other results such as confidence intervals. Let me know what you are most interested in.

**Python solution using scipy**

Here, I use the curve_fit function from scipy

import numpy as np from scipy.optimize import curve_fit xdata = np.array([-2,-1.64,-1.33,-0.7,0,0.45,1.2,1.64,2.32,2.9]) ydata = np.array([0.699369,0.700462,0.695354,1.03905,1.97389,2.41143,1.91091,0.919576,-0.730975,-1.42001]) def func(x, p1,p2): return p1*np.cos(p2*x) + p2*np.sin(p1*x) popt, pcov = curve_fit(func, xdata, ydata,p0=(1.0,0.2))

The variable popt contains the fit parameters

array([ 1.88184732, 0.70022901])

We need to do a little more work to get the sum of squared residuals

p1 = popt[0] p2 = popt[1] residuals = ydata - func(xdata,p1,p2) fres = sum(residuals**2)

which gives

0.053812696547933969

- I’ve put this in an Ipython notebook which can be downloaded here. There is also a pdf version of the notebook.