EuroSciPy 2010 lecture notes (PythonScientific.pdf) (EuroSciPy)

Python Scientiﬁc lecture notes
Release 2010

Contents

EuroScipy tutorial team Editors: Emmanuelle Gouillart, Gaël Varoquaux http://scipy-lectures.github.com

February 17, 2011

Getting started with Python for science 1.1 Scientiﬁc computing: why Python? . . . . . . . . . . . . . . . . 1.1.1 The scientist’s needs . . . . . . . . . . . . . . . . . . . 1.1.2 Speciﬁcations . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Existing solutions . . . . . . . . . . . . . . . . . . . . . 1.2 Building blocks of scientiﬁc computing with Python . . . . . . . 1.3 A (very short) introduction to Python . . . . . . . . . . . . . . . 1.3.1 First steps . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Basic types . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Assignment operator . . . . . . . . . . . . . . . . . . . 1.3.4 Control Flow . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Deﬁning functions . . . . . . . . . . . . . . . . . . . . 1.3.6 Reusing code: scripts and modules . . . . . . . . . . . . 1.3.7 Input and Output . . . . . . . . . . . . . . . . . . . . . 1.3.8 Standard Library . . . . . . . . . . . . . . . . . . . . . 1.3.9 Exceptions handling in Python . . . . . . . . . . . . . . 1.3.10 Object-oriented programming (OOP) . . . . . . . . . . . 1.4 NumPy: creating and manipulating numerical data . . . . . . . . 1.4.1 Creating NumPy data arrays . . . . . . . . . . . . . . . 1.4.2 Graphical data representation : matplotlib and Mayavi . 1.4.3 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Manipulating the shape of arrays . . . . . . . . . . . . . 1.4.6 Exercises : some simple array creations . . . . . . . . . 1.4.7 Real data: read/write arrays from/to ﬁles . . . . . . . . . 1.4.8 Simple mathematical and statistical operations on arrays 1.4.9 Fancy indexing . . . . . . . . . . . . . . . . . . . . . . 1.4.10 Broadcasting . . . . . . . . . . . . . . . . . . . . . . . 1.4.11 Synthesis exercises: framing Lena . . . . . . . . . . . . 1.5 Getting help and ﬁnding documentation . . . . . . . . . . . . . . 1.6 Matplotlib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 IPython . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 pylab . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Simple Plots . . . . . . . . . . . . . . . . . . . . . . . . 1.6.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.6 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 1 1 2 5 6 6 12 13 17 22 28 29 33 36 36 37 38 40 41 43 44 44 47 49 51 54 55 58 58 58 58 58 60 62

1.7

1.6.7 1.6.8 1.6.9 1.6.10 Scipy : 1.7.1 1.7.2 1.7.3 1.7.4 1.7.5 1.7.6 1.7.7 1.7.8 1.7.9 1.7.10 1.7.11 1.7.12

Ticks . . . . . . . . . . . . . . . . . . . . . . . Figures, Subplots, and Axes . . . . . . . . . . . Other Types of Plots . . . . . . . . . . . . . . . The Class Library . . . . . . . . . . . . . . . . . high-level scientiﬁc computing . . . . . . . . . . Scipy builds upon Numpy . . . . . . . . . . . . File input/output: scipy.io . . . . . . . . . . Signal processing: scipy.signal . . . . . . . Special functions: scipy.special . . . . . . Statistics and random numbers: scipy.stats Linear algebra operations: scipy.linalg . . Numerical integration: scipy.integrate . . Fast Fourier transforms: scipy.fftpack . . . Interpolation: scipy.interpolate . . . . . Optimization and ﬁt: scipy.optimize . . . . Image processing: scipy.ndimage . . . . . . Summary exercices on scientiﬁc computing . . .

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63 65 67 73 75 75 76 76 77 77 79 80 82 84 85 87 91 104 104 104 104 105 116 125 133 134 134 134 137 138 138 140 152 157 157 157 158 158 159 160 161 162 163 163 164 167 170 171 172

CHAPTER 1

Getting started with Python for science

Advanced topics 2.1 Advanced Numpy . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Advanced Numpy . . . . . . . . . . . . . . . . . . 2.1.3 Life of ndarray . . . . . . . . . . . . . . . . . . . 2.1.4 Universal functions . . . . . . . . . . . . . . . . . 2.1.5 Interoperability features . . . . . . . . . . . . . . . 2.1.6 Siblings: chararray, maskedarray, matrix 2.1.7 Summary . . . . . . . . . . . . . . . . . . . . . . 2.1.8 Hit list of the future for Numpy core . . . . . . . . 2.1.9 Contributing to Numpy/Scipy . . . . . . . . . . . 2.1.10 Python 2 and 3, single code base . . . . . . . . . . 2.2 Sparse Matrices in SciPy . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 2.2.2 Storage Schemes . . . . . . . . . . . . . . . . . . 2.2.3 Linear System Solvers . . . . . . . . . . . . . . . 2.2.4 Other Interesting Packages . . . . . . . . . . . . . 2.3 Sympy : Symbolic Mathematics in Python . . . . . . . . . 2.3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . 2.3.2 What is SymPy? . . . . . . . . . . . . . . . . . . 2.3.3 First Steps with SymPy . . . . . . . . . . . . . . . 2.3.4 Algebraic manipulations . . . . . . . . . . . . . . 2.3.5 Calculus . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Equation solving . . . . . . . . . . . . . . . . . . 2.3.7 Linear Algebra . . . . . . . . . . . . . . . . . . . 2.4 3D plotting with Mayavi . . . . . . . . . . . . . . . . . . . 2.4.1 A simple example . . . . . . . . . . . . . . . . . . 2.4.2 3D plotting functions . . . . . . . . . . . . . . . . 2.4.3 Figures and decorations . . . . . . . . . . . . . . . 2.4.4 Interaction . . . . . . . . . . . . . . . . . . . . . .

1.1 Scientiﬁc computing: why Python?
authors Fernando Perez, Emmanuelle Gouillart

1.1.1 The scientist’s needs
• Get data (simulation, experiment control) • Manipulate and process data. • Visualize results... to understand what we are doing! • Communicate on results: produce ﬁgures for reports or publications, write presentations.

1.1.2 Speciﬁcations
• Rich collection of already existing bricks corresponding to classical numerical methods or basic actions: we don’t want to re-program the plotting of a curve, a Fourier transform or a ﬁtting algorithm. Don’t reinvent the wheel! • Easy to learn: computer science neither is our job nor our education. We want to be able to draw a curve, smooth a signal, do a Fourier transform in a few minutes. • Easy communication with collaborators, students, customers, to make the code live within a labo or a company: the code should be as readable as a book. Thus, the language should contain as few syntax symbols or unneeded routines that would divert the reader from the mathematical or scientiﬁc understanding of the code. • Efﬁcient code that executes quickly... But needless to say that a very fast code becomes useless if we spend too much time writing it. So, we need both a quick development time and a quick execution time. • A single environment/language for everything, if possible, to avoid learning a new software for each new problem.

Bibliography Index

1.1.3 Existing solutions
Which solutions do the scientists use to work? Compiled languages: C, C++, Fortran, etc. ii 1

Python Scientiﬁc lecture notes, Release 2010

• Advantages: – Very fast. Very optimized compilers. For heavy computations, it’s difﬁcult to outperform these languages. – Some very optimized scientiﬁc libraries have been written for these languages. Ex: blas (vector/matrix operations) • Drawbacks: – Painful usage: no interactivity during development, mandatory compilation steps, verbose syntax (&, ::, }}, ; etc.), manual memory management (tricky in C). These are difﬁcult languages for non computer scientists. Scripting languages: Matlab • Advantages: – Very rich collection of libraries with numerous algorithms, for many different domains. Fast execution because these libraries are often written in a compiled language. – Pleasant development environment: comprehensive and well organized help, integrated editor, etc. – Commercial support is available. • Drawbacks: – Base language is quite poor and can become restrictive for advanced users. – Not free. Other script languages: Scilab, Octave, Igor, R, IDL, etc. • Advantages: – Open-source, free, or at least cheaper than Matlab. – Some features can be very advanced (statistics in R, ﬁgures in Igor, etc.) • Drawbacks: – fewer available algorithms than in Matlab, and the language is not more advanced. – Some softwares are dedicated to one domain. Ex: Gnuplot or xmgrace to draw curves. These programs are very powerful, but they are restricted to a single type of usage, such as plotting. What about Python? • Advantages: – Very rich scientiﬁc computing libraries (a bit less than Matlab, though) – Well-thought language, allowing to write very readable and well structured code: we “code what we think”. – Many libraries for other tasks than scientiﬁc computing (web server management, serial port access, etc.) – Free and open-source software, widely spread, with a vibrant community. • Drawbacks: – less pleasant development environment than, for example, Matlab. (More geek-oriented). – Not all the algorithms that can be found in more specialized softwares or toolboxes.

– Python language: data types (string, int), ﬂow control, data collections (lists, dictionaries), patterns, etc. – Modules of the standard library. – A large number of specialized modules or applications written in Python: web protocols, web framework, etc. ... and scientiﬁc computing. – Development tools (automatic tests, documentation generation) • IPython, an advanced Python shell http://ipython.scipy.org/moin/

In [1]: a = 2 In [2]: print "hello" hello In [3]: %run my_script.py

• Numpy : provides powerful numerical arrays objects, and routines to manipulate them.
>>> import numpy as np >>> t = np.arange(10) >>> t array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> print t [0 1 2 3 4 5 6 7 8 9] >>> signal = np.sin(t)

1.2 Building blocks of scientiﬁc computing with Python
author Emmanuelle Gouillart • Python, a generic and modern computing language 1.2. Building blocks of scientiﬁc computing with Python 2

http://www.scipy.org/ • Scipy : high-level data processing routines. Optimization, regression, interpolation, etc:
>>> import numpy as np >>> import scipy >>> t = np.arange(10)

1.2. Building blocks of scientiﬁc computing with Python

Python Scientiﬁc lecture notes, Release 2010

>>> t array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> signal = t**2 + 2*t + 2+ 1.e-2*np.random.random(10) >>> scipy.polyfit(t, signal, 2) array([ 1.00001151, 1.99920674, 2.00902748])

http://www.scipy.org/ • Matplotlib : 2-D visualization, “publication-ready” plots http://matplotlib.sourceforge.net/

• and many others.

1.3 A (very short) introduction to Python
authors Chris Burns, Christophe Combelles, Emmanuelle Gouillart, Gaël Varoquaux Python for scientiﬁc computing We introduce here the Python language. Only the bare minimum necessary for getting started with Numpy and Scipy is addressed here. To learn more about the language, consider going through the excellent tutorial http://docs.python.org/tutorial. Dedicated books are also available, such as http://diveintopython.org/.

Python is a programming language, as are C, Fortran, BASIC, PHP, etc. Some speciﬁc features of Python are as follows: • Mayavi : 3-D visualization http://code.enthought.com/projects/mayavi/ • an interpreted (as opposed to compiled) language. Contrary to e.g. C or Fortran, one does not compile Python code before executing it. In addition, Python can be used interactively: many Python interpreters are available, from which commands and scripts can be executed. • a free software released under an open-source license: Python can be used and distributed free of charge, even for building commercial software. • multi-platform: Python is available for all major operating systems, Windows, Linux/Unix, MacOS X, most likely your mobile phone OS, etc. • a very readable language with clear non-verbose syntax • a language for which a large variety of high-quality packages are available for various applications, from web frameworks to scientiﬁc computing. • a language very easy to interface with other languages, in particular C and C++. • Some other features of the language are illustrated just below. For example, Python is an object-oriented language, with dynamic typing (an object’s type can change during the course of a program). See http://www.python.org/about/ for more information about distinguishing features of Python.

1.2. Building blocks of scientiﬁc computing with Python

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

1.3.1 First steps
Start the Ipython shell (an enhanced interactive Python shell): • by typing “Ipython” from a Linux/Mac terminal, or from the Windows cmd shell, • or by starting the program from a menu, e.g. in the Python(x,y) or EPD menu if you have installed one these scientiﬁc-Python suites. If you don’t have Ipython installed on your computer, other Python shells are available, such as the plain Python shell started by typing “python” in a terminal, or the Idle interpreter. However, we advise to use the Ipython shell because of its enhanced features, especially for interactive scientiﬁc computing. Once you have started the interpreter, type
>>> print "Hello, world!" Hello, world!

>>> a=1.5+0.5j >>> a.real 1.5 >>> a.imag 0.5

and booleans:
>>> 3 > 4 False >>> test = (3 > 4) >>> test False >>> type(test) <type ’bool’>

The message “Hello, world!” is then displayed. You just executed your ﬁrst Python instruction, congratulations! To get yourself started, type the following stack of instructions
>>> a = 3 >>> b = 2*a >>> type(b) <type ’int’> >>> print b 6 >>> a*b 18 >>> b = ’hello’ >>> type(b) <type ’str’> >>> b + b ’hellohello’ >>> 2*b ’hellohello’

A Python shell can therefore replace your pocket calculator, with the basic arithmetic operations +, -, *, /, % (modulo) natively implemented:
>>> 7 * 3. 21.0 >>> 2**10 1024 >>> 8%3 2

Warning: Integer division
>>> 3/2 1

Trick: use ﬂoats:
>>> 3/2. 1.5 >>> >>> >>> 1 >>> 1.5 a = 3 b = 2 a/b a/float(b)

Two objects a and b have been deﬁned above. Note that one does not declare the type of an object before assigning its value. In C, conversely, one should write:
int a; a = 3;

In addition, the type of an object may change. b was ﬁrst an integer, but it became a string when it was assigned the value hello. Operations on integers (b=2*a) are coded natively in the Python standard library, and so are some operations on strings such as additions and multiplications, which amount respectively to concatenation and repetition.

• Scalar types: int, ﬂoat, complex, bool:
>>> type(1) <type ’int’> >>> type(1.) <type ’float’> >>> type(1. + 0j ) <type ’complex’> >>> a = 3 >>> type(a) <type ’int’>

1.3.2 Basic types
Numerical types Integer variables:
>>> 1 + 1 2 >>> a = 4

• Type conversion:
>>> float(1) 1.0

ﬂoats
>>> c = 2.1

complex (a native type in Python!)

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

Containers Python provides many efﬁcient types of containers, in which collections of objects can be stored.

Lists
A list is an ordered collection of objects, that may have different types. For example
>>> l = [1, 2, 3, 4, 5] >>> type(l) <type ’list’>

As the elements of a list can be of any type and size, accessing the i th element of a list has a complexity O(i). For collections of numerical data that all have the same type, it is more efﬁcient to use the array type provided by the Numpy module, which is a sequence of regularly-spaced chunks of memory containing ﬁxed-sized data istems. With Numpy arrays, accessing the i‘th‘ element has a complexity of O(1) because the elements are regularly spaced in memory. Python offers a large panel of functions to modify lists, or query them. Here are a few examples; for more details, see http://docs.python.org/tutorial/datastructures.html#more-on-lists Add and remove elements:
>>> >>> >>> [1, >>> 6 >>> [1, >>> >>> [1, >>> >>> [1, l = [1, 2, 3, 4, 5] l.append(6) l 2, 3, 4, 5, 6] l.pop() l 2, 3, 4, 5] l.extend([6, 7]) # extend l, in-place l 2, 3, 4, 5, 6, 7] l = l[:-2] l 2, 3, 4, 5]

• Indexing: accessing individual objects contained in the list:
>>> l[2] 3

Counting from the end with negative indices:
>>> l[-1] 5 >>> l[-2] 4

Warning: Indexing starts at 0 (as in C), not at 1 (as in Fortran or Matlab)! • Slicing: obtaining sublists of regularly-spaced elements
>>> [1, >>> [3, l 2, 3, 4, 5] l[2:4] 4]

Reverse l:
>>> r = l[::-1] >>> r [5, 4, 3, 2, 1]

Concatenate and repeat lists:
>>> [5, >>> [5, r + l 4, 3, 2, 1, 1, 2, 3, 4, 5] 2 * r 4, 3, 2, 1, 5, 4, 3, 2, 1]

Warning: Note that l[start:stop] contains the elements with indices i such as start<= i < stop (i ranging from start to stop-1). Therefore, l[start:stop] has (stop-start) elements. Slicing syntax: l[start:stop:stride] All slicing parameters are optional:
>>> [4, >>> [1, >>> [1, l[3:] 5] l[:3] 2, 3] l[::2] 3, 5]

Sort r (in-place):
>>> r.sort() >>> r [1, 2, 3, 4, 5]

Note: Methods and Object-Oriented Programming The notation r.method() (r.sort(), r.append(3), l.pop()) is our ﬁrst example of object-oriented programming (OOP). Being a list, the object r owns the method function that is called using the notation .. No further knowledge of OOP than understanding the notation . is necessary for going through this tutorial. Note: Discovering methods: In IPython: tab-completion (press tab)
In [28]: r. r.__add__ r.__class__ r.__contains__ r.__delattr__ r.__delitem__ r.__delslice__ r.__doc__ r.__eq__ r.__format__ r.__ge__ r.__iadd__ r.__imul__ r.__init__ r.__iter__ r.__le__ r.__len__ r.__lt__ r.__mul__ r.__ne__ r.__new__ r.__setattr__ r.__setitem__ r.__setslice__ r.__sizeof__ r.__str__ r.__subclasshook__ r.append r.count r.extend r.index

Lists are mutable objects and can be modiﬁed:
>>> l[0] = >>> l [28, 2, 3, >>> l[2:4] >>> l [28, 2, 3, 28 4, 5] = [3, 8] 8, 5]

Note: The elements of a list may have different types:
>>> >>> [3, >>> (2, l = [3, 2, ’hello’] l 2, ’hello’] l[1], l[2] ’hello’)

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

r.__getattribute__ r.__getitem__ r.__getslice__ r.__gt__ r.__hash__

r.__reduce__ r.__reduce_ex__ r.__repr__ r.__reversed__ r.__rmul__

r.insert r.pop r.remove r.reverse r.sort

console> in <module>() TypeError: ’str’ object does not support item assignment In [55]: a.replace(’l’, ’z’, 1) Out[55]: ’hezlo, world!’ In [56]: a.replace(’l’, ’z’) Out[56]: ’hezzo, worzd!’

Strings
Different string syntaxes (simple, double or triple quotes):
s = ’Hello, how are you?’ s = "Hi, what’s up" s = ’’’Hello, how are you’’’ s = """Hi, what’s up?’’’ In [1]: ’Hi, what’s up ? ’ -----------------------------------------------------------File "<ipython console>", line 1 ’Hi, what’s up?’ ^ SyntaxError: invalid syntax

Strings have many useful methods, such as a.replace as seen above. Remember the a. object-oriented notation and use tab completion or help(str) to search for new methods. Note: Python offers advanced possibilities for manipulating strings, looking for patterns or formatting. Due to lack of time this topic is not addressed here, but the interested reader is referred to http://docs.python.org/library/stdtypes.html#string-methods and http://docs.python.org/library/string.html#newstring-formatting • String substitution:
>>> ’An integer: %i; a float: %f; another string: %s’ % (1, 0.1, ’string’) ’An integer: 1; a float: 0.100000; another string: string’ >>> i = 102 >>> filename = ’processing_of_dataset_%03d.txt’%i >>> filename ’processing_of_dataset_102.txt’

The newline character is \n, and the tab characted is \t. Strings are collections as lists. Hence they can be indexed and sliced, using the same syntax and rules. Indexing:
>>> >>> ’h’ >>> ’e’ >>> ’o’ a = "hello" a[0] a[1] a[-1]

Dictionnaries
A dictionnary is basically a hash table that maps keys to values. It is therefore an unordered container:
>>> tel = {’emmanuelle’: 5752, ’sebastian’: 5578} >>> tel[’francis’] = 5915 >>> tel {’sebastian’: 5578, ’francis’: 5915, ’emmanuelle’: 5752} >>> tel[’sebastian’] 5578 >>> tel.keys() [’sebastian’, ’francis’, ’emmanuelle’] >>> tel.values() [5578, 5915, 5752] >>> ’francis’ in tel True

(Remember that Negative indices correspond to counting from the right end.) Slicing:
>>> a = "hello, world!" >>> a[3:6] # 3rd to 6th (excluded) elements: elements 3, 4, 5 ’lo,’ >>> a[2:10:2] # Syntax: a[start:stop:step] ’lo o’ >>> a[::3] # every three characters, from beginning to end ’hl r!’

This is a very convenient data container in order to store values associated to a name (a string for a date, a name, etc.). See http://docs.python.org/tutorial/datastructures.html#dictionaries for more information. A dictionnary can have keys (resp. values) with different types:
>>> d = {’a’:1, ’b’:2, 3:’hello’} >>> d {’a’: 1, 3: ’hello’, ’b’: 2}

Accents and special characters can also be handled http://docs.python.org/tutorial/introduction.html#unicode-strings).

Unicode

strings

(see

A string is an immutable object and it is not possible to modify its characters. One may however create new strings from an original one.
In [53]: a = "hello, world!" In [54]: a[2] = ’z’ --------------------------------------------------------------------------TypeError Traceback (most recent call last) /home/gouillar/travail/sgr/2009/talks/dakar_python/cours/gael/essai/source/<ipython

More container types
• Tuples Tuples are basically immutable lists. The elements of a tuple are written between brackets, or just separated by commas:

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

>>> t = 12345, 54321, ’hello!’ >>> t[0] 12345 >>> t (12345, 54321, ’hello!’) >>> u = (0, 2)

Out[4]: In [5]: Out[5]: In [6]: In [7]: Out[7]:

[1, 2, 3] a is b True b[1] = ’hi!’ a [1, ’hi!’, 3]

• Sets: non ordered, unique items:
>>> s = set((’a’, ’b’, ’c’, ’a’)) >>> s set([’a’, ’c’, ’b’]) >>> s.difference((’a’, ’b’)) set([’c’])

• to change a list in place, use indexing/slices:
In [1]: In [3]: Out[3]: In [4]: In [5]: Out[5]: In [6]: Out[6]: In [7]: In [8]: Out[8]: In [9]: Out[9]: a = [1, 2, 3] a [1, 2, 3] a = [’a’, ’b’, ’c’] # Creates another object. a [’a’, ’b’, ’c’] id(a) 138641676 a[:] = [1, 2, 3] # Modifies object in place. a [1, 2, 3] id(a) 138641676 # Same as in Out[6], yours will differ...

A bag of Ipython tricks • • • • Several Linux shell commands work in Ipython, such as ls, pwd, cd, etc. To get help about objects, functions, etc., type help object. Just type help() to get started. Use tab-completion as much as possible: while typing the beginning of an object’s name (variable, function, module), press the Tab key and Ipython will complete the expression to match available names. If many names are possible, a list of names is displayed. • History: press the up (resp. down) arrow to go through all previous (resp. next) instructions starting with the expression on the left of the cursor (put the cursor at the beginning of the line to go through all previous commands) • You may log your session by using the Ipython “magic command” %logstart. Your instructions will be saved in a ﬁle, that you can execute as a script in a different session.

• the key concept here is mutable vs. immutable – mutable objects can be changed in place – immutable objects cannot be modiﬁed once created A very good and detailed explanation of the above issues can be found in David M. Beazley’s article Types and Objects in Python.

In [1]: %logstart commandes.log Activating auto-logging. Current session state plus future input saved. Filename : commandes.log Mode : backup Output logging : False Raw input log : False Timestamping : False State : active

1.3.4 Control Flow
Controls the order in which the code is executed. if/elif/else
In [1]: if 2**2 == 4: ...: print(’Obvious!’) ...: Obvious!

1.3.3 Assignment operator
Python library reference says: Assignment statements are used to (re)bind names to values and to modify attributes or items of mutable objects. In short, it works as follows (simple assignment): 1. an expression on the right hand side is evaluated, the corresponding object is created/obtained 2. a name on the left hand side is assigned, or bound, to the r.h.s. object Things to note: • a single object can have several names bound to it:
In [1]: In [2]: In [3]: Out[3]: In [4]: a = [1, 2, 3] b = a a [1, 2, 3] b

Blocks are delimited by indentation Type the following lines in your Python interpreter, and be careful to respect the indentation depth. The Ipython shell automatically increases the indentation depth after a column : sign; to decrease the indentation depth, go four spaces to the left with the Backspace key. Press the Enter key twice to leave the logical block.
In [2]: a = 10 In [3]: if a == 1: ...: print(1) ...: elif a == 2: ...: print(2) ...: else: ...: print(’A lot’) ...: A lot

Indentation is compulsory in scripts as well. As an exercise, re-type the previous lines with the same indentation in a script condition.py, and execute the script with run condition.py in Ipython.

1.3. A (very short) introduction to Python

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for/range Iterating with an index:
In [4]: for i in range(4): ...: print(i) ...: 0 1 2 3

Evaluates to True: – any non-zero value – any sequence with a length > 0 Evaluates to False: – any zero value – any empty sequence • a == b Tests equality, with logics:
In [19]: 1 == 1. Out[19]: True

But most often, it is more readable to iterate over values:
In [5]: for word in (’cool’, ’powerful’, ’readable’): ...: print(’Python is %s’ % word) ...: Python is cool Python is powerful Python is readable

• a is b Tests identity: both objects are the same
In [20]: 1 is 1. Out[20]: False

while/break/continue Typical C-style while loop (Mandelbrot problem):
In [6]: z = 1 + 1j In [7]: while abs(z) < 100: ...: z = z**2 + 1 ...: In [8]: z Out[8]: (-134+352j)

In [21]: a = 1 In [22]: b = 1 In [23]: a is b Out[23]: True

• a in b For any collection b: b contains a
>>> b = [1, 2, 3] >>> 2 in b True >>> 5 in b False

More advanced features break out of enclosing for/while loop:
In [9]: z = 1 + 1j In [10]: while abs(z) < 100: ....: if z.imag == 0: ....: break ....: z = z**2 + 1 ....: ....:

If b is a dictionary, this tests that a is a key of b. Advanced iteration

Iterate over any sequence
• You can iterate over any sequence (string, list, dictionary, ﬁle, ...)
In [11]: vowels = ’aeiouy’ In [12]: for i in ’powerful’: ....: if i in vowels: ....: print(i), ....: ....: o e u >>> message = "Hello how are you?" >>> message.split() # returns a list [’Hello’, ’how’, ’are’, ’you?’] >>> for word in message.split(): ... print word ...

continue the next iteration of a loop.:
>>> a = >>> for ... ... ... ... 1.0 0.5 0.25 [1, 0, 2, 4] element in a: if element == 0: continue print 1. / element

Conditional Expressions • if object 1.3. A (very short) introduction to Python 14

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Hello how are you?

Exercise Compute the decimals of Pi using the Wallis formula:

Few languages (in particular, languages for scienﬁc computing) allow to loop over anything but integers/indices. With Python it is possible to loop exactly over the objects of interest without bothering with indices you often don’t care about. Warning: Not safe to modify the sequence you are iterating over.

Keeping track of enumeration number
Common task is to iterate over a sequence while keeping track of the item number. • Could use while loop with a counter as above. Or a for loop:
In [13]: for i in range(0, len(words)): ....: print(i, words[i]) ....: ....: 0 cool 1 powerful 2 readable

1.3.5 Deﬁning functions
Function deﬁnition
In [56]: def test(): ....: print(’in test function’) ....: ....: In [57]: test() in test function

• But Python provides enumerate for this:
>>> words = (’cool’, ’powerful’, ’readable’) >>> for index, item in enumerate(words): ... print index, item ... 0 cool 1 powerful 2 readable

Warning: Function blocks must be indented as other control-ﬂow blocks.

Return statement Functions can optionally return values.
In [6]: def disk_area(radius): ...: return 3.14 * radius * radius ...: In [8]: disk_area(1.5) Out[8]: 7.0649999999999995

Looping over a dictionary
Use iteritems:
In [15]: d = {’a’: 1, ’b’:1.2, ’c’:1j} In [15]: for key, val in d.iteritems(): ....: print(’Key: %s has value: %s’ % (key, val)) ....: ....: Key: a has value: 1 Key: c has value: 1j Key: b has value: 1.2

Note: By default, functions return None. Note: Note the syntax to deﬁne a function: • the def keyword; • is followed by the function’s name, then • the arguments of the function are given between brackets followed by a colon. • the function body ; • and return object for optionally returning values. Parameters Mandatory parameters (positional arguments)

List Comprehensions
In [16]: [i**2 for i in range(4)] Out[16]: [0, 1, 4, 9]

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In [81]: def double_it(x): ....: return x * 2 ....: In [82]: double_it(3) Out[82]: 6 In [83]: double_it() --------------------------------------------------------------------------TypeError Traceback (most recent call last) /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/<ipython console> in <module>() TypeError: double_it() takes exactly 1 argument (0 given)

In [106]: slicer(rhyme, start=1, stop=4, step=2) Out[106]: [’fish,’, ’fish,’]

The order of the keyword arguments does not matter:
In [107]: slicer(rhyme, step=2, start=1, stop=4) Out[107]: [’fish,’, ’fish,’]

but it is good practice to use the same ordering as the function’s deﬁnition. Keyword arguments are a very convenient feature for deﬁning functions with a variable number of arguments, especially when default values are to be used in most calls to the function. Passed by value Can you modify the value of a variable inside a function? Most languages (C, Java, ...) distinguish “passing by value” and “passing by reference”. In Python, such a distinction is somewhat artiﬁcial, and it is a bit subtle whether your variables are going to be modiﬁed or not. Fortunately, there exist clear rules. Parameters to functions are refereence to objects, which are passed by value. When you pass a variable to a function, python passes the reference to the object to which the variable refers (the value). Not the variable itself. If the value is immutable, the function does not modify the caller’s variable. If the value is mutable, the function may modify the caller’s variable in-place:
>>> def try_to_modify(x, y, z): ... x = 23 ... y.append(42) ... z = [99] # new reference ... print(x) ... print(y) ... print(z) ... >>> a = 77 # immutable variable >>> b = [99] # mutable variable >>> c = [28] >>> try_to_modify(a, b, c) 23 [99, 42] [99] >>> print(a) 77 >>> print(b) [99, 42] >>> print(c) [28]

Optional parameters (keyword or named arguments)
In [84]: def double_it(x=2): ....: return x * 2 ....: In [85]: double_it() Out[85]: 4 In [86]: double_it(3) Out[86]: 6

Keyword arguments allow you to specify default values. Warning: Default values are evaluated when the function is deﬁned, not when it is called.
In [124]: bigx = 10 In [125]: def double_it(x=bigx): .....: return x * 2 .....: In [126]: bigx = 1e9 In [128]: double_it() Out[128]: 20 # Now really big

More involved example implementing python’s slicing:
In [98]: def slicer(seq, start=None, stop=None, step=None): ....: """Implement basic python slicing.""" ....: return seq[start:stop:step] ....: In [101]: rhyme = ’one fish, two fish, red fish, blue fish’.split() In [102]: rhyme Out[102]: [’one’, ’fish,’, ’two’, ’fish,’, ’red’, ’fish,’, ’blue’, ’fish’] In [103]: slicer(rhyme) Out[103]: [’one’, ’fish,’, ’two’, ’fish,’, ’red’, ’fish,’, ’blue’, ’fish’] In [104]: slicer(rhyme, step=2) Out[104]: [’one’, ’two’, ’red’, ’blue’] In [105]: slicer(rhyme, 1, step=2) Out[105]: [’fish,’, ’fish,’, ’fish,’, ’fish’]

Functions have a local variable table. Called a local namespace. The variable x only exists within the function foo. Global variables Variables declared outside the function can be referenced within the function:
In [114]: x = 5 In [115]: def addx(y): .....: return x + y .....:

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In [116]: addx(10) Out[116]: 15

....: In [68]: funcname ? Type: function Base Class: <type ’function’> String Form: <function funcname at 0xeaa0f0> Namespace: Interactive File: /Users/cburns/src/scipy2009/.../<ipython console> Definition: funcname(params) Docstring: Concise one-line sentence describing the function. Extended summary which can contain multiple paragraphs.

But these “global” variables cannot be modiﬁed within the function, unless declared global in the function. This doesn’t work:
In [117]: def setx(y): .....: x = y .....: print(’x is %d’ % x) .....: .....: In [118]: setx(10) x is 10 In [120]: x Out[120]: 5

Note: Docstring guidelines For the sake of standardization, the Docstring Conventions webpage documents the semantics and conventions associated with Python docstrings. Also, the Numpy and Scipy modules have deﬁned a precised standard for documenting scientiﬁc functions, that you may want to follow for your own functions, with a Parameters section, an Examples section, etc. See http://projects.scipy.org/numpy/wiki/CodingStyleGuidelines#docstring-standard and http://projects.scipy.org/numpy/browser/trunk/doc/example.py#L37 Functions are objects Functions are ﬁrst-class objects, which means they can be: • assigned to a variable • an item in a list (or any collection) • passed as an argument to another function.
In [38]: va = variable_args

This works:
In [121]: def setx(y): .....: global x .....: x = y .....: print(’x is %d’ % x) .....: .....: In [122]: setx(10) x is 10 In [123]: x Out[123]: 10

Variable number of parameters Special forms of parameters: • *args: any number of positional arguments packed into a tuple • **kwargs: any number of keyword arguments packed into a dictionary
In [35]: def variable_args(*args, **kwargs): ....: print ’args is’, args ....: print ’kwargs is’, kwargs ....: In [36]: variable_args(’one’, ’two’, x=1, y=2, z=3) args is (’one’, ’two’) kwargs is {’y’: 2, ’x’: 1, ’z’: 3}

In [39]: va(’three’, x=1, y=2) args is (’three’,) kwargs is {’y’: 2, ’x’: 1}

Methods Methods are functions attached to objects. You’ve seen these in our examples on lists, dictionaries, strings, etc... Exercices

Exercice: Quicksort Implement the quicksort algorithm, as deﬁned by wikipedia:
function quicksort(array) var list less, greater if length(array) < 2 return array select and remove a pivot value pivot from array for each x in array if x < pivot + 1 then append x to less else append x to greater return concatenate(quicksort(less), pivot, quicksort(greater))

Docstrings Documention about what the function does and it’s parameters. General convention:
In [67]: def funcname(params): ....: """Concise one-line sentence describing the function. ....: ....: Extended summary which can contain multiple paragraphs. ....: """ ....: # function body ....: pass

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Exercice: Fibonacci sequence Write a function that displays the n ﬁrst terms of the Fibonacci sequence, deﬁned by: • u_0 = 1; u_1 = 1 • u_(n+2) = u_(n+1) + u_n

$ python file.py test arguments [’file.py’, ’test’, ’arguments’]

Note: Don’t implement option parsing yourself. Use modules such as optparse. Importing objects from modules

1.3.6 Reusing code: scripts and modules
For now, we have typed all instructions in the interpreter. For longer sets of instructions we need to change tack and write the code in text ﬁles (using a text editor), that we will call either scripts or modules. Use your favorite text editor (provided it offers syntax highlighting for Python), or the editor that comes with the Scientiﬁc Python Suite you may be using (e.g., Scite with Python(x,y)). Scripts Let us ﬁrst write a script, that is a ﬁle with a sequence of instructions that are executed each time the script is called. Instructions may be e.g. copied-and-pasted from the interpreter (but take care to respect indentation rules!). The extension for Python ﬁles is .py. Write or copy-and-paste the following lines in a ﬁle called test.py
message = "Hello how are you?" for word in message.split(): print word

In [1]: import os In [2]: os Out[2]: <module ’os’ from ’ / usr / lib / python2.6 / os.pyc ’ > In [3]: os.listdir(’.’) Out[3]: [’conf.py’, ’basic_types.rst’, ’control_flow.rst’, ’functions.rst’, ’python_language.rst’, ’reusing.rst’, ’file_io.rst’, ’exceptions.rst’, ’workflow.rst’, ’index.rst’]

And also:
In [4]: from os import listdir

Let us now execute the script interactively, that is inside the Ipython interpreter. This is maybe the most common use of scripts in scientiﬁc computing. • in Ipython, the syntax to execute a script is %run script.py. For example,
In [1]: %run test.py Hello how are you? In [2]: message Out[2]: ’Hello how are you?’

Importing shorthands:
In [5]: import numpy as np

Warning:
from os import *

The script has been executed. Moreover the variables deﬁned in the script (such as message) are now available inside the interpeter’s namespace. Other interpreters also offer the possibility to execute scripts (e.g., execfile in the plain Python interpreter, etc.). It is also possible In order to execute this script as a standalone program, by executing the script inside a shell terminal (Linux/Mac console or cmd Windows console). For example, if we are in the same directory as the test.py ﬁle, we can execute this in a console:
epsilon:~/sandbox$ python test.py Hello how are you?

Do not do it. • Makes the code harder to read and understand: where do symbols come from? • Makes it impossible to guess the functionality by the context and the name (hint: os.name is the name of the OS), and to proﬁt usefully from tab completion. • Restricts the variable names you can use: os.name might override name, or vise-versa. • Creates possible name clashes between modules. • Makes the code impossible to statically check for undeﬁned symbols. Modules are thus a good way to organize code in a hierarchical way. Actually, all the scientiﬁc computing tools we are going to use are modules:
>>> import numpy as np # data arrays >>> np.linspace(0, 10, 6) array([ 0., 2., 4., 6., 8., 10.]) >>> import scipy # scientific computing

In Python(x,y) software, Ipython(x,y) execute the following imports at startup:
>>> >>> >>> >>> import numpy import numpy as np from pylab import * import scipy

Standalone scripts may also take command-line arguments In file.py:
import sys print sys.argv

and it is not necessary to re-import these modules. 22 1.3. A (very short) introduction to Python 23

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Creating modules If we want to write larger and better organized programs (compared to simple scripts), where some objects are deﬁned, (variables, functions, classes) and that we want to reuse several times, we have to create our own modules. Let us create a module demo contained in the ﬁle demo.py:
" A demo module. " def print_b(): " Prints b " print(’b’) def print_a(): " Prints a " print(’a’)

’c’, ’d’, ’print_a’, ’print_b’]

In [8]: demo. demo.__builtins__ demo.__class__ demo.__delattr__ demo.__dict__ demo.__doc__ demo.__file__ demo.__format__ demo.__getattribute__ demo.__hash__

demo.__init__ demo.__name__ demo.__new__ demo.__package__ demo.__reduce__ demo.__reduce_ex__ demo.__repr__ demo.__setattr__ demo.__sizeof__

demo.__str__ demo.__subclasshook__ demo.c demo.d demo.print_a demo.print_b demo.py demo.pyc

c = 2 d = 2

Importing objects from modules into the main namespace
In [9]: from demo import print_a, print_b In [10]: whos Variable Type Data/Info -------------------------------demo module <module ’demo’ from ’demo.py’> print_a function <function print_a at 0xb7421534> print_b function <function print_b at 0xb74214c4> In [11]: print_a() a

In this ﬁle, we deﬁned two functions print_a and print_b. Suppose we want to call the print_a function from the interpreter. We could execute the ﬁle as a script, but since we just want to have access to the function test_a, we are rather going to import it as a module. The syntax is as follows.
In [1]: import demo

In [2]: demo.print_a() a In [3]: demo.print_b() b

Importing the module gives access to its objects, using the module.object syntax. Don’t forget to put the module’s name before the object’s name, otherwise Python won’t recognize the instruction. Introspection
In [4]: demo ? Type: module Base Class: <type ’module’> String Form: <module ’demo’ from ’demo.py’> Namespace: Interactive File: /home/varoquau/Projects/Python_talks/scipy_2009_tutorial/source/demo.py Docstring: A demo module.

Warning: Module caching Modules are cached: if you modify demo.py and re-import it in the old session, you will get the old one. Solution:
In [10]: reload(demo)

‘__main__’ and module loading File demo2.py:
import sys def print_a(): " Prints a " print(’a’) print sys.argv if __name__ == ’__main__’: print_a()

In [5]: who demo In [6]: whos Variable Type Data/Info -----------------------------demo module <module ’demo’ from ’demo.py’> In [7]: dir(demo) Out[7]: [’__builtins__’, ’__doc__’, ’__file__’, ’__name__’, ’__package__’,

Importing it:
In [11]: import demo2 b In [12]: import demo2

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In [13]: %run demo2 b a

This method is not very robust, however, because it makes the code less portable (user-dependent path) and because you have to add the directory to your sys.path each time you want to import from a module in this directory. See http://docs.python.org/tutorial/modules.html for more information about modules. Packages A directory that contains many modules is called a package. A package is a module with submodules (which can have submodules themselves, etc.). A special ﬁle called __init__.py (which may be empty) tells Python that the directory is a Python package, from which modules can be imported.
sd-2116 /usr/lib/python2.6/dist-packages/scipy $ ls [17:07] cluster/ io/ README.txt@ stsci/ __config__.py@ LATEST.txt@ setup.py@ __svn_version__.py@ __config__.pyc lib/ setup.pyc __svn_version__.pyc constants/ linalg/ setupscons.py@ THANKS.txt@ fftpack/ linsolve/ setupscons.pyc TOCHANGE.txt@ __init__.py@ maxentropy/ signal/ version.py@ __init__.pyc misc/ sparse/ version.pyc INSTALL.txt@ ndimage/ spatial/ weave/ integrate/ odr/ special/ interpolate/ optimize/ stats/ sd-2116 /usr/lib/python2.6/dist-packages/scipy $ cd ndimage [17:07] sd-2116 /usr/lib/python2.6/dist-packages/scipy/ndimage $ ls [17:07] doccer.py@ fourier.pyc interpolation.py@ morphology.pyc doccer.pyc info.py@ interpolation.pyc _nd_image.so setupscons.py@ filters.py@ info.pyc measurements.py@ _ni_support.py@ setupscons.pyc filters.pyc __init__.py@ measurements.pyc _ni_support.pyc fourier.py@ __init__.pyc morphology.py@ setup.py@

Scripts or modules? How to organize your code Note: Rule of thumb • Sets of instructions that are called several times should be written inside functions for better code reusability. • Functions (or other bits of code) that are called from several scripts should be written inside a module, so that only the module is imported in the different scripts (do not copy-and-paste your functions in the different scripts!). Note: How to import a module from a remote directory? Many solutions exist, depending mainly on your operating system. When the import mymodule statement is executed, the module mymodule is searched in a given list of directories. This list includes a list of installationdependent default path (e.g., /usr/lib/python) as well as the list of directories speciﬁed by the environment variable PYTHONPATH. The list of directories searched by Python is given by the sys.path variable
In [1]: import sys In [2]: sys.path Out[2]: [’’, ’/usr/bin’, ’/usr/local/include/enthought.traits-1.1.0’, ’/usr/lib/python2.6’, ’/usr/lib/python2.6/plat-linux2’, ’/usr/lib/python2.6/lib-tk’, ’/usr/lib/python2.6/lib-old’, ’/usr/lib/python2.6/lib-dynload’, ’/usr/lib/python2.6/dist-packages’, ’/usr/lib/pymodules/python2.6’, ’/usr/lib/pymodules/python2.6/gtk-2.0’, ’/usr/lib/python2.6/dist-packages/wx-2.8-gtk2-unicode’, ’/usr/local/lib/python2.6/dist-packages’, ’/usr/lib/python2.6/dist-packages’, ’/usr/lib/pymodules/python2.6/IPython/Extensions’, u’/home/gouillar/.ipython’]

setup.pyc

tests/

From Ipython:
In [1]: import scipy In [2]: scipy.__file__ Out[2]: ’/usr/lib/python2.6/dist-packages/scipy/__init__.pyc’ In [3]: import scipy.version In [4]: scipy.version.version Out[4]: ’0.7.0’ In [5]: import scipy.ndimage.morphology In [6]: from scipy.ndimage import morphology In [17]: morphology.binary_dilation ? Type: function Base Class: <type ’function’> String Form: <function binary_dilation at 0x9bedd84> Namespace: Interactive File: /usr/lib/python2.6/dist-packages/scipy/ndimage/morphology.py Definition: morphology.binary_dilation(input, structure=None, iterations=1, mask=None, output=None, border_value=0, origin=0, brute_force=False) Docstring: Multi-dimensional binary dilation with the given structure.

Modules must be located in the search path, therefore you can: • write your own modules within directories already deﬁned in the search path (e.g. ‘/usr/local/lib/python2.6/dist-packages’). You may use symbolic links (on Linux) to keep the code somewhere else. • modify the environment variable PYTHONPATH to include the directories containing the user-deﬁned modules. On Linux/Unix, add the following line to a ﬁle read by the shell at startup (e.g. /etc/proﬁle, .proﬁle)
export PYTHONPATH=$PYTHONPATH:/home/emma/user_defined_modules

On Windows, http://support.microsoft.com/kb/310519 explains how to handle environment variables. • or modify the sys.path variable itself within a Python script.
import sys new_path = ’/home/emma/user_defined_modules’ if new_path not in sys.path: sys.path.append(new_path)

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An output array can optionally be provided. The origin parameter controls the placement of the filter. If no structuring element is provided an element is generated with a squared connectivity equal to one. The dilation operation is repeated iterations times. If iterations is less than 1, the dilation is repeated until the result does not change anymore. If a mask is given, only those elements with a true value at the corresponding mask element are modified at each iteration.

>>> f = open(’workfile’, ’w’) # opens the workfile file >>> type(f) <type ’file’> >>> f.write(’This is a test \nand another test’) >>> f.close()

To read from a ﬁle
In [1]: f = open(’workfile’, ’r’) In [2]: s = f.read() In [3]: print(s) This is a test and another test In [4]: f.close()

Good practices Note: Good practices • Indentation: no choice! Indenting is compulsory in Python. Every commands block following a colon bears an additional indentation level with respect to the previous line with a colon. One must therefore indent after def f(): or while:. At the end of such logical blocks, one decreases the indentation depth (and re-increases it if a new block is entered, etc.) Strict respect of indentation is the price to pay for getting rid of { or ; characters that delineate logical blocks in other languages. Improper indentation leads to errors such as
-----------------------------------------------------------IndentationError: unexpected indent (test.py, line 2)

For more details: http://docs.python.org/tutorial/inputoutput.html Iterating over a ﬁle
In [6]: f = open(’workfile’, ’r’) In [7]: for line in f: ...: print line ...: ...: This is a test and another test In [8]: f.close()

All this indentation business can be a bit confusing in the beginning. However, with the clear indentation, and in the absence of extra characters, the resulting code is very nice to read compared to other languages. • Indentation depth: Inside your text editor, you may choose to indent with any positive number of spaces (1, 2, 3, 4, ...). However, it is considered good practice to indent with 4 spaces. You may conﬁgure your editor to map the Tab key to a 4-space indentation. In Python(x,y), the editor Scite is already conﬁgured this way. • Style guidelines Long lines: you should not write very long lines that span over more than (e.g.) 80 characters. Long lines can be broken with the \ character
>>> long_line = "Here is a very very long line \ ... that we break in two parts."

File modes
• Read-only: r • Write-only: w – Note: Create a new ﬁle or overwrite existing ﬁle. • Append a ﬁle: a • Read and Write: r+ • Binary mode: b – Note: Use for binary ﬁles, especially on Windows.

Spaces Write well-spaced code: put whitespaces after commas, around arithmetic operators, etc.:
>>> a = 1 # yes >>> a=1 # too cramped

A certain number of rules for writing “beautiful” code (and more importantly using the same conventions as anybody else!) are given in the Style Guide for Python Code. • Use meaningful object names

1.3.8 Standard Library 1.3.7 Input and Output
To be exhaustive, here are some informations about input and output in Python. Since we will use the Numpy methods to read and write ﬁles, you may skip this chapter at ﬁrst reading. We write or read strings to/from ﬁles (other types must be converted to strings). To write in a ﬁle: Note: Reference document for this section: • The Python Standard Library documentation: http://docs.python.org/library/index.html • Python Essential Reference, David Beazley, Addison-Wesley Professional os module: operating system functionality “A portable way of using operating system dependent functionality.” 1.3. A (very short) introduction to Python 28 1.3. A (very short) introduction to Python 29

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Directory and ﬁle manipulation
Current directory:
In [17]: os.getcwd() Out[17]: ’/Users/cburns/src/scipy2009/scipy_2009_tutorial/source’

In [70]: fp = open(’junk.txt’, ’w’) In [71]: fp.close() In [72]: a = os.path.abspath(’junk.txt’) In [73]: a Out[73]: ’/Users/cburns/src/scipy2009/scipy_2009_tutorial/source/junk.txt’ In [74]: os.path.split(a) Out[74]: (’/Users/cburns/src/scipy2009/scipy_2009_tutorial/source’, ’junk.txt’) In [78]: os.path.dirname(a) Out[78]: ’/Users/cburns/src/scipy2009/scipy_2009_tutorial/source’ In [79]: os.path.basename(a) Out[79]: ’junk.txt’ In [80]: os.path.splitext(os.path.basename(a)) Out[80]: (’junk’, ’.txt’) In [84]: os.path.exists(’junk.txt’) Out[84]: True In [86]: os.path.isfile(’junk.txt’) Out[86]: True In [87]: os.path.isdir(’junk.txt’) Out[87]: False In [88]: os.path.expanduser(’~/local’) Out[88]: ’/Users/cburns/local’ In [92]: os.path.join(os.path.expanduser(’~’), ’local’, ’bin’) Out[92]: ’/Users/cburns/local/bin’

List a directory:
In [31]: os.listdir(os.curdir) Out[31]: [’.index.rst.swo’, ’.python_language.rst.swp’, ’.view_array.py.swp’, ’_static’, ’_templates’, ’basic_types.rst’, ’conf.py’, ’control_flow.rst’, ’debugging.rst’, ...

Make a directory:
In [32]: os.mkdir(’junkdir’) In [33]: ’junkdir’ in os.listdir(os.curdir) Out[33]: True

Rename the directory:
In [36]: os.rename(’junkdir’, ’foodir’) In [37]: ’junkdir’ in os.listdir(os.curdir) Out[37]: False In [38]: ’foodir’ in os.listdir(os.curdir) Out[38]: True In [41]: os.rmdir(’foodir’) In [42]: ’foodir’ in os.listdir(os.curdir) Out[42]: False

Running an external command
In [8]: os.system(’ls *’) conf.py debug_file.py demo2.py~ demo.py demo.pyc conf.py~ demo2.py demo2.pyc demo.py~ my_file.py my_file.py~ pi_wallis_image.py

Delete a ﬁle:
In [44]: fp = open(’junk.txt’, ’w’) In [45]: fp.close() In [46]: ’junk.txt’ in os.listdir(os.curdir) Out[46]: True In [47]: os.remove(’junk.txt’) In [48]: ’junk.txt’ in os.listdir(os.curdir) Out[48]: False

Walking a directory
os.path.walk generates a list of ﬁlenames in a directory tree.
In [10]: for dirpath, dirnames, filenames in os.walk(os.curdir): ....: for fp in filenames: ....: print os.path.abspath(fp) ....: ....: /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/.index.rst.swo /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/.view_array.py.swp /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/basic_types.rst /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/conf.py /Users/cburns/src/scipy2009/scipy_2009_tutorial/source/control_flow.rst ...

os.path: path manipulations
os.path provides common operations on pathnames.

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

Environment variables:
In [9]: import os In [11]: os.environ.keys() Out[11]: [’_’, ’FSLDIR’, ’TERM_PROGRAM_VERSION’, ’FSLREMOTECALL’, ’USER’, ’HOME’, ’PATH’, ’PS1’, ’SHELL’, ’EDITOR’, ’WORKON_HOME’, ’PYTHONPATH’, ... In [12]: os.environ[’PYTHONPATH’] Out[12]: ’.:/Users/cburns/src/utils:/Users/cburns/src/nitools: /Users/cburns/local/lib/python2.5/site-packages/: /usr/local/lib/python2.5/site-packages/: /Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5’ In [16]: os.getenv(’PYTHONPATH’) Out[16]: ’.:/Users/cburns/src/utils:/Users/cburns/src/nitools: /Users/cburns/local/lib/python2.5/site-packages/: /usr/local/lib/python2.5/site-packages/: /Library/Frameworks/Python.framework/Versions/2.5/lib/python2.5’

In [117]: sys.platform Out[117]: ’darwin’ In [118]: sys.version Out[118]: ’2.5.2 (r252:60911, Feb 22 2008, 07:57:53) \n [GCC 4.0.1 (Apple Computer, Inc. build 5363)]’ In [119]: sys.prefix Out[119]: ’/Library/Frameworks/Python.framework/Versions/2.5’

• List of command line arguments passed to a Python script:
In [100]: sys.argv Out[100]: [’/Users/cburns/local/bin/ipython’]

sys.path is a list of strings that speciﬁes the search path for modules. Initialized from PYTHONPATH:
In [121]: sys.path Out[121]: [’’, ’/Users/cburns/local/bin’, ’/Users/cburns/local/lib/python2.5/site-packages/grin-1.1-py2.5.egg’, ’/Users/cburns/local/lib/python2.5/site-packages/argparse-0.8.0-py2.5.egg’, ’/Users/cburns/local/lib/python2.5/site-packages/urwid-0.9.7.1-py2.5.egg’, ’/Users/cburns/local/lib/python2.5/site-packages/yolk-0.4.1-py2.5.egg’, ’/Users/cburns/local/lib/python2.5/site-packages/virtualenv-1.2-py2.5.egg’, ...

pickle: easy persistence Useful to store arbritrary objects to a ﬁle. Not safe or fast!
In [1]: import pickle In [2]: l = [1, None, ’Stan’] In [3]: pickle.dump(l, file(’test.pkl’, ’w’)) In [4]: pickle.load(file(’test.pkl’)) Out[4]: [1, None, ’Stan’]

shutil: high-level ﬁle operations The shutil provides useful ﬁle operations: • shutil.rmtree: Recursively delete a directory tree. • shutil.move: Recursively move a ﬁle or directory to another location. • shutil.copy: Copy ﬁles or directories. glob: Pattern matching on ﬁles The glob module provides convenient ﬁle pattern matching. Find all ﬁles ending in .txt:
In [18]: import glob In [19]: glob.glob(’*.txt’) Out[19]: [’holy_grail.txt’, ’junk.txt’, ’newfile.txt’]

Exercise Write a program to search your PYTHONPATH for the module site.py. path_site

1.3.9 Exceptions handling in Python
It is highly unlikely that you haven’t yet raised Exceptions if you have typed all the previous commands of the tutorial. For example, you may have raised an exception if you entered a command with a typo. Exceptions are raised by different kinds of errors arising when executing Python code. In you own code, you may also catch errors, or deﬁne custom error types. Exceptions Exceptions are raised by errors in Python:

sys module: system-speciﬁc information System-speciﬁc information related to the Python interpreter. • Which version of python are you running and where is it installed:

1.3. A (very short) introduction to Python

Python Scientiﬁc lecture notes, Release 2010

In [1]: 1/0 --------------------------------------------------------------------------ZeroDivisionError: integer division or modulo by zero In [2]: 1 + ’e’ --------------------------------------------------------------------------TypeError: unsupported operand type(s) for +: ’int’ and ’str’ In [3]: d = {1:1, 2:2} In [4]: d[3] --------------------------------------------------------------------------KeyError: 3 In [5]: l = [1, 2, 3] In [6]: l[4] --------------------------------------------------------------------------IndexError: list index out of range In [7]: l.foobar --------------------------------------------------------------------------AttributeError: ’list’ object has no attribute ’foobar’

Easier to ask for forgiveness than for permission
In [11]: def print_sorted(collection): ....: try: ....: collection.sort() ....: except AttributeError: ....: pass ....: print(collection) ....: ....: In [12]: print_sorted([1, 3, 2]) [1, 2, 3] In [13]: print_sorted(set((1, 3, 2))) set([1, 2, 3]) In [14]: print_sorted(’132’) 132

Raising exceptions • Capturing and reraising an exception:
In [15]: def filter_name(name): ....: try: ....: name = name.encode(’ascii’) ....: except UnicodeError, e: ....: if name == ’Gaël’: ....: print(’OK, Gaël’) ....: else: ....: raise e ....: return name ....: In [16]: filter_name(’Gaël’) OK, Gaël Out[16]: ’Ga\xc3\xabl’

Different types of exceptions for different errors. Catching exceptions

try/except
In [8]: while True: ....: try: ....: x = int(raw_input(’Please enter a number: ’)) ....: break ....: except ValueError: ....: print(’That was no valid number. Try again...’) ....: ....: Please enter a number: a That was no valid number. Try again... Please enter a number: 1 In [9]: x Out[9]: 1

In [17]: filter_name(’Stéfan’) --------------------------------------------------------------------------UnicodeDecodeError: ’ascii’ codec can’t decode byte 0xc3 in position 2: ordinal not in rang

• Exceptions to pass messages between parts of the code:
In [17]: def achilles_arrow(x): ....: if abs(x - 1) < 1e-3: ....: raise StopIteration ....: x = 1 - (1-x)/2. ....: return x ....: In [18]: x = 0 In [19]: while True: ....: try: ....: x = achilles_arrow(x) ....: except StopIteration: ....: break ....: ....:

try/ﬁnally
In [10]: try: ....: x = int(raw_input(’Please enter a number: ’)) ....: finally: ....: print(’Thank you for your input’) ....: ....: Please enter a number: a Thank you for your input --------------------------------------------------------------------------ValueError: invalid literal for int() with base 10: ’a’

Important for resource management (e.g. closing a ﬁle) 1.3. A (very short) introduction to Python 34

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In [20]: x Out[20]: 0.9990234375

The array: the basic tool for scientiﬁc computing

Use exceptions to notify certain conditions are met (e.g. StopIteration) or not (e.g. custom error raising)

1.3.10 Object-oriented programming (OOP)
Python supports object-oriented programming (OOP). The goals of OOP are: • to organize the code, and • to re-use code in similar contexts. Here is a small example: we create a Student class, which is an object gathering several custom functions (methods) and variables (attributes), we will be able to use:
>>> ... ... ... ... ... ... ... >>> >>> >>> class Student(object): def __init__(self, name): self.name = name def set_age(self, age): self.age = age def set_major(self, major): self.major = major anna = Student(’anna’) anna.set_age(21) anna.set_major(’physics’)

Frequent manipulation of discrete sorted datasets : • discretized time of an experiment/simulation • signal recorded by a measurement device • pixels of an image, ... The Numpy module allows to • create such datasets in one shot • realize batch operations on data arrays (no loops on their items) Data arrays := numpy.ndarray

1.4.1 Creating NumPy data arrays
A small introductory example:
>>> import numpy as np >>> a = np.array([0, 1, 2]) >>> a array([0, 1, 2]) >>> print a [0 1 2] >>> b = np.array([[0., 1.], [2., 3.]]) >>> b array([[ 0., 1.], [ 2., 3.]])

In the previous example, the Student class has __init__, set_age and set_major methods. Its attributes are name, age and major. We can call these methods and attributes with the following notation: classinstance.method or classinstance.attribute. The __init__ constructor is a special method we call with: MyClass(init parameters if any). Now, suppose we want to create a new class MasterStudent with the same methods and attributes as the previous one, but with an additional internship attribute. We won’t copy the previous class, but inherit from it:
>>> class MasterStudent(Student): ... internship = ’mandatory, from March to June’ ... >>> james = MasterStudent(’james’) >>> james.internship ’mandatory, from March to June’ >>> james.set_age(23) >>> james.age 23

In practice, we rarely enter items one by one... • Evenly spaced values:
>>> import numpy as np >>> a = np.arange(10) # de 0 a n-1 >>> a array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> b = np.arange(1., 9., 2) # syntax : start, end, step >>> b array([ 1., 3., 5., 7.])

The MasterStudent class inherited from the Student attributes and methods. Thanks to classes and object-oriented programming, we can organize code with different classes corresponding to different objects we encounter (an Experiment class, an Image class, a Flow class, etc.), with their own methods and attributes. Then we can use inheritance to consider variations around a base class and re-use code. Ex : from a Flow base class, we can create derived StokesFlow, TurbulentFlow, PotentialFlow, etc.

or by specifying the number of points:
>>> c = >>> c array([ >>> d = >>> d array([ np.linspace(0, 1, 6) 0. , 0.2, 0.4, 0.6, 0.8, 1. ]) np.linspace(0, 1, 5, endpoint=False) 0. , 0.2, 0.4, 0.6, 0.8])

1.4 NumPy: creating and manipulating numerical data
authors Emmanuelle Gouillart, Didrik Pinte, Gaël Varoquaux

• Constructors for common arrays:
>>> a = np.ones((3,3)) >>> a array([[ 1., 1., 1.],

1.4. NumPy: creating and manipulating numerical data

Python Scientiﬁc lecture notes, Release 2010

[ 1., 1., 1.], [ 1., 1., 1.]]) >>> a.dtype dtype(’float64’) >>> b = np.ones(5, dtype=np.int) >>> b array([1, 1, 1, 1, 1]) >>> c = np.zeros((2,2)) >>> c array([[ 0., 0.], [ 0., 0.]]) >>> d = np.eye(3) >>> d array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]])

2D arrays (such as images)
In [48]: In [49]: In [50]: Out[50]: In [51]: In [52]: In [53]: Out[53]: In [54]: # 30x30 array with random floats btw 0 and 1 image = np.random.rand(30,30) imshow(image) <matplotlib.image.AxesImage object at 0x9e954ac> gray() hot() imshow(image, cmap=cm.gray) <matplotlib.image.AxesImage object at 0xa23972c> axis(’off’) # we remove ticks and labels

1.4.2 Graphical data representation : matplotlib and Mayavi
Now that we have our ﬁrst data arrays, we are going to visualize them. Matplotlib is a 2D plotting package. We can import its functions as below:
>>> import pylab >>> # or >>> from pylab import * # imports everything in the namespace

If you launched Ipython with python(x,y), or with ipython -pylab (under Linux), all the functions/objects of pylab are already imported, without needing from pylab import *. In the remainder of this tutorial, we assume you have already run from pylab import * or ipython -pylab: as a consequence, we won’t write pylab.function() but directly function. 1D curve plotting
In [6]: a = np.arange(20) In [7]: plot(a, a**2) # line plot Out[7]: [<matplotlib.lines.Line2D object at 0x95abd0c>] In [8]: plot(a, a**2, ’o’) # dotted plot Out[8]: [<matplotlib.lines.Line2D object at 0x95b1c8c>] In [9]: clf() # clear figure In [10]: loglog(a, a**2) Out[10]: [<matplotlib.lines.Line2D object at 0x95abf6c>] In [11]: xlabel(’x’) # a bit too small Out[11]: <matplotlib.text.Text object at 0x98923ec> In [12]: xlabel(’x’, fontsize=26) # bigger Out[12]: <matplotlib.text.Text object at 0x98923ec> In [13]: ylabel(’y’) Out[13]: <matplotlib.text.Text object at 0x9892b8c> In [14]: grid() In [15]: axvline(2) Out[15]: <matplotlib.lines.Line2D object at 0x9b633cc>

There are many other features in matplotlib: color choice, marker size, latex font, inclusions within ﬁgures, histograms, etc. To go further : • matplotlib documentation http://matplotlib.sourceforge.net/contents.html • an example gallery with corresponding sourcecode http://matplotlib.sourceforge.net/gallery.html 3D plotting For 3D visualization, we use another package: Mayavi. A quick example: start with relaunching iPython with these options: ipython -pylab -wthread
In [59]: from enthought.mayavi import mlab In [60]: mlab.figure() get fences failed: -1 param: 6, val: 0 Out[60]: <enthought.mayavi.core.scene.Scene object at 0xcb2677c> In [61]: mlab.surf(image) Out[61]: <enthought.mayavi.modules.surface.Surface object at 0xd0862fc>

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In [62]: mlab.axes() Out[62]: <enthought.mayavi.modules.axes.Axes object at 0xd07892c>

array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> a[0], a[2], a[-1] (0, 2, 9)

Warning! Indexes begin at 0, like other Python sequences (and C/C++). In Fortran or Matlab, indexes begin with 1. For multidimensional arrays, indexes are tuples of integers:
>>> a = np.diag(np.arange(5)) >>> a array([[0, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0], [0, 0, 0, 3, 0], [0, 0, 0, 0, 4]]) >>> a[1,1] 1 >>> a[2,1] = 10 # third line, second column >>> a array([[ 0, 0, 0, 0, 0], [ 0, 1, 0, 0, 0], [ 0, 10, 2, 0, 0], [ 0, 0, 0, 3, 0], [ 0, 0, 0, 0, 4]]) >>> a[1] array([0, 1, 0, 0, 0])

The mayavi/mlab window that opens is interactive : by clicking on the left mouse button you can rotate the image, zoom with the mouse wheel, etc.

Note that: • In 2D, the ﬁrst dimension corresponds to lines, the second to columns. • for an array a with more than one dimension,‘a[0]‘ is interpreted by taking all elements in the unspeciﬁed dimensions.

1.4.4 Slicing
Like indexing, it’s similar to Python sequences slicing:
>>> a = np.arange(10) >>> a array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> a[2:9:3] # [start:end:step] array([2, 5, 8])

Note that the last index is not included!:
>>> a[:4] array([0, 1, 2, 3])

start:end:step is a slice object which represents the set of indexes range(start, end, step). A slice can be explicitly created: For more information on Mayavi : http://code.enthought.com/projects/mayavi/docs/development/html/mayavi/index.html
>>> sl = slice(1, 9, 2) >>> a = np.arange(10) >>> b = 2*a + 1 >>> a, b (array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]), array([ 1, >>> a[sl], b[sl] (array([1, 3, 5, 7]), array([ 3, 7, 11, 15]))

1.4.3 Indexing
The items of an array can be accessed the same way as other Python sequences (list, tuple)
>>> a = np.arange(10) >>> a

9, 11, 13, 15, 17, 19]))

All three slice components are not required: by default, start is 0, end is the last and step is 1:

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Python Scientiﬁc lecture notes, Release 2010

>>> a[1:3] array([1, 2]) >>> a[::2] array([0, 2, 4, 6, 8]) >>> a[3:] array([3, 4, 5, 6, 7, 8, 9])

>>> b array([12, 2, 4, 6, 8]) >>> a # a a été modifié aussi ! array([12, 1, 2, 3, 4, 5, 6,

9])

This behaviour can be surprising at ﬁrst sight... but it allows to save a lot of memory.

Of course, it works with multidimensional arrays:
>>> a = np.eye(5) >>> a array([[ 1., 0., [ 0., 1., [ 0., 0., [ 0., 0., [ 0., 0., >>> a[2:4,:3] #3rd array([[ 0., 0., [ 0., 0.,

1.4.5 Manipulating the shape of arrays
The shape of an array can be retrieved with the ndarray.shape method which returns a tuple with the dimensions of the array:
>>> a = np.arange(10) >>> a.shape (10,) >>> b = np.ones((3,4)) >>> b.shape (3, 4) >>> b.shape[0] # the shape tuple elements can be accessed 3 >>> # an other way of doing the same >>> np.shape(b) (3, 4)

0., 0., 0.], 0., 0., 0.], 1., 0., 0.], 0., 1., 0.], 0., 0., 1.]]) and 4th lines, 3 first columns 1.], 0.]])

All elements speciﬁed by a slice can be easily modiﬁed:
>>> a[:3,:3] = 4 >>> a array([[ 4., 4., [ 4., 4., [ 4., 4., [ 0., 0., [ 0., 0.,

4., 4., 4., 0., 0.,

0., 0., 0., 1., 0.,

0.], 0.], 0.], 0.], 1.]])

Moreover, the length of the ﬁrst dimension can be queried with np.alen (by analogy with len for a list) and the total number of elements with ndarray.size:
>>> np.alen(b) 3 >>> b.size 12

A small illustrated summary of Numpy indexing and slicing...

Several NumPy functions allow to create an array with a different shape, from another array:
>>> a = np.arange(36) >>> b = a.reshape((6, 6)) >>> b array([[ 0, 1, 2, 3, 4, [ 6, 7, 8, 9, 10, [12, 13, 14, 15, 16, [18, 19, 20, 21, 22, [24, 25, 26, 27, 28, [30, 31, 32, 33, 34,

5], 11], 17], 23], 29], 35]])

ndarray.reshape returns a view, not a copy:
>>> b[0,0] = 10 >>> a array([10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35])

An array with a different number of elements can also be created with ndarray.resize: A slicing operation creates a view on the original array, which is just a way of accessing array data. Thus the original array is not copied in memory. When modifying the view, the original array is modiﬁed as well*:
>>> a = np.arange(10) >>> a array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) >>> b = a[::2]; b array([0, 2, 4, 6, 8]) >>> b[0] = 12 >>> a = np.arange(36) >>> a.resize((4,2)) >>> a array([[0, 1], [2, 3], [4, 5], [6, 7]]) >>> b = np.arange(4) >>> b.resize(3, 2)

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Python Scientiﬁc lecture notes, Release 2010

>>> b array([[0, 1], [2, 3], [0, 0]])

In [1]: mkdir python_scripts In [2]: cd python_scripts/ /home/gouillar/python_scripts In [3]: pwd Out[3]: ’/home/gouillar/python_scripts’ In [4]: ls In [5]: np.savetxt(’integers.txt’, np.arange(10)) In [6]: ls integers.txt

A large array can be tiled with a smaller one:
>>> a = np.arange(4).reshape((2,2)) >>> a array([[0, 1], [2, 3]]) >>> np.tile(a, (2,3)) array([[0, 1, 0, 1, 0, 1], [2, 3, 2, 3, 2, 3], [0, 1, 0, 1, 0, 1], [2, 3, 2, 3, 2, 3]])

• os (system routines) and os.path (path management) modules:
>>> import os, os.path >>> current_dir = os.getcwd() >>> current_dir ’/home/gouillar/sandbox’ >>> data_dir = os.path.join(current_dir, ’data’) >>> data_dir ’/home/gouillar/sandbox/data’ >>> if not(os.path.exists(data_dir)): ... os.mkdir(’data’) ... print "created ’data’ folder" ... >>> os.chdir(data_dir) # or in Ipython : cd data

1.4.6 Exercises : some simple array creations
By using miscellaneous constructors, indexing, slicing, and simple operations (+/-/x/:), large arrays with various patterns can be created. Example : create this array:
[[ 0 1 2 3 4] [ 5 6 7 8 9] [10 11 12 13 0] [15 16 17 18 19] [20 21 22 23 24]]

Solution
>>> a = np.arange(25).reshape((5,5)) >>> a[2, 4] = 0

IPython can actually be used like a shell, thanks to its integrated features and the os module. Writing a data array in a ﬁle
>>> a = np.arange(100) >>> a = a.reshape((10, 10))

Exercises : Create the following array with the simplest solution:
[[ [ [ [ 1. 1. 1. 1. 0 0 3 0 0 0 1. 1. 1. 6. 0 0 0 4 0 0 0 0 0 0 5 0 1. 1. 1. 1. 0] 0] 0] 0] 0] 6]] 1.] 1.] 2.] 1.]]

• Writing a text ﬁle (in ASCII):
>>> np.savetxt(’data_a.txt’, a)

• Writing a binary ﬁle (.npy extension, recommended format)
>>> np.save(’data_a.npy’, a)

[[0 [2 [0 [0 [0 [0

Loading a data array from a ﬁle • Reading from a text ﬁle:
>>> b = np.loadtxt(’data_a.txt’)

• Reading from a binary ﬁle:

1.4.7 Real data: read/write arrays from/to ﬁles
Often, our experiments or simulations write some results in ﬁles. These results must then be loaded in Python as NumPy arrays to be able to manipulate them. We also need to save some arrays into ﬁles. Going to the right folder To move in a folder hierarchy: • use the iPython commands: cd, pwd, tab-completion.

>>> c = np.load(’data_a.npy’)

To read matlab data ﬁles scipy.io.loadmat : the matlab structure of a .mat ﬁle is stored as a dictionary. Opening and saving images: imsave and imread
>>> import scipy >>> from pylab import imread, imsave, savefig >>> lena = scipy.lena()

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Python Scientiﬁc lecture notes, Release 2010

>>> imsave(’lena.png’, lena, cmap=cm.gray) >>> lena_reloaded = imread(’lena.png’) >>> imshow(lena_reloaded, cmap=gray) <matplotlib.image.AxesImage object at 0x989e14c> >>> savefig(’lena_figure.png’)

>>> # Note that the line is not always sorted >>> filelist.sort() >>> l2 = np.loadtxt(filelist[2])

Note: arrays can also be created from Excel/Calc ﬁles, HDF5 ﬁles, etc. (but with additional modules not described here: xlrd, pytables, etc.).

1.4.8 Simple mathematical and statistical operations on arrays
Some operations on arrays are natively available in NumPy (and are generally very efﬁcient):
>>> >>> 0 >>> 9 >>> 45 a = np.arange(10) a.min() # or np.min(a) a.max() # or np.max(a) a.sum() # or np.sum(a)

Operations can also be run along an axis, instead of on all elements:
>>> a = np.array([[1, 3], [9, 6]]) >>> a array([[1, 3], [9, 6]]) >>> a.mean(axis=0) # the array contains the mean of each column array([ 5. , 4.5]) >>> a.mean(axis=1) # the array contains the mean of each line array([ 2. , 7.5])

Many other operations are available. We will discover some of them in this course. Note: Arithmetic operations on arrays correspond to operations on each individual element. In particular, the multiplication is not a matrix multiplication (unlike Matlab)! The matrix multiplication is provided by np.dot:
>>> a = np.ones((2,2)) >>> a*a array([[ 1., 1.], [ 1., 1.]]) >>> np.dot(a,a) array([[ 2., 2.], [ 2., 2.]])

Selecting a ﬁle from a list Each line of a will be saved in a different ﬁle:
>>> for i, l in enumerate(a): ... print i, l ... np.savetxt(’line_’+str(i), l) ... 0 [0 1 2 3 4 5 6 7 8 9] 1 [10 11 12 13 14 15 16 17 18 19] 2 [20 21 22 23 24 25 26 27 28 29] 3 [30 31 32 33 34 35 36 37 38 39] 4 [40 41 42 43 44 45 46 47 48 49] 5 [50 51 52 53 54 55 56 57 58 59] 6 [60 61 62 63 64 65 66 67 68 69] 7 [70 71 72 73 74 75 76 77 78 79] 8 [80 81 82 83 84 85 86 87 88 89] 9 [90 91 92 93 94 95 96 97 98 99]

Example : diffusion simulation using a random walk algorithm

What is the typical distance from the origin of a random walker after t left or right jumps?

To get a list of all ﬁles beginning with line, we use the glob module which matches all paths corresponding to a pattern. Example:
>>> import glob >>> filelist = glob.glob(’line*’) >>> filelist [’line_0’, ’line_1’, ’line_2’, ’line_3’, ’line_4’, ’line_5’, ’line_6’, ’line_7’, ’line_8’, ’line_9’]

1.4. NumPy: creating and manipulating numerical data

Python Scientiﬁc lecture notes, Release 2010

5. Check that the number of lines of data equals the number of lines of the organisms. 6. What is the maximal percentage of women in all organisms, for all years taken together? 7. Create an array with the temporal mean of the percentage of women for each organism? (i.e. the mean of data along axis 1). 8. Which organism had the highest percentage of women in 2004? (hint: np.argmax) 9. Create a histogram of the percentage of women the different organisms in 2006 (hint: np.histogram, then matplotlib bar or plot for visulalization) 10. Create an array that contains the organism where the highest women’s percentage is found for the different years. Answers stat_recherche
>>> nreal = 1000 # number of walks >>> tmax = 200 # time during which we follow the walker >>> # We randomly choose all the steps 1 or -1 of the walk >>> walk = 2 * ( np.random.random_integers(0, 1, (nreal,tmax)) - 0.5 ) >>> np.unique(walk) # Verification : all steps are 1 or -1 array([-1., 1.]) >>> # We build the walks by summing steps along the time >>> cumwalk = np.cumsum(walk, axis=1) # axis = 1 : dimension of time >>> sq_distance = cumwalk**2 >>> # We get the mean in the axis of the steps >>> mean_sq_distance = np.mean(sq_distance, axis=0) In In In In [39]: [40]: [41]: [42]: figure() plot(mean_sq_distance) figure() plot(np.sqrt(mean_sq_distance))

1.4.9 Fancy indexing
Numpy arrays can be indexed with slices, but also with boolean or integer arrays (masks). This method is called fancy indexing. Masks
>>> np.random.seed(3) >>> a = np.random.random_integers(0, 20, 15) >>> a array([10, 3, 8, 0, 19, 10, 11, 9, 10, 6, 0, 20, 12, 7, 14]) >>> (a%3 == 0) array([False, True, False, True, False, False, False, True, False, True, True, False, True, False, False], dtype=bool) >>> mask = (a%3 == 0) >>> extract_from_a = a[mask] #one could directly write a[a%3==0] >>> extract_from_a # extract a sub-array with the mask array([ 3, 0, 9, 6, 0, 12])

Extracting a sub-array using a mask produces a copy of this sub-array, not a view:
>>> extract_from_a = -1 >>> a array([10, 3, 8, 0, 19, 10, 11,

9, 10,

0, 20, 12,

7, 14])

Indexing with a mask can be very useful to assign a new value to a sub-array:
>>> a[mask] = 0 >>> a array([10, 0, 8,

0, 19, 10, 11,

0, 10,

0, 20,

7, 14])

Indexing with an array of integers We ﬁnd again that the distance grows like the square root of the time! Exercise : statistics on the number of women in french research (INSEE data) 1. Get the following ﬁles organisms.txt and women_percentage.txt in the data directory. 2. Create a data array by opening the women_percentage.txt ﬁle with np.loadtxt. What is the shape of this array? 3. Columns correspond to year 2006 to 2001. Create a years array with integers corresponding to these years. 4. The different lines correspond to the research organisms whose names are stored in the organisms.txt ﬁle. Create a organisms array by opening this ﬁle. Beware that np.loadtxt creates ﬂoat arrays by default, and it must be speciﬁed to use strings instead: organisms = np.loadtxt(’organisms.txt’, dtype=str)
>>> a = np.arange(10) >>> a[::2] +=3 #to avoid having always the same np.arange(10)... >>> a array([ 3, 1, 5, 3, 7, 5, 9, 7, 11, 9]) >>> a[[2, 5, 1, 8]] # or a[np.array([2, 5, 1, 8])] array([ 5, 5, 1, 11])

Indexing can be done with an array of integers, where the same index is repeated several time:
>>> a[[2, 3, 2, 4, 2]] array([5, 3, 5, 7, 5])

New values can be assigned with this kind of indexing:

1.4. NumPy: creating and manipulating numerical data

Python Scientiﬁc lecture notes, Release 2010

>>> a[[9, 7]] = -10 >>> a array([ 3, 1, 5, 3, >>> a[[2, 3, 2, 4, 2]] +=1 >>> a array([ 3, 1, 6, 4,

Answers stat_recherche
7, 5, 9, -10, 11, -10])

1.4.10 Broadcasting
Basic operations on numpy arrays (addition, etc.) are done element by element, thus work on arrays of the same size. Nevertheless, it’s possible to do operations on arrays of different sizes if numpy can transform these arrays so that they all have the same size: this conversion is called broadcasting. The image below gives an example of broadcasting:

9, -10,

11, -10])

When a new array is created by indexing with an array of integers, the new array has the same shape than the array of integers:
>>> a = np.arange(10) >>> idx = np.array([[3, 4], [9, 7]]) >>> a[idx] array([[3, 4], [9, 7]]) >>> b = np.arange(10) >>> a = np.arange(12).reshape(3, 4) >>> a array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> i = np.array([0, 1, 1, 2]) >>> j = np.array([2, 1, 3, 3]) >>> a[i, j] array([ 2, 5, 7, 11]) >>> i = np.array([[0, 1], [1, 2]]) >>> j = np.array([[2, 1], [3, 3]]) >>> i array([[0, 1], [1, 2]]) >>> j array([[2, 1], [3, 3]]) >>> a[i, j] array([[ 2, 5], [ 7, 11]])

which gives the following in Ipython:
>>> a = np.arange(0, 40, 10) >>> b = np.arange(0, 3) >>> a = a.reshape((4,1)) # a must be changed into a vertical array >>> a + b array([[ 0, 1, 2], [10, 11, 12], [20, 21, 22], [30, 31, 32]])

We actually already used broadcasting without knowing it!:
>>> a = np.arange(20).reshape((4,5)) >>> a array([[ 0, 1, 2, 3, 4], [ 5, 6, 7, 8, 9], [10, 11, 12, 13, 14], [15, 16, 17, 18, 19]]) >>> a[0] = 1 # we assign an array of dimension 0 to an array of dimension 1 >>> a[:3] = np.arange(1,6) >>> a

Exercise Let’s take the same statistics about the percentage of women in the research (data and organisms arrays) 1. Create a sup30 array of the same size than data with a value of 1 if the value of data is greater than 30%, 0 otherwise. 2. Create an array containing the organisme having the greatest percentage of women of each year.

1.4. NumPy: creating and manipulating numerical data

Python Scientiﬁc lecture notes, Release 2010

array([[ 1, 2, 3, 4, 5], [ 1, 2, 3, 4, 5], [ 1, 2, 3, 4, 5], [15, 16, 17, 18, 19]])

A lot of grid-based or network-based problems can also use broadcasting. For instance, if we want to compute the distance from the origin of points on a 10x10 grid, we can do:
>>> x, y = np.arange(5), np.arange(5) >>> distance = np.sqrt(x**2 + y[:, np.newaxis]**2) >>> distance array([[ 0. , 1. , 2. , 3. , [ 1. , 1.41421356, 2.23606798, 3.16227766, [ 2. , 2.23606798, 2.82842712, 3.60555128, [ 3. , 3.16227766, 3.60555128, 4.24264069, [ 4. , 4.12310563, 4.47213595, 5. ,

We can even use fancy indexing and broadcasting at the same time. Take again the same example as above::
>>> a = np.arange(12).reshape(3,4) >>> a array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11]]) >>> i = np.array( [ [0,1], ... [1,2] ] ) >>> a[i, 2] # same as a[i, 2*np.ones((2,2), dtype=int)] array([[ 2, 6], [ 6, 10]])

4. ], 4.12310563], 4.47213595], 5. ], 5.65685425]])

The values of the distance array can be represented in colour, thanks to the pylab.imshow function (syntax: pylab.imshow(distance). See help for other options).

Broadcasting seems a bit magical, but it is actually quite natural to use it when we want to solve a problem whose output data is an array with more dimensions than input data. Example: let’s construct an array of distances (in miles) between cities of Route 66: Chicago, Springﬁeld, SaintLouis, Tulsa, Oklahoma City, Amarillo, Santa Fe, Albucquerque, Flagstaff and Los Angeles.
>>> mileposts = np.array([0, 198, 303, 736, 871, 1175, 1475, 1544, ... 1913, 2448]) >>> ditance_array = np.abs(mileposts - mileposts[:,np.newaxis]) >>> ditance_array array([[ 0, 198, 303, 736, 871, 1175, 1475, 1544, 1913, 2448], [ 198, 0, 105, 538, 673, 977, 1277, 1346, 1715, 2250], [ 303, 105, 0, 433, 568, 872, 1172, 1241, 1610, 2145], [ 736, 538, 433, 0, 135, 439, 739, 808, 1177, 1712], [ 871, 673, 568, 135, 0, 304, 604, 673, 1042, 1577], [1175, 977, 872, 439, 304, 0, 300, 369, 738, 1273], [1475, 1277, 1172, 739, 604, 300, 0, 69, 438, 973], [1544, 1346, 1241, 808, 673, 369, 69, 0, 369, 904], [1913, 1715, 1610, 1177, 1042, 738, 438, 369, 0, 535], [2448, 2250, 2145, 1712, 1577, 1273, 973, 904, 535, 0]])

Remark : the numpy.ogrid function allows to directly create vectors x and y of the previous example, with two “signiﬁcant dimensions”:
>>> x, y = np.ogrid[0:5, 0:5] >>> x, y (array([[0], [1], [2], [3], [4]]), array([[0, 1, 2, 3, 4]])) >>> x.shape, y.shape ((5, 1), (1, 5)) >>> distance = np.sqrt(x**2 + y**2)

So, np.ogrid is very useful as soon as we have to handle computations on a network. On the other hand, np.mgrid directly provides matrices full of indices for cases where we can’t (or don’t want to) beneﬁt from broadcasting:
>>> x, y = >>> x array([[0, [1, [2, [3, >>> y array([[0, [0, [0, [0, np.mgrid[0:4, 0:4] 0, 1, 2, 3, 1, 1, 1, 1, 0, 1, 2, 3, 2, 2, 2, 2, 0], 1], 2], 3]]) 3], 3], 3], 3]])

Warning: Good practices In the previous example, we can note some good (and bad) practices: • Give explicit variable names (no need of a comment to explain what is in the variable) • Put spaces after commas, around =, etc. A certain number of rules for writing “beautiful” code (and more importantly using the same conventions as anybody else!) are given in the Style Guide for Python Code and the Docstring Conventions page (to manage help strings). • Except some rare cases, write variable names and comments in english.

1.4. NumPy: creating and manipulating numerical data

Python Scientiﬁc lecture notes, Release 2010

1.4.11 Synthesis exercises: framing Lena
Let’s do some manipulations on numpy arrays by starting with the famous image of Lena (http://www.cs.cmu.edu/~chuck/lennapg/). scipy provides a 2D array of this image with the scipy.lena function:
>>> import scipy >>> lena = scipy.lena()

Conclusion : what do you need to know about numpy arrays to start? • Know how to create arrays : array, arange, ones, zeros. • Know the shape of the array with array.shape, then use slicing to obtain different views of the array: array[::2], etc. Change the shape of the array using reshape. • Obtain a subset of the elements of an array and/or modify their values with masks:
>>> a[a<0] = 0

Here are a few images we will be able to obtain with our manipulations: use different colormaps, crop the image, change some parts of the image.

• Know miscellaneous operations on arrays, like ﬁnding the mean or max (array.max(), array.mean()). No need to retain everything, but have the reﬂex to search in the documentation (see Getting help and ﬁnding documentation (page 55)) !! • For advanced use: master the indexing with arrays of integers, as well as broadcasting. Know more functions of numpy allowing to handle array operations.

1.5 Getting help and ﬁnding documentation
author Emmanuelle Gouillart • Let’s use the imshow function of pylab to display the image.
In [3]: import pylab In [4]: lena = scipy.lena() In [5]: pylab.imshow(lena)

Rather than knowing all functions in Numpy and Scipy, it is important to ﬁnd rapidly information throughout the documentation and the available help. Here are some ways to get information: • In Ipython, help function opens the docstring of the function. Only type the beginning of the function’s name and use tab completion to display the matching functions.
In [204]: help np.v np.vander np.vdot np.var np.vectorize In [204]: help np.vander np.version np.void np.void0 np.vsplit np.vstack

• Lena is then displayed in false colors. A colormap must be speciﬁed for her to be displayed in grey.
In [6]: pylab.imshow(lena, pl.cm.gray) In [7]: # ou In [8]: gray()

• Create an array of the image with a narrower centring : for example, remove 30 pixels from all the borders of the image. To check the result, display this new array with imshow.
In [9]: crop_lena = lena[30:-30,30:-30]

In Ipython it is not possible to open a separated window for help and documentation; however one can always open a second Ipython shell just to display help and docstrings... • Numpy’s and Scipy’s documentations can be browsed online on http://docs.scipy.org/doc. The search button is quite useful inside the reference documentation of the two packages (http://docs.scipy.org/doc/numpy/reference/ and http://docs.scipy.org/doc/scipy/reference/). Tutorials on various topics as well as the complete API with all docstrings are found on this website.

• We will now frame Lena’s face with a black locket. For this, we need to – create a mask corresponding to the pixels we want to be black. The mask is deﬁned by this condition (y-256)**2 + (x-256)**2
In [15]: In [16]: Out[16]: In [17]: In [18]: y, x = np.ogrid[0:512,0:512] # x and y indices of pixels y.shape, x.shape ((512, 1), (1, 512)) centerx, centery = (256, 256) # center of the image mask = ((y - centery)**2 + (x - centerx)**2)> 230**2

then • assign the value 0 to the pixels of the image corresponding to the mask. The syntax is extremely simple and intuitive:
In [19]: lena[mask]=0 In [20]: imshow(lena) Out[20]: <matplotlib.image.AxesImage object at 0xa36534c>

• Subsidiary question : copy all instructions of this exercise in a script called lena_locket.py then execute this script in iPython with %run lena_locket.py.

1.4. NumPy: creating and manipulating numerical data

1.5. Getting help and ﬁnding documentation

Python Scientiﬁc lecture notes, Release 2010

• Numpy’s and Scipy’s documentation is enriched and updated on a regular basis by users on a wiki http://docs.scipy.org/numpy/. As a result, some docstrings are clearer or more detailed on the wiki, and you may want to read directly the documentation on the wiki instead of the ofﬁcial documentation website. Note that anyone can create an account on the wiki and write better documentation; this is an easy way to contribute to an open-source project and improve the tools you are using!

Finally, two more “technical” possibilities are useful as well: • In Ipython, the magical function %psearch search for objects matching patterns. This is useful if, for example, one does not know the exact name of a function.
In [3]: import numpy as np In [4]: %psearch np.diag* np.diag np.diagflat np.diagonal

• Scipy’s cookbook http://www.scipy.org/Cookbook gives recipes on many common problems frequently encountered, such as ﬁtting data points, solving ODE, etc. • Matplotlib’s website http://matplotlib.sourceforge.net/ features a very nice gallery with a large number of plots, each of them shows both the source code and the resulting plot. This is very useful for learning by example. More standard documentation is also available.

• numpy.lookfor looks for keywords inside the docstrings of speciﬁed modules.
In [45]: numpy.lookfor(’convolution’) Search results for ’convolution’ -------------------------------numpy.convolve Returns the discrete, linear convolution of two one-dimensional sequences. numpy.bartlett Return the Bartlett window. numpy.correlate Discrete, linear correlation of two 1-dimensional sequences. In [46]: numpy.lookfor(’remove’, module=’os’) Search results for ’remove’ --------------------------os.remove remove(path) os.removedirs removedirs(path) os.rmdir rmdir(path) os.unlink unlink(path) os.walk Directory tree generator.

• Mayavi’s website http://code.enthought.com/projects/mayavi/docs/development/html/mayavi/ also has a very nice gallery of examples http://code.enthought.com/projects/mayavi/docs/development/html/mayavi/auto/examples.html in which one can browse for different visualization solutions.

• If everything listed above fails (and Google doesn’t have the answer)... don’t despair! Write to the mailinglist suited to your problem: you should have a quick answer if you describe your problem well. Experts on scientiﬁc python often give very enlightening explanations on the mailing-list. – Numpy discussion (numpy-discussion@scipy.org): all about numpy arrays, manipulating them, indexation questions, etc.

1.5. Getting help and ﬁnding documentation

Python Scientiﬁc lecture notes, Release 2010

– SciPy Users List (scipy-user@scipy.org): scientiﬁc computing with Python, high-level data processing, in particular with the scipy package. – matplotlib-users@lists.sourceforge.net for plotting with matplotlib.

The numbers form 0 through 9 are plotted:

1.6 Matplotlib
author Mike Müller

1.6.1 Introduction
matplotlib is probably the single most used Python package for 2D-graphics. It provides both a very quick way to visualize data from Python and publication-quality ﬁgures in many formats. We are going to explore matplotlib in interactive mode covering most common cases. We also look at the class library which is provided with an object-oriented interface.

1.6.2 IPython
IPython is an enhanced interactive Python shell that has lots of interesting features including named inputs and outputs, access to shell commands, improved debugging and many more. When we start it with the command line argument -pylab, it allows interactive matplotlib sessions that has Matlab/Mathematica-like functionality.

Now we can interactively add features to or plot:
In [2]: xlabel(’measured’) Out[2]: <matplotlib.text.Text instance at 0x01A9D210> In [3]: ylabel(’calculated’) Out[3]: <matplotlib.text.Text instance at 0x01A9D918> In [4]: title(’Measured vs. calculated’) Out[4]: <matplotlib.text.Text instance at 0x01A9DF80> In [5]: grid(True) In [6]:

1.6.3 pylab
pylab provides a procedural interface to the matplotlib object-oriented plotting library. It is modeled closely after Matlab(TM). Therefore, the majority of plotting commands in pylab has Matlab(TM) analogs with similar arguments. Important commands are explained with interactive examples.

1.6.4 Simple Plots
Let’s start an interactive session:
$python ipython.py -pylab

We get a reference to our plot:
In [6]: my_plot = gca()

and to our line we plotted, which is the ﬁrst in the plot:
In [7]: line = my_plot.lines[0]

This brings us to the IPython prompt:
IPython ? %magic help object? 0.8.1 -- An enhanced Interactive Python. -> Introduction to IPython’s features. -> Information about IPython’s ’magic’ % functions. -> Python’s own help system. -> Details about ’object’. ?object also works, ?? prints more.

Now we can set properties using set_something methods:
In [8]: line.set_marker(’o’)

or the setp function:
In [9]: setp(line, color=’g’) Out[9]: [None]

Welcome to pylab, a matplotlib-based Python environment. For more information, type ’help(pylab)’. In [1]:

To apply the new properties we need to redraw the screen:
In [10]: draw()

Now we can make our ﬁrst, really simple plot:
In [1]: plot(range(10)) Out[1]: [<matplotlib.lines.Line2D instance at 0x01AA26E8>] In [2]:

We can also add several lines to one plot:
In [1]: x = arange(100) In [2]: linear = arange(100) In [3]: square = [v * v for v in arange(0, 10, 0.1)] In [4]: lines = plot(x, linear, x, square)

1.6. Matplotlib

Python Scientiﬁc lecture notes, Release 2010 Property alpha antialiased color data_clipping label linestyle linewidth marker markeredgewidth markeredgecolor markerfacecolor markersize Symbol – -. : . , o ^ v < > s + x D d 1 2 3 4 h H p | _ steps

Python Scientiﬁc lecture notes, Release 2010 Value alpha transparency on 0-1 scale True or False - use antialised rendering matplotlib color arg whether to use numeric to clip data string optionally used for legend one of - : -. ﬂoat, the line width in points one of + , o . s v x > <, etc line width around the marker symbol edge color if a marker is used face color if a marker is used size of the marker in points

Let’s add a legend:
In [5]: legend((’linear’, ’square’)) Out[5]: <matplotlib.legend.Legend instance at 0x01BBC170>

This does not look particularly nice. We would rather like to have it at the left. So we clean the old graph:
In [6]: clf()

and print it anew providing new line styles (a green dotted line with crosses for the linear and a red dashed line with circles for the square graph):
In [7]: lines = plot(x, linear, ’g:+’, x, square, ’r--o’)

Now we add the legend at the upper left corner:
In [8]: l = legend((’linear’, ’square’), loc=’upper left’)

There are many line styles that can be speciﬁed with symbols: Description solid line dashed line dash-dot line dotted line points pixels circle symbols triangle up symbols triangle down symbols triangle left symbols triangle right symbols square symbols plus symbols cross symbols diamond symbols thin diamond symbols tripod down symbols tripod up symbols tripod left symbols tripod right symbols hexagon symbols rotated hexagon symbols pentagon symbols vertical line symbols horizontal line symbols use gnuplot style ‘steps’ # kwarg only

The result looks like this:

Exercises 1. Plot a simple graph of a sinus function in the range 0 to 3 with a step size of 0.01. 2. Make the line red. Add diamond-shaped markers with size of 5. 3. Add a legend and a grid to the plot.

1.6.5 Properties
So far we have used properties for the lines. There are three possibilities to set them: 1) as keyword arguments at creation time: plot(x, linear, ’g:+’, x, square, ’r--o’). 2. with the function setp: setp(line, color=’g’). 3. using the set_something methods: line.set_marker(’o’) Lines have several properties as shown in the following table:

Colors can be given in many ways: one-letter abbreviations, gray scale intensity from 0 to 1, RGB in hex and tuple format as well as any legal html color name. The one-letter abbreviations are very handy for quick work. With following you can get quite a few things done: Abbreviation b g r c m y k w Color blue green red cyan magenta yellow black white

Other objects also have properties. The following table list the text properties:

1.6. Matplotlib

Python Scientiﬁc lecture notes, Release 2010 Property alpha color family fontangle horizontalalignment multialignment name position variant rotation size style text verticalalignment weight Exercise 1. Apply different line styles to a plot. Change line color and thickness as well as the size and the kind of the marker. Experiment with different styles. Value alpha transparency on 0-1 scale matplotlib color arg set the font family, eg sans-serif, cursive, fantasy the font slant, one of normal, italic, oblique left, right or center left, right or center only for multiline strings font name, eg, Sans, Courier, Helvetica x,y location font variant, eg normal, small-caps angle in degrees for rotated text fontsize in points, eg, 8, 10, 12 font style, one of normal, italic, oblique set the text string itself top, bottom or center font weight, e.g. normal, bold, heavy, light

Python Scientiﬁc lecture notes, Release 2010

In [4]: ax.annotate(’Here is something special’, xy = (1, 1))

We will write the text at the position (1, 1) in terms of data. There are many optional arguments that help to customize the position of the text. The arguments textcoords and xycoords speciﬁes what x and y mean: argument ﬁgure points ﬁgure pixels ﬁgure fraction axes points axes pixels axes fraction data coordinate system points from the lower left corner of the ﬁgure pixels from the lower left corner of the ﬁgure 0,0 is lower left of ﬁgure and 1,1 is upper, right points from lower left corner of axes pixels from lower left corner of axes 0,1 is lower left of axes and 1,1 is upper right use the axes data coordinate system

If we do not supply xycoords, the text will be written at xy. Furthermore, we can use an arrow whose appearance can also be described in detail:
In [14]: plot(arange(10)) Out[14]: [<matplotlib.lines.Line2D instance at 0x01BB15D0>] In [15]: ax = gca() In [16]: ax.annotate(’Here is something special’, xy = (2, 1), xytext=(1,5)) Out[16]: <matplotlib.text.Annotation instance at 0x01BB1648> In [17]: ax.annotate(’Here is something special’, xy = (2, 1), xytext=(1,5), ....: arrowprops={’facecolor’: ’r’})

1.6.6 Text
We’ve already used some commands to add text to our ﬁgure: xlabel ylabel, and title. There are two functions to put text at a deﬁned position. text adds the text with data coordinates:
In [2]: plot(arange(10)) In [3]: t1 = text(5, 5, ’Text in the middle’)

Exercise 1. Annotate a line at two places with text. Use green and red arrows and align it according to figure points and data.

figtext uses ﬁgure coordinates form 0 to 1:
In [4]: t2 = figtext(0.8, 0.8, ’Upper right text’)

1.6.7 Ticks
Where and What Well formated ticks are an important part of publishing-ready ﬁgures. matplotlib provides a totally conﬁgurable system for ticks. There are tick locators to specify where ticks should appear and tick formatters to make ticks look like the way you want. Major and minor ticks can be located and formated independently from each other. Per default minor ticks are not shown, i.e. there is only an empty list for them because it is as NullLocator (see below). Tick Locators There are several locators for different kind of requirements: Class NullLocator IndexLocator LinearLocator LogLocator MultipleLocator AutoLocator Description no ticks locator for index plots (e.g. where x = range(len(y)) evenly spaced ticks from min to max logarithmically ticks from min to max ticks and range are a multiple of base; either integer or ﬂoat choose a MultipleLocator and dynamically reassign

matplotlib supports TeX mathematical expression. So r’$\pi$’ will show up as: π If you want to get more control over where the text goes, you use annotations:

All of these locators derive from the base class matplotlib.ticker.Locator. You can make your own locator deriving from it.

1.6. Matplotlib

Python Scientiﬁc lecture notes, Release 2010

Handling dates as ticks can be especially tricky. Therefore, matplotlib provides special locators in ‘‘matplotlib.dates: Class MinuteLocator HourLocator DayLocator WeekdayLocator MonthLocator YearLocator RRuleLocator Tick Formatters Similarly to locators, there are formatters: Class NullFormatter FixedFormatter FuncFormatter FormatStrFormatter IndexFormatter ScalarFormatter LogFormatter DateFormatter Description no labels on the ticks set the strings manually for the labels user deﬁned function sets the labels use a sprintf format string cycle through ﬁxed strings by tick position default formatter for scalars; autopick the fmt string formatter for log axes use an strftime string to format the date Description locate minutes locate hours locate speciﬁed days of the month locate days of the week, e.g. MO, TU locate months, e.g. 10 for October locate years that are multiples of base locate using a matplotlib.dates.rrule

1.6.8 Figures, Subplots, and Axes
The Hierarchy So far we have used implicit ﬁgure and axes creation. This is handy for fast plots. We can have more control over the display using figure, subplot, and axes explicitly. A figure in matplotlib means the whole window in the user interface. Within this figure there can be subplots. While subplot positions the plots in a regular grid, axes allows free placement within the figure. Both can be useful depending on your intention. We’ve already work with ﬁgures and subplots without explicitly calling them. When we call plot matplotlib calls gca() to get the current axes and gca in turn calls gcf() to get the current ﬁgure. If there is none it calls figure() to make one, strictly speaking, to make a subplot(111). Let’s look at the details. Figures A figure is the windows in the GUI that has “Figure #” as title. Figures are numbered starting from 1 as opposed to the normal Python way starting from 0. This is clearly MATLAB-style. There are several parameters that determine how the ﬁgure looks like: Argument num figsize dpi facecolor edgecolor frameon Default 1 figure.figsize figure.dpi figure.facecolor figure.edgecolor True Description number of ﬁgure ﬁgure size in in inches (width, height) resolution in dots per inch color of the drawing background color of edge around the drawing background draw ﬁgure frame or not

All of these formatters derive from the base class matplotlib.ticker.Formatter. You can make your own formatter deriving from it. Now we set our major locator to 2 and the minor locator to 1. We also format the numbers as decimals using the FormatStrFormatter:
In [5]: major_locator = MultipleLocator(2) In [6]: major_formatter = FormatStrFormatter(’%5.2f’) In [7]: minor_locator = MultipleLocator(1) In [8]: ax.xaxis.set_major_locator(major_locator) In [9]: ax.xaxis.set_minor_locator(minor_locator) In [10]: ax.xaxis.set_major_formatter(major_formatter) In [10]: draw()

The defaults can be speciﬁed in the resource ﬁle and will be used most of the time. Only the number of the ﬁgure is frequently changed. When you work with the GUI you can close a ﬁgure by clicking on the x in the upper right corner. But you can close a ﬁgure programmatically by calling close. Depending on the argument it closes (1) the current ﬁgure (no argument), (2) a speciﬁc ﬁgure (ﬁgure number or ﬁgure instance as argument), or (3) all ﬁgures (all as argument). As with other objects, you can set ﬁgure properties also setp or with the set_something methods. Subplots With subplot you can arrange plots in regular grid. You need to specify the number of rows and columns and the number of the plot. A plot with two rows and one column is created with subplot(211) and subplot(212). The result looks like this:

After we redraw the ﬁgure our x axis should look like this:

Exercises 1. Plot a graph with dates for one year with daily values at the x axis using the built-in module datetime. 2. Format the dates in such a way that only the ﬁrst day of the month is shown. 3. Display the dates with and without the year. Show the month as number and as ﬁrst three letters of the month name. If you want two plots side by side, you create one row and two columns with subplot(121) and subplot(112). The result looks like this:

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1.6.9 Other Types of Plots
Many More So far we have used only line plots. matplotlib offers many more types of plots. We will have a brief look at some of them. All functions have many optional arguments that are not shown here. Bar Charts The function bar creates a new bar chart:
bar([1, 2, 3], [4, 3, 7])

You can arrange as many ﬁgures as you want. A two-by-two arrangement can be created with subplot(221), subplot(222), subplot(223), and subplot(224). The result looks like this:

Now we have three bars starting at 1, 2, and 3 with height of 4, 3, 7 respectively:

Frequently, you don’t want all subplots to have ticks or labels. You can set the xticklabels or the yticklabels to an empty list ([]). Every subplot deﬁnes the methods ’is_first_row, is_first_col, is_last_row, is_last_col. These can help to set ticks and labels only for the outer pots. Axes Axes are very similar to subplots but allow placement of plots at any location in the ﬁgure. So if we want to put a smaller plot inside a bigger one we do so with axes:
In [22]: plot(x) Out[22]: [<matplotlib.lines.Line2D instance at 0x02C9CE90>] In [23]: a = axes([0.2, 0.5, 0.25, 0.25]) In [24]: plot(x)

The default column width is 0.8. It can be changed with common methods by setting width. As it can be color and bottom, we can also set an error bar with yerr or xerr. Horizontal Bar Charts The function barh creates an vertical bar chart. Using the same data:
barh([1, 2, 3], [4, 3, 7])

We get:

The result looks like this:

Broken Horizontal Bar Charts Exercises 1. Draw two ﬁgures, one 5 by 5, one 10 by 10 inches. 2. Add four subplots to one ﬁgure. Add labels and ticks only to the outermost axes. 3. Place a small plot in one bigger plot. We can also have discontinuous vertical bars with broken_barh. We specify start and width of the range in y-direction and all start-width pairs in x-direction:
yrange = (2, 1) xranges = ([0, 0.5], [1, 1], [4, 1]) broken_barh(xranges, yrange)

We changes the extension of the y-axis to make plot look nicer:
ax = gca() ax.set_ylim(0, 5) (0, 5) draw()

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and get this:

Now we can make a simple contour plot:
contour(x, x, z)

Our plot looks like this:

Box and Whisker Plots We can draw box and whisker plots:
boxplot((arange(2, 10), arange(1, 5)))

We can also ﬁll the area. We just use numbers form 0 to 9 for the values v:
v = x contourf(x, x, z, v)

We want to have the whiskers well within the plot and therefore increase the y axis:
ax = gca() ax.set_ylim(0, 12) draw()

Now our plot area is ﬁlled:

Our plot looks like this:

Histograms We can make histograms. Let’s get some normally distributed random numbers from numpy: The range of the whiskers can be determined with the argument whis, which defaults to 1.5. The range of the whiskers is between the most extreme data point within whis*(75%-25%) of the data. Contour Plots We can also do contour plots. We deﬁne arrays for the x and y coordinates:
x = y = arange(10) import numpy as N r_numbers = N.random.normal(size= 1000)

Now we make a simple histogram:
hist(r_numbers)

With 100 numbers our ﬁgure looks pretty good:

and also a 2D array for z:
z = ones((10, z[5,5] = 7 z[2,1] = 3 z[8,7] = 4 z array([[ 1., [ 1., [ 1., [ 1., [ 1., [ 1., [ 1., [ 1., [ 1., [ 1., 10))

1., 1., 3., 1., 1., 1., 1., 1., 1., 1.,

1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,

1., 1., 1., 1., 1., 7., 1., 1., 1., 1.,

1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,

1., 1., 1., 1., 1., 1., 1., 1., 4., 1.,

1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,

1.], 1.], 1.], 1.], 1.], 1.], 1.], 1.], 1.], 1.]])

Loglog Plots Plots with logarithmic scales are easy:
loglog(arange(1000))

We set the mayor and minor grid:

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grid(True) grid(True, which=’minor’)

Arrow Plots Plotting arrows in 2D plane can be achieved with quiver. We deﬁne the x and y coordinates of the arrow shafts:
x = y = arange(10)

Now we have loglog plot:

The x and y components of the arrows are speciﬁed as 2D arrays:
u = ones((10, 10)) v = ones((10, 10)) u[4, 4] = 3 v[1, 1] = -1

Now we can plot the arrows: If we want only one axis with a logarithmic scale we can use semilogx or semilogy. Pie Charts Pie charts can also be created with a few lines:
data = [500, 700, 300] labels = [’cats’, ’dogs’, ’other’] pie(data, labels=labels) quiver(x, y, u, v)

All arrows point to the upper right, except two. The one at the location (4, 4) has 3 units in x-direction and the other at location (1, 1) has -1 unit in y direction:

The result looks as expected:

Scatter Plots Scatter plots print x vs. y diagrams. We deﬁne x and y and make some point in y random:
x = arange(10) y = arange(10) y[1] = 7 y[4] = 2 y[8] = 3

Polar Plots Polar plots are also possible. Let’s deﬁne our r from 0 to 360 and our theta from 0 to 360 degrees. We need to convert them to radians:
r = arange(360) theta = r / (180/pi)

Now make a scatter plot:
scatter(x, y)

The three different values for y break out of the line:

Now plot in polar coordinates:
polar(theta, r)

We get a nice spiral:

Sparsity Pattern Plots Working with sparse matrices, it is often of interest as how the matrix looks like in terms of sparsity. We take an identity matrix as an example:

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i = identity(10) i array([[1, 0, 0, [0, 1, 0, [0, 0, 1, [0, 0, 0, [0, 0, 0, [0, 0, 0, [0, 0, 0, [0, 0, 0, [0, 0, 0, [0, 0, 0,

datetime.datetime(2000, 4, 30, 0, 0), datetime.datetime(2000, 5, 15, 0, 0)] 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], 0], 0], 0], 0], 0], 0], 0], 0], 1]])

We reuse our data from the scatter plot (see above):
y array([0, 7, 2, 3, 2, 5, 6, 7, 3, 9])

Now we can plot the dates at the x axis:
plot_date(dates, y)

Now we look at it more visually:
spy(i)

1.6.10 The Class Library
So far we have used the pylab interface only. As the name suggests it is just wrapper around the class library. All pylabb commands can be invoked via the class library using an object-oriented approach. Stem Plots Stem plots vertical lines at the given x location up to the speciﬁed y location. Let’s reuse x and y from our scatter (see above):
stem(x, y)

The Figure Class The class Figure lives in the module matplotlib.figure. Its constructor takes these arguments:
figsize=None, dpi=None, facecolor=None, edgecolor=None, linewidth=1.0, frameon=True, subplotpars=None

Comparing this with the arguments of figure in pylab shows signiﬁcant overlap:
num=None, figsize=None, dpi=None, facecolor=None edgecolor=None, frameon=True

Figure provides lots of methods, many of them have equivalents in pylab. The methods add_axes and add_subplot are called if new axes or subplot are created with axes or subplot in pylab. Also the method gca maps directly to pylab as do legend, text and many others. Date Plots There is a function for creating date plots. Let’s deﬁne 10 dates starting at January 1st 2000 at 15.day intervals:
import datetime d1 = datetime.datetime(2000, 1, 1) delta = datetime.timedelta(15) dates = [d1 + x * delta for x in range(1 dates [datetime.datetime(2000, 1, 1, 0, 0), datetime.datetime(2000, 1, 16, 0, 0), datetime.datetime(2000, 1, 31, 0, 0), datetime.datetime(2000, 2, 15, 0, 0), datetime.datetime(2000, 3, 1, 0, 0), datetime.datetime(2000, 3, 16, 0, 0), datetime.datetime(2000, 3, 31, 0, 0), datetime.datetime(2000, 4, 15, 0, 0),

There are also several set_something method such as set_facecolor or set_edgecolor that will be called through pylab to set properties of the ﬁgure. Figure also implements get_something methods such as get_axes or get_facecolor to get properties of the ﬁgure. The Classes Axes and Subplot In the class Axes we ﬁnd most of the ﬁgure elements such as Axis, Tick, Line2D, or Text. It also sets the coordinate system. The class Subplot inherits from Axes and adds some more functionality to arrange the plots in a grid. Analogous to Figure, it has methods to get and set properties and methods already encountered as functions in pylab such as annotate. In addition, Axes has methods for all types of plots shown in the previous section. Other Classes Other classes such as text, Legend or Ticker are setup very similarly. They can be understood mostly by comparing to the pylab interface. 72 1.6. Matplotlib 73

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Example Let’s look at an example for using the object-oriented API:
#file matplotlib/oo.py from matplotlib.figure import Figure figsize = (8, 5) fig = Figure(figsize=figsize) ax = fig.add_subplot(111) line = ax.plot(range(10))[0] ax.set_title(’Plotted with OO interface’) ax.set_xlabel(’measured’) ax.set_ylabel(’calculated’) ax.grid(True) line.set_marker(’o’) #1 #2 #3 #4 #5 #6

Exercises 1. Use the object-oriented API of matplotlib to create a png-ﬁle with a plot of two lines, one linear and square with a legend in it.

1.7 Scipy : high-level scientiﬁc computing
authors Adrien Chauve, Andre Espaze, Emmanuelle Gouillart, Gaël Varoquaux Scipy The scipy package contains various toolboxes dedicated to common issues in scientiﬁc computing. Its different submodules correspond to different applications, such as interpolation, integration, optimization, image processing, statistics, special functions, etc. scipy can be compared to other standard scientiﬁc-computing libraries, such as the GSL (GNU Scientiﬁc Library for C and C++), or Matlab’s toolboxes. scipy is the core package for scientiﬁc routines in Python; it is meant to operate efﬁciently on numpy arrays, so that numpy and scipy work hand in hand. Before implementing a routine, if is worth checking if the desired data processing is not already implemented in Scipy. As non-professional programmers, scientists often tend to re-invent the wheel, which leads to buggy, non-optimal, difﬁcult-to-share and unmaintainable code. By contrast, Scipy‘s routines are optimized and tested, and should therefore be used when possible. Warning: This tutorial is far from an introduction to numerical computing. As enumerating the different submodules and functions in scipy would be very boring, we concentrate instead on a few examples to give a general idea of how to use scipy for scientiﬁc computing. To begin with
>>> import numpy as np >>> import scipy

#7 #8

from matplotlib.backends.backend_agg import FigureCanvasAgg #9 canvas = FigureCanvasAgg(fig) #10 canvas.print_figure("oo.png", dpi=80) #11

import Tkinter as Tk #12 from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg #13 root = Tk.Tk() #13 canvas2 = FigureCanvasTkAgg(fig, master=root) #14 canvas2.show() #15 canvas2.get_tk_widget().pack(side=Tk.TOP, fill=Tk.BOTH, expand=1) #16 Tk.mainloop() #17 from matplotlib import _pylab_helpers import pylab pylab_fig = pylab.figure(1, figsize=figsize) figManager = _pylab_helpers.Gcf.get_active() figManager.canvas.figure = fig pylab.show() #18 #19 #20 #21 #22 #23

scipy is mainly composed of task-speciﬁc sub-modules: cluster fftpack integrate interpolate io linalg maxentropy ndimage odr optimize signal sparse spatial special stats Vector quantization / Kmeans Fourier transform Integration routines Interpolation Data input and output Linear algebra routines Routines for ﬁtting maximum entropy models n-dimensional image package Orthogonal distance regression Optimization Signal processing Sparse matrices Spatial data structures and algorithms Any special mathematical functions Statistics

Since we are not in the interactive pylab-mode, we need to import the class Figure explicitly (#1). We set the size of our ﬁgure to be 8 by 5 inches (#2). Now we initialize a new ﬁgure (#3) and add a subplot to the ﬁgure (#4). The 111 says one plot at position 1, 1 just as in MATLAB. We create a new plot with the numbers from 0 to 9 and at the same time get a reference to our line (#5). We can add several things to our plot. So we set a title and labels for the x and y axis (#6). We also want to see the grid (#7) and would like to have little ﬁlled circles as markers (#8). There are many different backends for rendering our ﬁgure. We use the Anti-Grain Geometry toolkit (http://www.antigrain.com) to render our ﬁgure. First, we import the backend (#9), then we create a new canvas that renders our ﬁgure (#10). We save our ﬁgure in a png-ﬁle with a resolution of 80 dpi (#11). We can use several GUI toolkits directly. So we import Tkinter (#12) as well as the corresponding backend (#13). Now we have to do some basic GUI programming work. We make a root object for our GUI (#13) and feed it together with our ﬁgure to the backend to get our canvas (14). We call the show method (#15), pack our widget (#16), and call the Tkinter mainloop to start the application (#17). You should see GUI window with the ﬁgure on your screen. After closing the screen, the next part, the script, will be executed. We would like to create a screen display just as we would use pylab. Therefore we import a helper (#18) and pylab itself (#19). We create a normal ﬁgure with pylab‘ (‘‘20) and get the corresponding ﬁgure manager (#21). Now let’s set our ﬁgure we created above to be the current ﬁgure (#22) and let pylab show the result (#23). The lower part of the ﬁgure might be cover by the toolbar. If so, please adjust the figsize for pylab accordingly. 1.6. Matplotlib 74

1.7.1 Scipy builds upon Numpy
Numpy is required for running Scipy but also for using it. The most important type introduced to Python is the N dimensional array, and it can be seen that Scipy uses the same:

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>>> scipy.ndarray is np.ndarray True

t = np.linspace(0, 5, 100) x = np.sin(t) pl.plot(t, x, linewidth=3) pl.plot(t[::2], signal.resample(x, 50), ’ko’)
1.5 1.0 0.5

Moreover most of the Scipy usual functions are provided by Numpy:
>>> scipy.cos is np.cos True

If you would like to know the objects used from Numpy, have a look at the scipy.__file__[:-1] ﬁle. On version ‘0.6.0’, the whole Numpy namespace is imported by the line from numpy import *.

1.7.2 File input/output: scipy.io
• Loading and saving matlab ﬁles:
>>> from scipy import io >>> struct = io.loadmat(’file.mat’, struct_as_record=True) >>> io.savemat(’file.mat’, struct)

0.0 0.5 1.0 1.50 1 2 3 4 5

See also: • Load text ﬁles:
np.loadtxt/np.savetxt

Notice how on the side of the window the resampling is less accurate and has a rippling effect. • Signal has many window function: hamming, bartlett, blackman... • Signal has ﬁltering (Gaussian, median ﬁlter, Wiener), but we will discuss this in the image paragraph.

• Clever loading of text/csv ﬁles:
np.genfromtxt/np.recfromcsv

• Fast an efﬁcient, but numpy-speciﬁc, binary format:
np.save/np.load

1.7.4 Special functions: scipy.special
Special functions are transcendal functions. The docstring of the module is well-written and we will not list them. Frequently used ones are: • Bessel function, such as special.jn (nth integer order Bessel function) • Elliptic function (special.ellipj for the Jacobian elliptic function, ...) • Gamma function: special.gamma, alos note special.gammaln which will give the log of Gamma to a higher numerical precision. • Erf, the area under a Gaussian curve: special.erf

1.7.3 Signal processing: scipy.signal
>>> from scipy import signal

• Detrend: remove linear trend from signal:
t = np.linspace(0, 5, 100) x = t + np.random.normal(size=100) pl.plot(t, x, linewidth=3) pl.plot(t, signal.detrend(x), linewidth=3)
6 5 4 3 2 1 0 1 2 30 1 2 3 4 5

1.7.5 Statistics and random numbers: scipy.stats
The module scipy.stats contains statistical tools and probabilistic description of random processes. Random number generators for various random process can be found in numpy.random. Histogram and probability density function Given observations of a random process, their histogram is an estimator of the random process’s PDF (probability density function):
>>> a = np.random.normal(size=1000) >>> bins = np.arange(-4, 5) >>> bins array([-4, -3, -2, -1, 0, 1, 2, 3, 4]) >>> histogram = np.histogram(a, bins=bins, normed=True)[0] >>> bins = 0.5*(bins[1:] + bins[:-1]) >>> bins array([-3.5, -2.5, -1.5, -0.5, 0.5, 1.5, 2.5, 3.5]) >>> from scipy import stats >>> b = stats.norm.pdf(bins)

• Resample: resample a signal to n points using FFT.

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In [1]: pl.plot(bins, histogram) In [2]: pl.plot(bins, b)
0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 6 4 2 0 2 4 6

• The T statistic value: it is a number the sign of which is proportional to the difference between the two random processes and the magnitude is related to the signiﬁcance of this difference. • the p value: the probability of both process being identical. If it is close to 1, the two process are almost certainly identical. The closer it is to zero, the more likely it is that the processes have different mean.

1.7.6 Linear algebra operations: scipy.linalg
First, the linalg module provides standard linear algebra operations. The det function computes the determinant of a square matrix:
>>> from scipy import linalg >>> arr = np.array([[1, 2], ... [3, 4]]) >>> linalg.det(arr) -2.0 >>> arr = np.array([[3, 2], ... [6, 4]]) >>> linalg.det(arr) 0.0 >>> linalg.det(np.ones((3, 4))) Traceback (most recent call last): ... ValueError: expected square matrix

If we know that the random process belongs to a given family of random processes, such as normal processes, we can do a maximum-likelihood ﬁt of the observations to estimate the parameters of the underlying distribution. Here we ﬁt a normal process to the observed data:
>>> loc, std = stats.norm.fit(a) >>> loc 0.003738964114102075 >>> std 0.97450996668871193

The inv function computes the inverse of a square matrix:
>>> arr = np.array([[1, 2], ... [3, 4]]) >>> iarr = linalg.inv(arr) >>> iarr array([[-2. , 1. ], [ 1.5, -0.5]]) >>> np.allclose(np.dot(arr, iarr), np.eye(2)) True

Percentiles The median is the value with half of the observations below, and half above:
>>> np.median(a) 0.0071645570292782519

It is also called the percentile 50, because 50% of the observation are below it:
>>> stats.scoreatpercentile(a, 50) 0.0071645570292782519

Note that in case you use the matrix type, the inverse is computed when requesting the I attribute:
>>> ma = np.matrix(arr, copy=False) >>> np.allclose(ma.I, iarr) True

Similarly, we can calculate the percentile 90:
>>> stats.scoreatpercentile(a, 90) 1.2729556087871292

Finally computing the inverse of a singular matrix (its determinant is zero) will raise LinAlgError:
>>> arr = np.array([[3, 2], ... [6, 4]]) >>> linalg.inv(arr) Traceback (most recent call last): ... LinAlgError: singular matrix

The percentile is an estimator of the CDF: cumulative distribution function. Statistical tests A statistical test is a decision indicator. For instance, if we have 2 sets of observations, that we assume are generated from Gaussian processes, we can use a T-test to decide whether the two sets of observations are signiﬁcantly different:
>>> a = np.random.normal(0, 1, size=100) >>> b = np.random.normal(1, 1, size=10) >>> stats.ttest_ind(a, b) (-2.389876434401887, 0.018586471712806949)

More advanced operations are available like singular-value decomposition (SVD):
>>> arr = np.arange(12).reshape((3, 4)) + 1 >>> uarr, spec, vharr = linalg.svd(arr)

The resulting array spectrum is:
>>> spec array([ 2.54368356e+01, 1.72261225e+00, 5.14037515e-16])

The resulting output is composed of:

For the recomposition, an alias for manipulating matrix will ﬁrst be deﬁned:

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>>> asmat = np.asmatrix

The solver requires more iterations at start. The ﬁnal trajectory is seen on the Matplotlib ﬁgure computed with odeint-introduction.py.

then the steps are:
>>> sarr = np.zeros((3, 4)) >>> sarr.put((0, 5, 10), spec) >>> svd_mat = asmat(uarr) * asmat(sarr) * asmat(vharr) >>> np.allclose(svd_mat, arr) True

SVD is commonly used in statistics or signal processing. Many other standard decompositions (QR, LU, Cholesky, Schur), as well as solvers for linear systems, are available in scipy.linalg.

1.7.7 Numerical integration: scipy.integrate
The most generic integration routine is scipy.integrate.quad:
>>> from scipy.integrate import quad >>> res, err = quad(np.sin, 0, np.pi/2) >>> np.allclose(res, 1) True >>> np.allclose(err, 1 - res) True

Others integration schemes are available with fixed_quad, quadrature, romberg. scipy.integrate also features routines for Ordinary differential equations (ODE) integration. In particular, scipy.integrate.odeint is a general-purpose integrator using LSODA (Livermore solver for ordinary differential equations with automatic method switching for stiff and nonstiff problems), see the ODEPACK Fortran library for more details. odeint solves ﬁrst-order ODE systems of the form:
‘‘dy/dt = rhs(y1, y2, .., t0,...)‘‘

As an introduction, let us solve the ODE dy/dt = -2y between t = 0..4, with the initial condition y(t=0) = 1. First the function computing the derivative of the position needs to be deﬁned:
>>> def calc_derivative(ypos, time, counter_arr): ... counter_arr += 1 ... return -2*ypos ...

Another example with odeint will be a damped spring-mass oscillator (2nd order oscillator). The position of a mass attached to a spring obeys the 2nd order ODE y” + 2 eps wo y’ + wo^2 y = 0 with wo^2 = k/m being k the spring constant, m the mass and eps=c/(2 m wo) with c the damping coefﬁcient. For a computing example, the parameters will be:
>>> mass = 0.5 # kg >>> kspring = 4 # N/m >>> cviscous = 0.4 # N s/m

An extra argument counter_arr has been added to illustrate that the function may be called several times for a single time step, until solver convergence. The counter array is deﬁned as:
>>> counter = np.zeros((1,), np.uint16)

so the system will be underdamped because:
>>> eps = cviscous / (2 * mass * np.sqrt(kspring/mass)) >>> eps < 1 True

The trajectory will now be computed:
>>> from scipy.integrate import odeint >>> time_vec = np.linspace(0, 4, 40) >>> yvec, info = odeint(calc_derivative, 1, time_vec, ... args=(counter,), full_output=True)

For the odeint solver the 2nd order equation needs to be transformed in a system of two ﬁrst-order equations for the vector Y=(y, y’). It will be convenient to deﬁne nu = 2 eps wo = c / m and om = wo^2 = k/m:
>>> nu_coef = cviscous/mass >>> om_coef = kspring/mass

Thus the derivative function has been called more than 40 times:
>>> counter array([129], dtype=uint16)

Thus the function will calculate the velocity and acceleration by:
>>> def calc_deri(yvec, time, nuc, omc): ... return (yvec[1], -nuc * yvec[1] - omc * yvec[0]) ... >>> time_vec = np.linspace(0, 10, 100) >>> yarr = odeint(calc_deri, (1, 0), time_vec, args=(nu_coef, om_coef))

and the cumulative iterations number for the 10 ﬁrst convergences can be obtained by:
>>> info[’nfe’][:10] array([31, 35, 43, 49, 53, 57, 59, 63, 65, 69], dtype=int32)

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The ﬁnal position and velocity are shown on a Matplotlib ﬁgure built with the odeint-damped-spring-mass.py script.

Thus the signal frequency can be found by: There is no Partial Differential Equations (PDE) solver in scipy. Some PDE packages are written in Python, such as ﬁpy or SfePy.
>>> freq = freqs[power.argmax()] >>> np.allclose(freq, 1./period) True

Now only the main signal component will be extracted from the Fourier transform:

1.7.8 Fast Fourier transforms: scipy.fftpack
The fftpack module allows to compute fast Fourier transforms. As an illustration, an input signal may look like:
>>> >>> >>> >>> ... time_step = 0.1 period = 5. time_vec = np.arange(0, 20, time_step) sig = np.sin(2 * np.pi / period * time_vec) + \ np.cos(10 * np.pi * time_vec)

>>> sig_fft[np.abs(sample_freq) > freq] = 0

The resulting signal can be computed by the ifft function:
>>> main_sig = fftpack.ifft(sig_fft)

The result is shown on the Matplotlib ﬁgure generated by the fftpack-illustration.py script.

However the observer does not know the signal frequency, only the sampling time step of the signal sig. But the signal is supposed to come from a real function so the Fourier transform will be symmetric. The fftfreq function will generate the sampling frequencies and fft will compute the fast fourier transform:
>>> from scipy import fftpack >>> sample_freq = fftpack.fftfreq(sig.size, d=time_step) >>> sig_fft = fftpack.fft(sig)

Nevertheless only the positive part will be used for ﬁnding the frequency because the resulting power is symmetric:
>>> pidxs = np.where(sample_freq > 0) >>> freqs = sample_freq[pidxs] >>> power = np.abs(sig_fft)[pidxs]

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1.7.9 Interpolation: scipy.interpolate
The scipy.interpolate is useful for ﬁtting a function from experimental data and thus evaluating points where no measure exists. The module is based on the FITPACK Fortran subroutines from the netlib project. By imagining experimental data close to a sinus function:
>>> measured_time = np.linspace(0, 1, 10) >>> noise = (np.random.random(10)*2 - 1) * 1e-1 >>> measures = np.sin(2 * np.pi * measured_time) + noise

scipy.interpolate.interp2d is similar to interp1d, but for 2-D arrays. Note that for the interp family, the computed time must stay within the measured time range. See the summary exercice on ‘Maximum wind speed prediction at the Sprogø station‘_ for a more advance spline interpolation example.

1.7.10 Optimization and ﬁt: scipy.optimize
Optimization is the problem of ﬁnding a numerical solution to a minimization or equality. The scipy.optimize module provides useful algorithms for function minimization (scalar or multidimensional), curve ﬁtting and root ﬁnding. Example: Minimizing a scalar function using different algorithms Let’s deﬁne the following function:
>>> def f(x): ... return x**2 + 10*np.sin(x)

The interp1d class can built a linear interpolation function:
>>> from scipy.interpolate import interp1d >>> linear_interp = interp1d(measured_time, measures)

Then the linear_interp instance needs to be evaluated on time of interest:
>>> computed_time = np.linspace(0, 1, 50) >>> linear_results = linear_interp(computed_time)

and plot it:
>>> x = np.arange(-10,10,0.1) >>> plt.plot(x, f(x)) >>> plt.show()

A cubic interpolation can also be selected by providing the kind optional keyword argument:
>>> cubic_interp = interp1d(measured_time, measures, kind=’cubic’) >>> cubic_results = cubic_interp(computed_time)

The results are now gathered on a Matplotlib ﬁgure generated by the script scipy-interpolation.py.

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This approach take 20 ms on our computer. This simple alorithm becomes very slow as the size of the grid grows, so you should use optimize.brent instead for scalar functions:
>>> optimize.brent(f) -1.3064400120612139

To ﬁnd the local minimum, let’s add some constraints on the variable using optimize.fminbound:
>>> # search the minimum only between 0 and 10 >>> optimize.fminbound(f, 0, 10) array([ 3.83746712])

You can ﬁnd algorithms with the same functionalities for multi-dimensional problems in scipy.optimize. See the summary exercise on Non linear least squares curve ﬁtting: application to point extraction in topographical lidar data (page 95) for a more advanced example.

1.7.11 Image processing: scipy.ndimage
The submodule dedicated to image processing in scipy is scipy.ndimage.
>>> from scipy import ndimage

Image processing routines may be sorted according to the category of processing they perform. Geometrical transformations on images Changing orientation, resolution, .. This function has a global minimum around -1.3 and a local minimum around 3.8. Local (convex) optimization The general and efﬁcient way to ﬁnd a minimum for this function is to conduct a gradient descent starting from a given initial point. The BFGS algorithm is a good way of doing this:
>>> optimize.fmin_bfgs(f, 0) Optimization terminated successfully. Current function value: -7.945823 Iterations: 5 Function evaluations: 24 Gradient evaluations: 8 array([-1.30644003]) >>> import scipy >>> lena = scipy.lena() >>> shifted_lena = ndimage.shift(lena, (50, 50)) >>> shifted_lena2 = ndimage.shift(lena, (50, 50), mode=’nearest’) >>> rotated_lena = ndimage.rotate(lena, 30) >>> cropped_lena = lena[50:-50, 50:-50] >>> zoomed_lena = ndimage.zoom(lena, 2) >>> zoomed_lena.shape (1024, 1024)

This resolution takes 4.11ms on our computer. The problem with this approach is that, if the function has local minima (is not convex), the algorithm may ﬁnd these local minima instead of the global minimum depending on the initial point. If we don’t know the neighborhood of the global minima to choose the initial point, we need to resort to costlier global opimtization. Global optimization To ﬁnd the global minimum, the simplest algorithm is the brute force algorithm, in which the function is evaluated on each point of a given grid:
>>> from scipy import optimize >>> grid = (-10, 10, 0.1) >>> optimize.brute(f, (grid,)) array([-1.30641113])

In [35]: subplot(151) Out[35]: <matplotlib.axes.AxesSubplot object at 0x925f46c> In [36]: imshow(shifted_lena, cmap=cm.gray) Out[36]: <matplotlib.image.AxesImage object at 0x9593f6c> In [37]: axis(’off’) Out[37]: (-0.5, 511.5, 511.5, -0.5) In [39]: # etc.

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Image ﬁltering
>>> >>> >>> >>> >>> >>> >>> >>> lena = scipy.lena() import numpy as np noisy_lena = np.copy(lena) noisy_lena += lena.std()*0.5*np.random.standard_normal(lena.shape) blurred_lena = ndimage.gaussian_filter(noisy_lena, sigma=3) median_lena = ndimage.median_filter(blurred_lena, size=5) import scipy.signal wiener_lena = scipy.signal.wiener(blurred_lena, (5,5))

array([[0, 1, 0], [1, 1, 1], [0, 1, 0]])

• Erosion
>>> a = np.zeros((7,7), dtype=np.int) >>> a[1:6, 2:5] = 1 >>> a array([[0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0]]) >>> ndimage.binary_erosion(a).astype(a.dtype) array([[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]]) >>> #Erosion removes objects smaller than the structure >>> ndimage.binary_erosion(a, structure=np.ones((5,5))).astype(a.dtype) array([[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0]])

And many other ﬁlters in scipy.ndimage.filters and scipy.signal can be applied to images Exercise Compare histograms for the different ﬁltered images.

Mathematical morphology Mathematical morphology is a mathematical theory that stems from set theory. It characterizes and transforms geometrical structures. Binary (black and white) images, in particular, can be transformed using this theory: the sets to be transformed are the sets of neighboring non-zero-valued pixels. The theory was also extended to gray-valued images.

• Dilation
>>> a = np.zeros((5, 5)) >>> a[2, 2] = 1 >>> a array([[ 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0.], [ 0., 0., 1., 0., 0.], [ 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0.]]) >>> ndimage.binary_dilation(a).astype(a.dtype) array([[ 0., 0., 0., 0., 0.], [ 0., 0., 1., 0., 0.], [ 0., 1., 1., 1., 0.], [ 0., 0., 1., 0., 0.], [ 0., 0., 0., 0., 0.]])

• Opening Elementary mathematical-morphology operations use a structuring element in order to modify other geometrical structures. Let us ﬁrst generate a structuring element
>>> el = ndimage.generate_binary_structure(2, 1) >>> el array([[False, True, False], [ True, True, True], [False, True, False]], dtype=bool) >>> el.astype(np.int) >>> a = np.zeros((5,5), dtype=np.int) >>> a[1:4, 1:4] = 1; a[4, 4] = 1 >>> a array([[0, 0, 0, 0, 0], [0, 1, 1, 1, 0], [0, 1, 1, 1, 0], [0, 1, 1, 1, 0], [0, 0, 0, 0, 1]]) >>> # Opening removes small objects >>> ndimage.binary_opening(a, structure=np.ones((3,3))).astype(np.int) array([[0, 0, 0, 0, 0],

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[0, 1, 1, 1, 0], [0, 1, 1, 1, 0], [0, 1, 1, 1, 0], [0, 0, 0, 0, 0]]) >>> # Opening can also smooth corners >>> ndimage.binary_opening(a).astype(np.int) array([[0, 0, 0, 0, 0], [0, 0, 1, 0, 0], [0, 1, 1, 1, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0]])

[0, [0, [0, [0, [0, [0,

0, 0, 0, 0, 0, 0,

0, 1, 1, 3, 0, 0,

0, 1, 1, 2, 0, 0,

0, 0, 0, 0, 0, 0,

0], 0], 0], 0], 0], 0]])

Measurements on images Let us ﬁrst generate a nice synthetic binary image.
>>> x, y = np.indices((100, 100)) >>> sig = np.sin(2*np.pi*x/50.)*np.sin(2*np.pi*y/50.)*(1+x*y/50.**2)**2 >>> mask = sig > 1

• Closing: ndimage.binary_closing Exercise Check that opening amounts to eroding, then dilating. An opening operation removes small structures, while a closing operation ﬁlls small holes. Such operation can therefore be used to “clean” an image.
>>> >>> >>> >>> >>> >>> a = np.zeros((50, 50)) a[10:-10, 10:-10] = 1 a += 0.25*np.random.standard_normal(a.shape) mask = a>=0.5 opened_mask = ndimage.binary_opening(mask) closed_mask = ndimage.binary_closing(opened_mask)

Now we look for various information about the objects in the image:
>>> labels, nb = ndimage.label(mask) >>> nb 8 >>> areas = ndimage.sum(mask, labels, xrange(1, labels.max()+1)) >>> areas [190.0, 45.0, 424.0, 278.0, 459.0, 190.0, 549.0, 424.0] >>> maxima = ndimage.maximum(sig, labels, xrange(1, labels.max()+1)) >>> maxima [1.8023823799830032, 1.1352760475048373, 5.5195407887291426, 2.4961181804217221, 6.7167361922608864, 1.8023823799830032, 16.765472169131161, 5.5195407887291426] >>> ndimage.find_objects(labels==4) [(slice(30, 48, None), slice(30, 48, None))] >>> sl = ndimage.find_objects(labels==4) >>> imshow(sig[sl[0]])

Exercise Check that the area of the reconstructed square is smaller than the area of the initial square. (The opposite would occur if the closing step was performed before the opening). For gray-valued images, eroding (resp. dilating) amounts to replacing a pixel by the minimal (resp. maximal) value among pixels covered by the structuring element centered on the pixel of interest.
>>> a = np.zeros((7,7), dtype=np.int) >>> a[1:6, 1:6] = 3 >>> a[4,4] = 2; a[2,3] = 1 >>> a array([[0, 0, 0, 0, 0, 0, 0], [0, 3, 3, 3, 3, 3, 0], [0, 3, 3, 1, 3, 3, 0], [0, 3, 3, 3, 3, 3, 0], [0, 3, 3, 3, 2, 3, 0], [0, 3, 3, 3, 3, 3, 0], [0, 0, 0, 0, 0, 0, 0]]) >>> ndimage.grey_erosion(a, size=(3,3)) array([[0, 0, 0, 0, 0, 0, 0],

See the summary exercise on Image processing application: counting bubbles and unmolten grains (page 99) for a more advanced example.

1.7.12 Summary exercices on scientiﬁc computing
The summary exercices use mainly Numpy, Scipy and Matplotlib. They ﬁrst aim at providing real life examples on scientiﬁc computing with Python. Once the groundwork is introduced, the interested user is invited to try some exercices.

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Maximum wind speed prediction at the Sprogø station The exercice goal is to predict the maximum wind speed occuring every 50 years even if no measure exists for such a period. The available data are only measured over 21 years at the Sprogø meteorological station located in Denmark. First, the statistical steps will be given and then illustrated with functions from the scipy.interpolate module. At the end the interested readers are invited to compute results from raw data and in a slightly different approach.

The quantile function is now going to be evaluated from the full range of probabilties:
>>> nprob = np.linspace(0, 1, 1e2) >>> fitted_max_speeds = quantile_func(nprob)

2% In the current model, the maximum wind speed occuring every 50 years is deﬁned as the upper 2% quantile. As a result, the cumulative probability value will be:
>>> fifty_prob = 1. - 0.02

Statistical approach
The annual maxima are supposed to ﬁt a normal probability density function. However such function is not going to be estimated because it gives a probability from a wind speed maxima. Finding the maximum wind speed occuring every 50 years requires the opposite approach, the result needs to be found from a deﬁned probabilty. That is the quantile function role and the exercice goal will be to ﬁnd it. In the current model, it is supposed that the maximum wind speed occuring every 50 years is deﬁned as the upper 2$%$ quantile. By deﬁnition, the quantile function is the inverse of the cumulative distribution function. The latter describes the probability distribution of an annual maxima. In the exercice, the cumulative probabilty p_i for a given year i is deﬁned as p_i = i/(N+1) with N = 21, the number of measured years. Thus it will be possible to calculate the cumulative probability of every measured wind speed maxima. From those experimental points, the scipy.interpolate module will be very useful for ﬁtting the quantile function. Finally the 50 years maxima is going to be evaluated from the cumulative probability of the 2% quantile.

So the storm wind speed occuring every 50 years can be guessed by:
>>> fifty_wind = quantile_func(fifty_prob) >>> fifty_wind array([ 32.97989825])

The results are now gathered on a Matplotlib ﬁgure.

Computing the cumulative probabilites
The annual wind speeds maxima have already been computed and saved in the numpy format in the ﬁle maxspeeds.npy, thus they will be loaded by using numpy:
>>> import numpy as np >>> max_speeds = np.load(’data/max-speeds.npy’) >>> years_nb = max_speeds.shape[0]

Following the cumulative probability deﬁnition p_i from the previous section, the corresponding values will be:
>>> cprob = (np.arange(years_nb, dtype=np.float32) + 1)/(years_nb + 1)

and they are assumed to ﬁt the given wind speeds:
>>> sorted_max_speeds = np.sort(max_speeds)

Prediction with UnivariateSpline
In this section the quantile function will be estimated by using the UnivariateSpline class which can represent a spline from points. The default behavior is to build a spline of degree 3 and points can have different weights according to their reliability. Variantes are InterpolatedUnivariateSpline and LSQUnivariateSpline on which errors checking is going to change. In case a 2D spline is wanted, the BivariateSpline class family is provided. All thoses classes for 1D and 2D splines use the FITPACK Fortran subroutines, that’s why a lower library access is available through the splrep and splev functions for respectively representing and evaluating a spline. Moreover interpolation functions without the use of FITPACK parameters are also provided for simpler use (see interp1d, interp2d, barycentric_interpolate and so on). For the Sprogø maxima wind speeds, the UnivariateSpline will be used because a spline of degree 3 seems to correctly ﬁt the data:
>>> from scipy.interpolate import UnivariateSpline >>> quantile_func = UnivariateSpline(cprob, sorted_max_speeds)

All those steps have been gathered in the script cumulative-wind-speed-prediction.py.

Exercice with the Gumbell distribution
The interested readers are now invited to make an exercice by using the wind speeds measured over 21 years. The measurement period is around 90 minutes (the original period was around 10 minutes but the ﬁle size has been reduced for making the exercice setup easier). The data are stored in numpy format inside the ﬁle sprogwindspeeds.npy. • The ﬁrst step will be to ﬁnd the annual maxima by using numpy and plot them as a matplotlib bar ﬁgure.

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• The second step will be to use the Gumbell distribution on cumulative probabilities p_i deﬁned as -log( -log(p_i) ) for ﬁtting a linear quantile function (remember that you can deﬁne the degree of the UnivariateSpline). Plotting the annual maxima versus the Gumbell distribution should give you the following ﬁgure.

• The last step will be to ﬁnd 34.23 m/s for the maximum wind speed occuring every 50 years. Once done, you may compare your code with a solution example available in the script gumbell-wind-speedprediction.py. Non linear least squares curve ﬁtting: application to point extraction in topographical lidar data The goal of this exercise is to ﬁt a model to some data. The data used in this tutorial are lidar data and are described in details in the following introductory paragraph. If you’re impatient and want to practise now, please skip it ang go directly to Loading and visualization (page 96).

Introduction
Lidars systems are optical rangeﬁnders that analyze property of scattered light to measure distances. Most of them emit a short light impulsion towards a target and record the reﬂected signal. This signal is then processed to extract the distance between the lidar sytem and the target. Topographical lidar systems are such systems embedded in airborne platforms. They measure distances between the platform and the Earth, so as to deliver information on the Earth’s topography (see [Mallet09] (page 171) for more details). In this tutorial, the goal is to analyze the waveform recorded by the lidar system 1 . Such a signal contains peaks whose center and amplitude permit to compute the position and some characteristics of the hit target. When the footprint of the laser beam is around 1m on the Earth surface, the beam can hit multiple targets during the two-way propagation (for example the ground and the top of a tree or building). The sum of the contributions of each target hit by the laser beam then produces a complex signal with multiple peaks, each one containing information about one target.
1 The data used for this tutorial are part of the demonstration data available for the FullAnalyze software and were kindly provided by the GIS DRAIX.

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One state of the art method to extract information from these data is to decompose them in a sum of Gaussian functions where each function represents the contribution of a target hit by the laser beam. Therefore, we use the scipy.optimize module to ﬁt a waveform to one or a sum of Gaussian functions.

• call scipy.optimize.leastsq Model A gaussian function deﬁned by B + A exp − t−µ σ
2

Loading and visualization
Load the ﬁrst waveform using:
>>> import numpy as np >>> waveform_1 = np.load(’data/waveform_1.npy’)

can be deﬁned in python by:
>>> def model(t, coeffs): ... return coeffs[0] + coeffs[1] * np.exp( - ((t-coeffs[2])/coeffs[3])**2 )

and visualize it:
>>> >>> >>> >>> import matplotlib.pyplot as plt t = np.arange(len(waveform_1)) plt.plot(t, waveform_1) plt.show()

where • coeffs[0] is B (noise) • coeffs[1] is A (amplitude) • coeffs[2] is µ (center) • coeffs[3] is σ (width) Initial solution An approximative initial solution that we can ﬁnd from looking at the graph is for instance:
>>> x0 = np.array([3, 30, 15, 1], dtype=float)

Fit scipy.optimize.leastsq minimizes the sum of squares of the function given as an argument. Basically, the function to minimize is the residuals (the difference between the data and the model):
>>> def residuals(coeffs, y, t): ... return y - model(t, coeffs)

So let’s get our solution by calling scipy.optimize.leastsq with the following arguments: • the function to minimize • an initial solution • the additional arguments to pass to the function
>>> from scipy.optimize import leastsq >>> x, flag = leastsq(residuals, x0, args=(waveform_1, t)) >>> print x [ 2.70363341 27.82020742 15.47924562 3.05636228]

And visualize the solution:
>>> plt.plot(t, waveform_1, t, model(t, x)) >>> plt.legend([’waveform’, ’model’]) >>> plt.show()

As you can notice, this waveform is a 80-bin-length signal with a single peak.

Fitting a waveform with a simple Gaussian model
The signal is very simple and can be modelled as a single Gaussian function and an offset corresponding to the background noise. To ﬁt the signal with the function, we must: • deﬁne the model • propose an initial solution

Remark: from scipy v0.8 and above, you should rather use scipy.optimize.curve_fit which takes the model and the data as arguments, so you don’t need to deﬁne the residuals any more.

Going further
• Try with a more complex waveform (for instance data/waveform_2.npy) that contains three significant peaks. You must adapt the model which is now a sum of Gaussian functions instead of only one Gaussian peak.

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Image processing application: counting bubbles and unmolten grains

• In some cases, writing an explicit function to compute the Jacobian is faster than letting leastsq estimate it numerically. Create a function to compute the Jacobian of the residuals and use it as an input for leastsq. • When we want to detect very small peaks in the signal, or when the initial guess is too far from a good solution, the result given by the algorithm is often not satisfying. Adding constraints to the parameters of the model enables to overcome such limitations. An example of a priori knowledge we can add is the sign of our variables (which are all positive). With the following initial solution:
>>> x0 = np.array([3, 50, 20, 1], dtype=float)

Statement of the problem
1. Open the image ﬁle MV_HFV_012.jpg and display it. Browse through the keyword arguments in the docstring of imshow to display the image with the “right” orientation (origin in the bottom left corner, and not the upper left corner as for standard arrays). This Scanning Element Microscopy image shows a glass sample (light gray matrix) with some bubbles (on black) and unmolten sand grains (dark gray). We wish to determine the fraction of the sample covered by these three phases, and to estimate the typical size of sand grains and bubbles, their sizes, etc. 2. Crop the image to remove the lower panel with measure information. 3. Slightly ﬁlter the image with a median ﬁlter in order to reﬁne its histogram. Check how the histogram changes. 4. Using the histogram of the ﬁltered image, determine thresholds that allow to deﬁne masks for sand pixels, glass pixels and bubble pixels. Other option (homework): write a function that determines automatically the thresholds from the minima of the histogram. 5. Display an image in which the three phases are colored with three different colors. 6. Use mathematical morphology to clean the different phases. 7. Attribute labels to all bubbles and sand grains, and remove from the sand mask grains that are smaller than 10 pixels. To do so, use ndimage.sum or np.bincount to compute the grain sizes. 8. Compute the mean size of bubbles.

compare the result of scipy.optimize.leastsq and what scipy.optimize.fmin_slsqp when adding boundary constraints.

you

can

get

with

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Proposed solution
Example of solution for the image processing exercise: unmolten grains in glass

1. Open the image ﬁle MV_HFV_012.jpg and display it. Browse through the keyword arguments in the docstring of imshow to display the image with the “right” orientation (origin in the bottom left corner, and not the upper left corner as for standard arrays).
>>> dat = imread(’MV_HFV_012.jpg’)

4. Using the histogram of the ﬁltered image, determine thresholds that allow to deﬁne masks for sand pixels, glass pixels and bubble pixels. Other option (homework): write a function that determines automatically the thresholds from the minima of the histogram.
>>> void = filtdat <= 50 >>> sand = np.logical_and(filtdat>50, filtdat<=114) >>> glass = filtdat > 114

2. Crop the image to remove the lower panel with measure information.
>>> dat = dat[60:]

3. Slightly ﬁlter the image with a median ﬁlter in order to reﬁne its histogram. Check how the histogram changes.
>>> filtdat = ndimage.median_filter(dat, size=(7,7)) >>> hi_dat = np.histogram(dat, bins=np.arange(256)) >>> hi_filtdat = np.histogram(filtdat, bins=np.arange(256))

5. Display an image in which the three phases are colored with three different colors.
>>> phases = void.astype(np.int) + 2*glass.astype(np.int) +\ 3*sand.astype(np.int)

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>>> mean_bubble_size, median_bubble_size (1699.875, 65.0)

6. Use mathematical morphology to clean the different phases.
>>> sand_op = ndimage.binary_opening(sand, iterations=2)

7. Attribute labels to all bubbles and sand grains, and remove from the sand mask grains that are smaller than 10 pixels. To do so, use ndimage.sum or np.bincount to compute the grain sizes.
>>> >>> ... >>> >>> sand_labels, sand_nb = ndimage.label(sand_op) sand_areas = np.array(ndimage.sum(sand_op, sand_labels,\ np.arange(sand_labels.max()+1))) mask = sand_areas>100 remove_small_sand = mask[sand_labels.ravel()].reshape(sand_labels.shape)

8. Compute the mean size of bubbles.
>>> >>> >>> >>> bubbles_labels, bubbles_nb = ndimage.label(void) bubbles_areas = np.bincount(bubbles_labels.ravel())[1:] mean_bubble_size = bubbles_areas.mean() median_bubble_size = np.median(bubbles_areas)

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Contents • • • • • • • • • Advanced Numpy (page 104) Life of ndarray (page 105) Universal functions (page 116) Interoperability features (page 125) Siblings: chararray, maskedarray, matrix (page 133) Summary (page 134) Hit list of the future for Numpy core (page 134) Contributing to Numpy/Scipy (page 134) Python 2 and 3, single code base (page 137)

Advanced topics

2.1.3 Life of ndarray
It’s...

2.1 Advanced Numpy
2.1.1 Abstract
Numpy is at the base of Python’s scientiﬁc stack of tools. Its purpose is simple: implementing efﬁcient operations on many items in a block of memory. Understanding how it works in detail helps in making efﬁcient use of its ﬂexibility, taking useful shortcuts, and in building new work based on it. This tutorial aims to cover: • Anatomy of Numpy arrays, and its consequences. Tips and tricks. • Universal functions: what, why, and what to do if you want a new one. • Integration with other tools: Numpy offers several ways to wrap any data in an ndarray, without unnecessary copies. • Recently added features, and what’s in them for me: PEP 3118 buffers, generalized ufuncs, ...

ndarray = block of memory + indexing scheme + data type descriptor • raw data • how to locate an element • how to interpret an element

2.1.2 Advanced Numpy
July 8, @ EuroScipy 2010 Pauli Virtanen
typedef struct PyArrayObject { PyObject_HEAD /* Block of memory */ char *data; /* Data type descriptor */ PyArray_Descr *descr; /* Indexing scheme */ int nd; npy_intp *dimensions; npy_intp *strides; /* Other stuff */ PyObject *base; int flags; PyObject *weakreflist; } PyArrayObject;

Prerequisites
• Numpy (>= 1.2; preferably newer...) • Cython (>= 0.12, for the Ufunc example) • PIL (I’ll use it in a couple of examples) • Source codes: – http://pav.iki.ﬁ/tmp/advnumpy-ex.zip (updated recently)
>>> import numpy as np

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Block of memory
>>> x = np.array([1, 2, 3, 4], dtype=np.int32) >>> x.data <read-write buffer for 0xa37bfd8, size 16, offset 0 at 0xa4eabe0> >>> str(x.data) ’\x01\x00\x00\x00\x02\x00\x00\x00\x03\x00\x00\x00\x04\x00\x00\x00’

Example: reading .wav ﬁles
The .wav ﬁle header: chunk_id chunk_size format fmt_id fmt_size audio_fmt num_channels sample_rate byte_rate block_align bits_per_sample data_id data_size "RIFF" 4-byte unsigned little-endian integer "WAVE" "fmt " 4-byte unsigned little-endian integer 2-byte unsigned little-endian integer 2-byte unsigned little-endian integer 4-byte unsigned little-endian integer 4-byte unsigned little-endian integer 2-byte unsigned little-endian integer 2-byte unsigned little-endian integer "data" 4-byte unsigned little-endian integer

Memory address of the data:
>>> x.__array_interface__[’data’][0] 159755776

Reminder: two ndarrays may share the same memory:
>>> x = np.array([1,2,3,4]) >>> y = x[:-1] >>> x[0] = 9 >>> y array([9, 2, 3])

• 44-byte block of raw data (in the beginning of the ﬁle) • ... followed by data_size bytes of actual sound data. The .wav ﬁle header as a Numpy structured data type:
>>> wav_header_dtype = np.dtype([ ("chunk_id", (str, 4)), # flexible-sized scalar type, item size 4 ("chunk_size", "<u4"), # little-endian unsigned 32-bit integer ("format", "S4"), # 4-byte string ("fmt_id", "S4"), ("fmt_size", "<u4"), ("audio_fmt", "<u2"), # ("num_channels", "<u2"), # .. more of the same ... ("sample_rate", "<u4"), # ("byte_rate", "<u4"), ("block_align", "<u2"), ("bits_per_sample", "<u2"), ("data_id", ("S1", (2, 2))), # sub-array, just for fun! ("data_size", "u4"), # # the sound data itself cannot be represented here: # it does not have a fixed size ])

Memory does not need to be owned by an ndarray:
>>> x = ’1234’ >>> y = np.frombuffer(x, dtype=np.int8) >>> y.data <read-only buffer for 0xa588ba8, size 4, offset 0 at 0xa55cd60> >>> y.base is x True >>> y.flags C_CONTIGUOUS : True F_CONTIGUOUS : True OWNDATA : False WRITEABLE : False ALIGNED : True UPDATEIFCOPY : False

The owndata and writeable ﬂags indicate status of the memory block. Data types

The descriptor
dtype describes a single item in the array: type itemsize byteorder ﬁelds shape scalar type of the data, one of: int8, int16, ﬂoat64, et al. (ﬁxed size) str, unicode, void (ﬂexible size) size of the data block byte order: big-endian > / little-endian < / not applicable | sub-dtypes, if it’s a structured data type shape of the array, if it’s a sub-array

See Also: wavreader.py
>>> wav_header_dtype[’format’] dtype(’|S4’) >>> wav_header_dtype.fields <dictproxy object at 0x85e9704> >>> wav_header_dtype.fields[’format’] (dtype(’|S4’), 8)

• The ﬁrst element is the sub-dtype in the structured data, corresponding to the name format • The second one is its offset (in bytes) from the beginning of the item Note: Mini-exercise, make a “sparse” dtype by using offsets, and only some of the ﬁelds:
>>> wav_header_dtype = np.dtype(dict( names=[’format’, ’sample_rate’, ’data_id’], offsets=[offset_1, offset_2, offset_3], # counted from start of structure in bytes formats=list of dtypes for each of the fields, ))

>>> np.dtype(int).type <type ’numpy.int32’> >>> np.dtype(int).itemsize 4 >>> np.dtype(int).byteorder ’=’

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and use that to read the sample rate, and data_id (as sub-array).
>>> f = open(’test.wav’, ’r’) >>> wav_header = np.fromfile(f, dtype=wav_header_dtype, count=1) >>> f.close() >>> print(wav_header) [ (’RIFF’, 17402L, ’WAVE’, ’fmt ’, 16L, 1, 1, 16000L, 32000L, 2, 16, [[’d’, ’a’], [’t’, ’a’]], 17366L)] >>> wav_header[’sample_rate’] array([16000], dtype=uint32)

array([1, 2, 3, 4], dtype=int8) >>> y + 256.0 array([ 257., 258., 259., 260.]) >>> y + np.array([256], dtype=np.int32) array([258, 259, 260, 261])

• Casting on setitem: dtype of the array is not changed on item assignment
>>> y[:] = y + 1.5 >>> y array([2, 3, 4, 5], dtype=int8)

Let’s try accessing the sub-array:
>>> wav_header[’data_id’] array([[[’d’, ’a’], [’t’, ’a’]]], dtype=’|S1’) >>> wav_header.shape (1,) >>> wav_header[’data_id’].shape (1, 2, 2)

Note: Exact rules: see documentation: http://docs.scipy.org/doc/numpy/reference/ufuncs.html#casting-rules

Re-interpretation / viewing
• Data block in memory (4 bytes) 0x01 || 0x02 || 0x03 || 0x04 – 4 of uint8, OR, – 4 of int8, OR, – 2 of int16, OR, – 1 of int32, OR, – 1 of ﬂoat32, OR, – ... How to switch from one to another? 1. Switch the dtype:
>>> x = np.array([1, 2, 3, 4], dtype=np.uint8) >>> x.dtype = "<i2" >>> x array([ 513, 1027], dtype=int16) >>> 0x0201, 0x0403 (513, 1027)

When accessing sub-arrays, the dimensions get added to the end! Note: There are existing modules such as wavfile, audiolab, etc. for loading sound data...

Casting and re-interpretation/views
casting • on assignment • on array construction • on arithmetic • etc. • and manually: .astype(dtype) data re-interpretation • manually: .view(dtype)

0x01

0x02

0x03

0x04

Casting
• Casting in arithmetic, in nutshell: – only type (not value!) of operands matters – largest “safe” type able to represent both is picked – scalars can “lose” to arrays in some situations • Casting in general copies data
>>> x = np.array([1, 2, 3, 4], dtype=np.float) >>> x array([ 1., 2., 3., 4.]) >>> y = x.astype(np.int8) >>> y array([1, 2, 3, 4], dtype=int8) >>> y + 1 array([2, 3, 4, 5], dtype=int8) >>> y + 256

Note: little-endian: least signiﬁcant byte is on the left in memory 2. Create a new view:
>>> y = x.view("<i4") >>> y array([67305985]) >>> 0x04030201 67305985

0x01 Note:

0x02

0x03

0x04

• .view() makes views, does not copy (or alter) the memory block • only changes the dtype (and adjusts array shape)
>>> x[1] = 5 >>> y array([328193]) >>> y.base is x True

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Mini-exercise: data re-interpretation
See Also: view-colors.py You have RGBA data in an array
>>> >>> >>> >>> >>> x = np.zeros((10, 10, 4), dtype=np.int8) x[:,:,0] = 1 x[:,:,1] = 2 x[:,:,2] = 3 x[:,:,3] = 4

Indexing scheme: strides

Main point
The question
>>> x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.int8) >>> str(x.data) ’\x01\x02\x03\x04\x05\x06\x07\x08\t’

where the last three dimensions are the R, B, and G, and alpha channels. How to make a (10, 10) structured array with ﬁeld names ‘r’, ‘g’, ‘b’, ‘a’ without copying data?
>>> y = ... >>> >>> >>> >>> assert assert assert assert (y[’r’] (y[’g’] (y[’b’] (y[’a’] == == == == 1).all() 2).all() 3).all() 4).all()

At which byte in x.data does the item x[1,2] begin? The answer (in Numpy) • strides: the number of bytes to jump to ﬁnd the next element • 1 stride per dimension
>>> x.strides (3, 1) >>> byte_offset = 3*1 + 1*2 >>> x.data[byte_offset] ’\x06’ >>> x[1,2] 6

# to find x[1,2]

Solution
>>> y = x.view([(’r’, ’i1’), (’g’, ’i1’), (’b’, ’i1’), (’a’, ’i1’)] )[:,:,0]

• simple, ﬂexible

C and Fortran order
>>> x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.int16, order=’C’) >>> x.strides (6, 2) >>> str(x.data) ’\x01\x00\x02\x00\x03\x00\x04\x00\x05\x00\x06\x00\x07\x00\x08\x00\t\x00’

Warning: Another array taking exactly 4 bytes of memory:
>>> y = np.array([[1, 3], [2, 4]], dtype=np.uint8).transpose() >>> x = y.copy() >>> x array([[1, 2], [3, 4]], dtype=uint8) >>> y array([[1, 2], [3, 4]], dtype=uint8) >>> x.view(np.int16) array([[ 513], [1027]], dtype=int16) >>> 0x0201, 0x0403 (513, 1027) >>> y.view(np.int16) array([[ 769, 1026]], dtype=int16)

• Need to jump 6 bytes to ﬁnd the next row • Need to jump 2 bytes to ﬁnd the next column
>>> y = np.array(x, order=’F’) >>> y.strides (2, 6) >>> str(y.data) ’\x01\x00\x04\x00\x07\x00\x02\x00\x05\x00\x08\x00\x03\x00\x06\x00\t\x00’

• What happened? • ... we need to look into what x[0,1] actually means
>>> 0x0301, 0x0402 (769, 1026)

• Need to jump 2 bytes to ﬁnd the next row • Need to jump 6 bytes to ﬁnd the next column • Similarly to higher dimensions: – C: last dimensions vary fastest (= smaller strides) – F: ﬁrst dimensions vary fastest

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shape = (d1 , d2 , ..., dn ) strides = (s1 , s2 , ..., sn ) sC = dj+1 dj+2 ...dn × itemsize j sF = d1 d2 ...dj−1 × itemsize j

Warning: as_strided does not check that you stay inside the memory block bounds...
>>> x = np.array([1, 2, 3, 4], dtype=np.int16) >>> as_strided(x, strides=(2*2,), shape=(2,)) array([1, 3], dtype=int16) >>> x[::2] array([1, 3], dtype=int16)

Note: Now we can understand the behavior of .view():
>>> y = np.array([[1, 3], [2, 4]], dtype=np.uint8).transpose() >>> x = y.copy()

See Also: stride-fakedims.py Exercise
array([1, 2, 3, 4], dtype=np.int8) -> array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]], dtype=np.int8)

Transposition does not affect the memory layout of the data, only strides
>>> (2, >>> (1, x.strides 1) y.strides 2)

>>> str(x.data) ’\x01\x02\x03\x04’ >>> str(y.data) ’\x01\x03\x02\x04’

using only as_strided.:
Hint: byte_offset = stride[0]*index[0] + stride[1]*index[1] + ...

• the results are different when interpreted as 2 of int16 • .copy() creates new arrays in the C order (by default)

Spoiler Stride can also be 0:
>>> x = np.array([1, 2, 3, 4], dtype=np.int8) >>> y = as_strided(x, strides=(0, 1), shape=(3, 4)) >>> y array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]], dtype=int8) >>> y.base.base is x True

Slicing with integers
• Everything can be represented by changing only shape, strides, and possibly adjusting the data pointer! • Never makes copies of the data
>>> x = np.array([1, 2, 3, 4, 5, 6], dtype=np.int32) >>> y = x[::-1] >>> y array([6, 5, 4, 3, 2, 1]) >>> y.strides (-4,) >>> y = x[2:] >>> y.__array_interface__[’data’][0] - x.__array_interface__[’data’][0] 8 >>> x = np.zeros((10, 10, 10), dtype=np.float) >>> x.strides (800, 80, 8) >>> x[::2,::3,::4].strides (1600, 240, 32)

Broadcasting
• Doing something useful with it: outer product of [1, 2, 3, 4] and [5, 6, 7]
>>> x = np.array([1, 2, 3, 4], dtype=np.int16) >>> x2 = as_strided(x, strides=(0, 1*2), shape=(3, 4)) >>> x2 array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]], dtype=int16) >>> y = np.array([5, 6, 7], dtype=np.int16) >>> y2 = as_strided(y, strides=(1*2, 0), shape=(3, 4)) >>> y2 array([[5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7]], dtype=int16) >>> x2 * array([[ [ [ y2 5, 10, 15, 20], 6, 12, 18, 24], 7, 14, 21, 28]], dtype=int16)

Example: fake dimensions with strides Stride manipulation
>>> from numpy.lib.stride_tricks import as_strided >>> help(as_strided) as_strided(x, shape=None, strides=None) Make an ndarray from the given array with the given shape and strides

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... seems somehow familiar ...
>>> x = np.array([1, 2, 3, 4], dtype=np.int16) >>> y = np.array([5, 6, 7], dtype=np.int16) >>> x[np.newaxis,:] * y[:,np.newaxis] array([[ 5, 10, 15, 20], [ 6, 12, 18, 24], [ 7, 14, 21, 28]], dtype=int16)

Compute the tensor trace
>>> x = np.arange(5*5*5*5).reshape(5,5,5,5) >>> s = 0 >>> for i in xrange(5): >>> for j in xrange(5): >>> s += x[j,i,j,i]

by striding, and using sum() on the result.
>>> y = as_strided(x, shape=(5, 5), strides=(TODO, TODO)) >>> s2 = ... >>> assert s == s2

• Internally, array broadcasting is indeed implemented using 0-strides.

More tricks: diagonals
See Also: stride-diagonals.py Challenge Pick diagonal entries of the matrix: (assume C memory order)
>>> x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.int32) >>> x_diag = as_strided(x, shape=(3,), strides=(???,))

Solution
>>> y = as_strided(x, shape=(5, 5), strides=((5*5*5+5)*x.itemsize, (5*5+1)*x.itemsize)) >>> s2 = y.sum()

CPU cache effects
Memory layout can affect performance:
In [1]: x = np.zeros((20000,))

Pick the ﬁrst super-diagonal entries [2, 6]. And the sub-diagonals? (Hint to the last two: slicing ﬁrst moves the point where striding starts from.) Solution Pick diagonals:
>>> x_diag = as_strided(x, shape=(3,), strides=((3+1)*x.itemsize,)) >>> x_diag array([1, 5, 9])

In [2]: y = np.zeros((20000*67,))[::67] In [3]: x.shape, y.shape ((20000,), (20000,)) In [4]: %timeit x.sum() 100000 loops, best of 3: 0.180 ms per loop In [5]: %timeit y.sum() 100000 loops, best of 3: 2.34 ms per loop In [6]: x.strides, y.strides ((8,), (536,))

Slice ﬁrst, to adjust the data pointer:
>>> as_strided(x[0,1:], shape=(2,), strides=((3+1)*x.itemsize,)) array([2, 6]) >>> as_strided(x[1:,0], shape=(2,), strides=((3+1)*x.itemsize,)) array([4, 8])

Smaller strides are faster?

Note:
>>> y = np.diag(x, k=1) >>> y array([2, 6])

However,
>>> y.flags.owndata True

• CPU pulls data from main memory to its cache in blocks • If many array items consecutively operated on ﬁt in a single block (small stride): – ⇒ fewer transfers needed See Also: – ⇒ faster

It makes a copy?! Room for improvement... (bug #xxx) See Also: stride-diagonals.py Challenge 2.1. Advanced Numpy 114

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Much more about this in the next session (Francesc Alted) today!

np.add, np.subtract, scipy.special.*, ...

Example: inplace operations (caveat emptor)
• Sometimes,
>>> a -= b

• Automatically support: broadcasting, casting, ... • The author of an ufunc only has to supply the elementwise operation, Numpy takes care of the rest. • The elementwise operation needs to be implemented in C (or, e.g., Cython)

is not the same as
>>> a -= b.copy() >>> x = np.array([[1, 2], [3, 4]]) >>> x -= x.transpose() >>> x array([[ 0, -1], [ 4, 0]]) >>> y = np.array([[1, 2], [3, 4]]) >>> y -= y.T.copy() >>> y array([[ 0, -1], [ 1, 0]])

Parts of an Ufunc
1. Provided by user
void ufunc_loop(void **args, int *dimensions, int *steps, void *data) { /* * int8 output = elementwise_function(int8 input_1, int8 input_2) * * This function must compute the ufunc for many values at once, * in the way shown below. */ char *input_1 = (char*)args[0]; char *input_2 = (char*)args[1]; char *output = (char*)args[2]; int i; for (i = 0; i < dimensions[0]; ++i) { *output = elementwise_function(*input_1, *input_2); input_1 += steps[0]; input_2 += steps[1]; output += steps[2]; } }

• x and x.transpose() share data • x -= x.transpose() modiﬁes the data element-by-element... • because x and x.transpose() have different striding, modiﬁed data re-appears on the RHS Findings in dissection

2. The Numpy part, built by
char types[3] types[0] = NPY_BYTE types[1] = NPY_BYTE types[2] = NPY_BYTE /* type of first input arg */ /* type of second input arg */ /* type of third input arg */

• memory block: may be shared, .base, .data • data type descriptor: structured data, sub-arrays, byte order, casting, viewing, .astype(), .view() • strided indexing: strides, C/F-order, slicing w/ integers, as_strided, broadcasting, stride tricks, diag, CPU cache coherence

PyObject *python_ufunc = PyUFunc_FromFuncAndData( ufunc_loop, NULL, types, 1, /* ntypes */ 2, /* num_inputs */ 1, /* num_outputs */ identity_element, name, docstring, unused)

• A ufunc can also support multiple different input-output type combinations.

2.1.4 Universal functions
What they are? • Ufunc performs and elementwise operation on all elements of an array. Examples: 2.1. Advanced Numpy 116

... can be made slightly easier
3. ufunc_loop is of very generic form, and Numpy provides pre-made ones

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PyUfunc_f_f float elementwise_func(float input_1) PyUfunc_ff_ffloat elementwise_func(float input_1, float input_2) PyUfunc_d_d double elementwise_func(double input_1) PyUfunc_dd_ddouble elementwise_func(double input_1, double input_2) PyUfunc_D_D elementwise_func(npy_cdouble *input, npy_cdouble* output) PyUfunc_DD_Delementwise_func(npy_cdouble *in1, npy_cdouble *in2, npy_cdouble* out) • Only elementwise_func needs to be supplied • ... except when your elementwise function is not in one of the above forms Exercise: building an ufunc from scratch The Mandelbrot fractal is deﬁned by the iteration z ← z2 + c where c = x + iy is a complex number. This iteration is repeated – if z stays ﬁnite no matter how long the iteration runs, c belongs to the Mandelbrot set. • Make ufunc called mandel(z0, c) that computes:
z = z0 for k in range(iterations): z = z*z + c

# # # # # # # # # # # #

Some points of note: - It’s *NOT* allowed to call any Python functions here. The Ufunc loop runs with the Python Global Interpreter Lock released. Hence, the ‘‘nogil‘‘. - And so all local variables must be declared with ‘‘cdef‘‘ - Note also that this function receives *pointers* to the data

cdef double complex z = z_in[0] cdef double complex c = c_in[0] cdef int k # the integer we use in the for loop # # TODO: write the Mandelbrot iteration for one point here, # as you would write it in Python. # # Say, use 100 as the maximum number of iterations, and 1000 # as the cutoff for z.real**2 + z.imag**2. # TODO: mandelbrot iteration should go here # Return the answer for this point z_out[0] = z

say, 100 iterations or until z.real**2 + z.imag**2 > 1000. Use it to determine which c are in the Mandelbrot set. • Our function is a simple one, so make use of the PyUFunc_* helpers. • Write it in Cython See Also: mandel.pyx, mandelplot.py
# # Fix the parts marked by TODO # # # Compile this file by (Cython >= 0.12 required because of the complex vars) # # cython mandel.pyx # python setup.py build_ext -i # # and try it out with, in this directory, # # >>> import mandel # >>> mandel.mandel(0, 1 + 2j) # # # The elementwise function # -----------------------cdef void mandel_single_point(double complex *z_in, double complex *c_in, double complex *z_out) nogil: # # The Mandelbrot iteration # # # # # #

Boilerplate Cython definitions The litany below is particularly long, but you don’t really need to read this part; it just pulls in stuff from the Numpy C headers. ----------------------------------------------------------

cdef extern from "numpy/arrayobject.h": void import_array() ctypedef int npy_intp cdef enum NPY_TYPES: NPY_DOUBLE NPY_CDOUBLE NPY_LONG cdef extern from "numpy/ufuncobject.h": void import_ufunc() ctypedef void (*PyUFuncGenericFunction)(char**, npy_intp*, npy_intp*, void*) object PyUFunc_FromFuncAndData(PyUFuncGenericFunction* func, void** data, char* types, int ntypes, int nin, int nout, int identity, char* name, char* doc, int c) # List of pre-defined loop functions void void void void void void void void void PyUFunc_f_f_As_d_d(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_d_d(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_f_f(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_g_g(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_F_F_As_D_D(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_F_F(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_D_D(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_G_G(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_ff_f_As_dd_d(char** args, npy_intp* dimensions, npy_intp* steps, void* func)

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void void void void void void void

PyUFunc_ff_f(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_dd_d(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_gg_g(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_FF_F_As_DD_D(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_DD_D(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_FF_F(char** args, npy_intp* dimensions, npy_intp* steps, void* func) PyUFunc_GG_G(char** args, npy_intp* dimensions, npy_intp* steps, void* func)

0, # ‘identity‘ element, never mind this "mandel", # function name "mandel(z, c) -> computes z*z + c", # docstring 0 # unused )

Reminder: some pre-made Ufunc loops: PyUfunc_f_f float elementwise_func(float input_1) PyUfunc_ff_f float elementwise_func(float input_1, float input_2) PyUfunc_d_d double elementwise_func(double input_1) PyUfunc_dd_d double elementwise_func(double input_1, double input_2) PyUfunc_D_D elementwise_func(complex_double *input, complex_double* output) PyUfunc_DD_D elementwise_func(complex_double *in1, complex_double *in2, complex_double* out) Type codes:
NPY_BOOL, NPY_BYTE, NPY_UBYTE, NPY_SHORT, NPY_USHORT, NPY_INT, NPY_UINT, NPY_LONG, NPY_ULONG, NPY_LONGLONG, NPY_ULONGLONG, NPY_FLOAT, NPY_DOUBLE, NPY_LONGDOUBLE, NPY_CFLOAT, NPY_CDOUBLE, NPY_CLONGDOUBLE, NPY_DATETIME, NPY_TIMEDELTA, NPY_OBJECT, NPY_STRING, NPY_UNICODE, NPY_VOID

# Required module initialization # -----------------------------import_array() import_ufunc()

# The actual ufunc declaration # ---------------------------cdef PyUFuncGenericFunction loop_func[1] cdef char input_output_types[3] cdef void *elementwise_funcs[1] # # # # # # # # # # # # # # # # # # # # # # Reminder: some pre-made Ufunc loops: ================ ‘‘PyUfunc_f_f‘‘ ‘‘PyUfunc_ff_f‘‘ ‘‘PyUfunc_d_d‘‘ ‘‘PyUfunc_dd_d‘‘ ‘‘PyUfunc_D_D‘‘ ‘‘PyUfunc_DD_D‘‘ ================ ======================================================= ‘‘float elementwise_func(float input_1)‘‘ ‘‘float elementwise_func(float input_1, float input_2)‘‘ ‘‘double elementwise_func(double input_1)‘‘ ‘‘double elementwise_func(double input_1, double input_2)‘‘ ‘‘elementwise_func(complex_double *input, complex_double* complex_double)‘‘ ‘‘elementwise_func(complex_double *in1, complex_double *in2, complex_double* out)‘‘ =======================================================

Solution: building an ufunc from scratch
# The elementwise function # -----------------------cdef void mandel_single_point(double complex *z_in, double complex *c_in, double complex *z_out) nogil: # # The Mandelbrot iteration # # # # # # # # # # # # # #

The full list is above.

Type codes: NPY_BOOL, NPY_BYTE, NPY_UBYTE, NPY_SHORT, NPY_USHORT, NPY_INT, NPY_UINT, NPY_LONG, NPY_ULONG, NPY_LONGLONG, NPY_ULONGLONG, NPY_FLOAT, NPY_DOUBLE, NPY_LONGDOUBLE, NPY_CFLOAT, NPY_CDOUBLE, NPY_CLONGDOUBLE, NPY_DATETIME, NPY_TIMEDELTA, NPY_OBJECT, NPY_STRING, NPY_UNICODE, NPY_VOID

loop_func[0] = ... TODO: suitable PyUFunc_* ... input_output_types[0] = ... TODO ... ... TODO: fill in rest of input_output_types ... # This thing is passed as the ‘‘data‘‘ parameter for the generic # PyUFunc_* loop, to let it know which function it should call. elementwise_funcs[0] = <void*>mandel_single_point # Construct the ufunc:

cdef double complex z = z_in[0] cdef double complex c = c_in[0] cdef int k # the integer we use in the for loop # Straightforward iteration

mandel = PyUFunc_FromFuncAndData( loop_func, elementwise_funcs, input_output_types, 1, # number of supported input types TODO, # number of input args TODO, # number of output args

for k in range(100): z = z*z + c if z.real**2 + z.imag**2 > 1000: break # Return the answer for this point

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z_out[0] = z

y = np.linspace(-1.4, 1.4, 1000) c = x[None,:] + 1j*y[:,None] z = mandel.mandel(c, c) import matplotlib.pyplot as plt plt.imshow(abs(z)**2 < 1000, extent=[-1.7, 0.6, -1.4, 1.4]) plt.gray() plt.show()

# # # # #

Boilerplate Cython definitions You don’t really need to read this part, it just pulls in stuff from the Numpy C headers. ----------------------------------------------------------

cdef extern from "numpy/arrayobject.h": void import_array() ctypedef int npy_intp cdef enum NPY_TYPES: NPY_CDOUBLE cdef extern from "numpy/ufuncobject.h": void import_ufunc() ctypedef void (*PyUFuncGenericFunction)(char**, npy_intp*, npy_intp*, void*) object PyUFunc_FromFuncAndData(PyUFuncGenericFunction* func, void** data, char* types, int ntypes, int nin, int nout, int identity, char* name, char* doc, int c) void PyUFunc_DD_D(char**, npy_intp*, npy_intp*, void*) # Required module initialization # -----------------------------import_array() import_ufunc()

Note: Most of the boilerplate could be automated by these Cython modules: http://wiki.cython.org/MarkLodato/CreatingUfuncs

Several accepted input types
E.g. supporting both single- and double-precision versions
cdef void mandel_single_point(double complex *z_in, double complex *c_in, double complex *z_out) nogil: ... cdef void mandel_single_point_singleprec(float complex *z_in, float complex *c_in, float complex *z_out) nogil: ... cdef PyUFuncGenericFunction loop_funcs[2] cdef char input_output_types[3*2] cdef void *elementwise_funcs[1*2] loop_funcs[0] = PyUFunc_DD_D input_output_types[0] = NPY_CDOUBLE input_output_types[1] = NPY_CDOUBLE input_output_types[2] = NPY_CDOUBLE elementwise_funcs[0] = <void*>mandel_single_point loop_funcs[1] = PyUFunc_FF_F input_output_types[3] = NPY_CFLOAT input_output_types[4] = NPY_CFLOAT input_output_types[5] = NPY_CFLOAT elementwise_funcs[1] = <void*>mandel_single_point_singleprec mandel = PyUFunc_FromFuncAndData( loop_func, elementwise_funcs,

# The actual ufunc declaration # ---------------------------cdef PyUFuncGenericFunction loop_func[1] cdef char input_output_types[3] cdef void *elementwise_funcs[1] loop_func[0] = PyUFunc_DD_D input_output_types[0] = NPY_CDOUBLE input_output_types[1] = NPY_CDOUBLE input_output_types[2] = NPY_CDOUBLE elementwise_funcs[0] = <void*>mandel_single_point mandel = PyUFunc_FromFuncAndData( loop_func, elementwise_funcs, input_output_types, 1, # number of supported input types 2, # number of input args 1, # number of output args 0, # ‘identity‘ element, never mind this "mandel", # function name "mandel(z, c) -> computes iterated z*z + c", # docstring 0 # unused ) import numpy as np import mandel x = np.linspace(-1.7, 0.6, 1000)

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input_output_types, 2, # number of supported input types <---------------2, # number of input args 1, # number of output args 0, # ‘identity‘ element, never mind this "mandel", # function name "mandel(z, c) -> computes iterated z*z + c", # docstring 0 # unused )

Generalized ufunc loop
Matrix multiplication (m,n),(n,p) -> (m,p)
void gufunc_loop(void **args, int *dimensions, int *steps, void *data) { char *input_1 = (char*)args[0]; /* these are as previously */ char *input_2 = (char*)args[1]; char *output = (char*)args[2]; int int int int int int input_1_stride_m = steps[3]; /* strides for the core dimensions */ input_1_stride_n = steps[4]; /* are added after the non-core */ input_2_strides_n = steps[5]; /* steps */ input_2_strides_p = steps[6]; output_strides_n = steps[7]; output_strides_p = steps[8];

Generalized ufuncs ufunc output = elementwise_function(input) Both output and input can be a single array element only. generalized ufunc output and input can be arrays with a ﬁxed number of dimensions For example, matrix trace (sum of diag elements):
input shape = (n, n) output shape = () (n, n) -> () i.e. scalar

int m = dimension[1]; /* core dimensions are added after */ int n = dimension[2]; /* the main dimension; order as in */ int p = dimension[3]; /* signature */ int i; for (i = 0; i < dimensions[0]; ++i) { matmul_for_strided_matrices(input_1, input_2, output, strides for each array...); input_1 += steps[0]; input_2 += steps[1]; output += steps[2]; } }

Matrix product:
input_1 shape = (m, n) input_2 shape = (n, p) output shape = (m, p) (m, n), (n, p) -> (m, p)

2.1.5 Interoperability features
Sharing multidimensional, typed data Suppose you

• This is called the “signature” of the generalized ufunc • The dimensions on which the g-ufunc acts, are “core dimensions”

Status in Numpy
• g-ufuncs are in Numpy already ... • new ones can be created with PyUFunc_FromFuncAndDataAndSignature • ... but we don’t ship with public g-ufuncs, except for testing, ATM
>>> import numpy.core.umath_tests as ut >>> ut.matrix_multiply.signature ’(m,n),(n,p)->(m,p)’ >>> x = np.ones((10, 2, 4)) >>> y = np.ones((10, 4, 5)) >>> ut.matrix_multiply(x, y).shape (10, 2, 5)

1. Write a library than handles (multidimensional) binary data, 2. Want to make it easy to manipulate the data with Numpy, or whatever other library, 3. ... but would not like to have Numpy as a dependency. Currently, 3 solutions: 1. the “old” buffer interface 2. the array interface 3. the “new” buffer interface (PEP 3118) The old buffer protocol • Only 1-D buffers • No data type information • C-level interface; PyBufferProcs tp_as_buffer in the type object • But it’s integrated into Python (e.g. strings support it)

• the last two dimensions became core dimensions, and are modiﬁed as per the signature • otherwise, the g-ufunc operates “elementwise” • matrix multiplication this way could be useful for operating on many small matrices at once

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Mini-exercise using PIL (Python Imaging Library): See Also: pilbuffer.py
>>> import Image >>> x = np.zeros((200, 200, 4), dtype=np.int8)

• TODO: RGBA images consist of 32-bit integers whose bytes are [RR,GG,BB,AA]. – Fill x with opaque red [255,0,0,255] – Mangle it to (200,200) 32-bit integer array so that PIL accepts it
>>> img = Image.frombuffer("RGBA", (200, 200), data) >>> img.save(’test.png’)

Q: Check what happens if x is now modiﬁed, and img saved again. The old buffer protocol
import numpy as np import Image # Let’s make a sample image, RGBA format x = np.zeros((200, 200, 4), dtype=np.int8) x[:,:,0] = 254 # red x[:,:,3] = 255 # opaque data = x.view(np.int32) # Check that you understand why this is OK! img = Image.frombuffer("RGBA", (200, 200), data) img.save(’test.png’) # # Modify the original data, and save again. # # It turns out that PIL, which knows next to nothing about Numpy, # happily shares the same data. # x[:,:,1] = 254 img.save(’test2.png’)

Array interface protocol • Multidimensional buffers • Data type information present • Numpy-speciﬁc approach; slowly deprecated (but not going away) • Not integrated in Python otherwise See Also: Documentation: http://docs.scipy.org/doc/numpy/reference/arrays.interface.html
>>> x = np.array([[1, 2], [3, 4]]) >>> x.__array_interface__ {’data’: (171694552, False), # memory address of data, is readonly? ’descr’: [(’’, ’<i4’)], # data type descriptor ’typestr’: ’<i4’, # same, in another form ’strides’: None, # strides; or None if in C-order ’shape’: (2, 2), ’version’: 3, } >>> import Image >>> img = Image.open(’test.png’)

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>>> img.__array_interface__ {’data’: a very long string, ’shape’: (200, 200, 4), ’typestr’: ’|u1’} >>> x = np.asarray(img) >>> x.shape (200, 200, 4) >>> x.dtype dtype(’uint8’)

>>> import array >>> x = array.array(’h’, b’1212’) >>> y = np.asarray(x) >>> y array([12849, 12849], dtype=int16) >>> y.base <memory at 0xa225acc>

# 2 of int16, but no Numpy!

Note: A more C-friendly variant of the array interface is also deﬁned. The new buffer protocol: PEP 3118 • Multidimensional buffers • Data type information present • C-level interface; extensions to tp_as_buffer • Integrated into Python (>= 2.6) • Replaces the “old” buffer interface in Python 3 • Next releases of Numpy (>= 1.5) will also implement it fully • Demo in Python 3 / Numpy dev version:
Python 3.1.2 (r312:79147, Apr 15 2010, 12:35:07) [GCC 4.4.3] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> import numpy as np >>> np.__version__ ’2.0.0.dev8469+15dcfb’

PEP 3118 – details • Suppose: you have a new Python extension type MyObject (deﬁned in C) • ... and want it to interoperable with a (2, 2) array of int32 See Also: myobject.c, myobject_test.py, setup_myobject.py
static PyBufferProcs myobject_as_buffer = { (getbufferproc)myobject_getbuffer, (releasebufferproc)myobject_releasebuffer, }; /* .. and stick this to the type object */ static int myobject_getbuffer(PyObject *obj, Py_buffer *view, int flags) { PyMyObjectObject *self = (PyMyObjectObject*)obj; /* Called when something requests that a MyObject-type object provides a buffer interface */ view->buf = self->buffer; view->readonly = 0; view->format = "i"; view->shape = malloc(sizeof(Py_ssize_t) * 2); view->strides = malloc(sizeof(Py_ssize_t) * 2);; view->suboffsets = NULL; TODO: Just fill in view->len, view->itemsize, view->strides[0] and 1, view->shape[0] and 1, and view->ndim. /* Note: if correct interpretation *requires* strides or shape, you need to check flags for what was requested, and raise appropriate errors. The same if the buffer is not readable. */ view->obj = (PyObject*)self; Py_INCREF(self); return 0; } static void myobject_releasebuffer(PyMemoryViewObject *self, Py_buffer *view) { if (view->shape) { free(view->shape); view->shape = NULL; }

• Memoryview object exposes some of the stuff Python-side
>>> x = np.array([[1, 2], [3, 4]]) >>> y = memoryview(x) >>> y.format ’l’ >>> y.itemsize 4 >>> y.ndim 2 >>> y.readonly False >>> y.shape (2, 2) >>> y.strides (8, 4)

• Roundtrips work
>>> z = np.asarray(y) >>> z array([[1, 2], [3, 4]]) >>> x[0,0] = 9 >>> z array([[9, 2], [3, 4]])

• Interoperability with the built-in Python array module

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if (view->strides) { free(view->strides); view->strides = NULL; } } /* * Sample implementation of a custom data type that exposes an array * interface. (And does nothing else :) * * Requires Python >= 3.1 */ /* * Mini-exercises: * * - make the array strided * - change the data type * */ #define PY_SSIZE_T_CLEAN #include <Python.h> #include "structmember.h"

return 0; } static void myobject_releasebuffer(PyMemoryViewObject *self, Py_buffer *view) { if (view->shape) { free(view->shape); view->shape = NULL; } if (view->strides) { free(view->strides); view->strides = NULL; } }

static PyBufferProcs myobject_as_buffer = { (getbufferproc)myobject_getbuffer, (releasebufferproc)myobject_releasebuffer, }; /* * Standard stuff follows */ PyTypeObject PyMyObject_Type; static PyObject * myobject_new(PyTypeObject *subtype, PyObject *args, PyObject *kwds) { PyMyObjectObject *self; static char *kwlist[] = {NULL}; if (!PyArg_ParseTupleAndKeywords(args, kwds, "", kwlist)) { return NULL; } self = (PyMyObjectObject *) PyObject_New(PyMyObjectObject, &PyMyObject_Type); self->buffer[0] = 1; self->buffer[1] = 2; self->buffer[2] = 3; self->buffer[3] = 4; return (PyObject*)self; }

typedef struct { PyObject_HEAD int buffer[4]; } PyMyObjectObject;

static int myobject_getbuffer(PyObject *obj, Py_buffer *view, int flags) { PyMyObjectObject *self = (PyMyObjectObject*)obj; /* Called when something requests that a MyObject-type object provides a buffer interface */ view->buf = self->buffer; view->readonly = 0; view->format = "i"; view->len = 4; view->itemsize = sizeof(int); view->ndim = 2; view->shape = malloc(sizeof(Py_ssize_t) * 2); view->shape[0] = 2; view->shape[1] = 2; view->strides = malloc(sizeof(Py_ssize_t) * 2);; view->strides[0] = 2*sizeof(int); view->strides[1] = sizeof(int); view->suboffsets = NULL; /* Note: if correct interpretation *requires* strides or shape, you need to check flags for what was requested, and raise appropriate errors. The same if the buffer is not readable. */ view->obj = (PyObject*)self; Py_INCREF(self);

PyTypeObject PyMyObject_Type = { PyVarObject_HEAD_INIT(NULL, 0) "MyObject", sizeof(PyMyObjectObject), 0, /* methods */ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

/* tp_itemsize */ /* /* /* /* /* /* /* /* /* /* tp_dealloc */ tp_print */ tp_getattr */ tp_setattr */ tp_reserved */ tp_repr */ tp_as_number */ tp_as_sequence */ tp_as_mapping */ tp_hash */

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0, 0, 0, 0, &myobject_as_buffer, Py_TPFLAGS_DEFAULT, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, myobject_new, 0, 0, 0, 0, 0, 0, 0, 0, 0, }; struct PyModuleDef moduledef = { PyModuleDef_HEAD_INIT, "myobject", NULL, -1, NULL, NULL, NULL, NULL, NULL }; PyObject *PyInit_myobject(void) { PyObject *m, *d; if (PyType_Ready(&PyMyObject_Type) < 0) { return NULL; } m = PyModule_Create(&moduledef); d = PyModule_GetDict(m);

/* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /* /*

tp_call */ tp_str */ tp_getattro */ tp_setattro */ tp_as_buffer */ tp_flags */ tp_doc */ tp_traverse */ tp_clear */ tp_richcompare */ tp_weaklistoffset */ tp_iter */ tp_iternext */ tp_methods */ tp_members */ tp_getset */ tp_base */ tp_dict */ tp_descr_get */ tp_descr_set */ tp_dictoffset */ tp_init */ tp_alloc */ tp_new */ tp_free */ tp_is_gc */ tp_bases */ tp_mro */ tp_cache */ tp_subclasses */ tp_weaklist */ tp_del */ tp_version_tag */

2.1.6 Siblings: chararray, maskedarray, matrix chararray — vectorized string operations
>>> x = np.array([’a’, ’ bbb’, ’ ccc’]).view(np.chararray) >>> x.lstrip(’ ’) chararray([’a’, ’bbb’, ’ccc’], dtype=’|S5’) >>> x.upper() chararray([’A’, ’ BBB’, ’ CCC’], dtype=’|S5’)

Note: .view() has a second meaning: it can make an ndarray an instance of a specialized ndarray subclass

MaskedArray — dealing with (propagation of) missing data
• For ﬂoats one could use NaN’s, but masks work for all types
>>> x = np.ma.array([1, 2, 3, 4], mask=[0, 1, 0, 1]) >>> x masked_array(data = [1 -- 3 --], mask = [False True False True], fill_value = 999999) >>> y = np.ma.array([1, 2, 3, 4], mask=[0, 1, 1, 1]) >>> x + y masked_array(data = [2 -- -- --], mask = [False True True True], fill_value = 999999)

• Masking versions of common functions
>>> np.ma.sqrt([1, -1, 2, -2]) masked_array(data = [1.0 -- 1.41421356237 --], mask = [False True False True], fill_value = 1e+20)

Note: There’s more to them!

recarray — purely convenience
>>> arr = np.array([(’a’, 1), (’b’, 2)], dtype=[(’x’, ’S1’), (’y’, int)]) >>> arr2 = arr.view(np.recarray) >>> arr2.x chararray([’a’, ’b’], dtype=’|S1’) >>> arr2.y array([1, 2])

matrix — convenience?
• always 2-D • * is the matrix product, not the elementwise one
>>> np.matrix([[1, 0], [0, 1]]) * np.matrix([[1, 2], [3, 4]]) matrix([[1, 2], [3, 4]])

Py_INCREF(&PyMyObject_Type); PyDict_SetItemString(d, "MyObject", (PyObject *)&PyMyObject_Type); return m; }

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2.1.7 Summary
• Anatomy of the ndarray: data, dtype, strides. • Universal functions: elementwise operations, how to make new ones • Ndarray subclasses • Various buffer interfaces for integration with other tools • Recent additions: PEP 3118, generalized ufuncs

– No replies in a week or so? Just ﬁle a bug ticket.

Good bug report
Title: numpy.random.permutations fails for non-integer arguments I’m trying to generate random permutations, using numpy.random.permutations When calling numpy.random.permutation with non-integer arguments it fails with a cryptic error message: >>> np.random.permutation(12) array([10, 9, 4, 7, 3, 8, 0, 6, 5, 1, 11, 2]) >>> np.random.permutation(long(12)) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "mtrand.pyx", line 3311, in mtrand.RandomState.permutation File "mtrand.pyx", line 3254, in mtrand.RandomState.shuffle TypeError: len() of unsized object This also happens with long arguments, and so np.random.permutation(X.shape[0]) where X is an array fails on 64 bit windows (where shape is a tuple of longs). It would be great if it could cast to integer or at least raise a proper error for non-integer types. I’m using Numpy 1.4.1, built from the official tarball, on Windows 64 with Visual studio 2008, on Python.org 64-bit Python.

2.1.8 Hit list of the future for Numpy core
• Python 3 – no wait, it already works! (Numpy 1.5, sooner or later) • PEP 3118 – hey, also that’s there! (Numpy 1.5) • Breaking backwards binary compatibility, we need to clean up a bit (Numpy 2.0, later) • Refactoring Python out of innards of Numpy (Numpy 2.0 or later, I’d guess) • Numpy on PyPy? (a GSoC project is working on this currently) • Memory access pattern optimizations for more efﬁcient CPU cache usage? • We should really have some public generalized ufuncs (e.g. multiplying many small matrices is often asked for) • Fixing the inplace operations so that
>>> a += b

is guaranteed to produce the same a as
>>> a = a + b

0. What are you trying to do? 1. Small code snippet reproducing the bug (if possible) • What actually happens • What you’d expect 2. Platform (Windows / Linux / OSX, 32/64 bits, x86/PPC, ...) 3. Version of Numpy/Scipy
>>> print numpy.__version__

Preferably without sacriﬁcing too much performance-wise...

2.1.9 Contributing to Numpy/Scipy
Get this tutorial: http://www.euroscipy.org/talk/882 Why • “There’s a bug?” • “I don’t understand what this is supposed to do?” • “I have this fancy code. Would you like to have it?” • “I’d like to help! What can I do?” Reporting bugs • Bug tracker (prefer this) – http://projects.scipy.org/numpy – http://projects.scipy.org/scipy – Click the “Register” link to get an account • Mailing lists ( scipy.org/Mailing_Lists ) – If you’re unsure 2.1. Advanced Numpy 134

Check that the following is what you expect
>>> print numpy.__file__

In case you have old/broken Numpy installations lying around. If unsure, try to remove existing Numpy installations, and reinstall... Contributing to documentation 1. Documentation editor • http://docs.scipy.org/numpy • Registration – Register an account – Subscribe to scipy-dev ML (subscribers-only) – Problem with mailing lists: you get mail 2.1. Advanced Numpy 135

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* But: you can turn mail delivery off * “change your subscription options”, at the bottom of http://mail.scipy.org/mailman/listinfo/scipy-dev – Send a mail @ scipy-dev mailing list; ask for activation:
To: scipy-dev@scipy.org Hi, I’d like to edit Numpy/Scipy docstrings. My account is XXXXX Cheers, N. N.

How to help, in general • Bug ﬁxes always welcome! – What irks you most – Browse the tracker • Documentation work – API docs: improvements to docstrings * Know some Scipy module well? – User guide * Needs to be done eventually. * Want to think? Come up with a Table of Contents http://scipy.org/Developer_Zone/UG_Toc • Ask on communication channels: – numpy-discussion list – scipy-dev list – or now @ Euroscipy :)

• Check the style guide: – http://docs.scipy.org/numpy/ – Don’t be intimidated; to ﬁx a small thing, just ﬁx it • Edit 2. Edit sources and send patches (as for bugs) 3. Complain on the ML Contributing features 0. Ask on ML, if unsure where it should go 1. Write a patch, add an enhancement ticket on the bug tracket 2. OR, create a Git branch implementing the feature + add enhancement ticket. • Especially for big/invasive additions • http://projects.scipy.org/numpy/wiki/GitMirror • http://www.spheredev.org/wiki/Git_for_the_lazy
# Clone numpy repository git clone --origin svn http://projects.scipy.org/git/numpy.git numpy cd numpy # Create a feature branch git checkout -b name-of-my-feature-branch <edit stuff> git commit -a

2.1.10 Python 2 and 3, single code base
• The brave new future? • You can do it! The case of Numpy • 150 000+ SLOC • Numpy is complicated (PyString, PyInt, buffer interfaces, ...)! Porting: • Python 2 and 3, with a single code base • Took ca. 2-3 man-weeks • git log --grep="3K" -M -C -p | filterdiff -x ’*/mtrand.c’ | diffstat | tail -n1 207 files changed, 5626 insertions(+), 2836 deletions(-) The case of Scipy • WIP • But is much easier: very little string handing etc. How? • http://projects.scipy.org/numpy/browser/trunk/doc/Py3K.txt (A bit out of date, but illustrative.) • Python changes:

svn/trunk

• Create account on http://github.com (or anywhere) • Create a new repository @ Github • Push your work to github
git remote add github git@github:YOURUSERNAME/YOURREPOSITORYNAME.git git push github name-of-my-feature-branch

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– 2to3 on setup.py build – Compatibility package numpy.compat.py3k – str vs. bytes, cyclic imports, types module, ... • C changes: – Compatibility header npy_3kcompat.h – Module & type initialization – Buffer interface (provider) – Strings: (i) Unicode on 2+3, (ii) UC on 3, Bytes on 2, (iii) Bytes on 2+3 – Division, comparison – I/O – PyCObject -> PyCapsule – ...

• pros: huge memory savings • cons: depends on actual storage scheme, (*) usually does not hold Typical Applications • solution of partial differential equations (PDEs) – the ﬁnite element method – mechanical engineering, electrotechnics, physics, ... • graph theory – nonzero at (i, j) means that node i is connected to node j • ... Prerequisites recent versions of • numpy • scipy • matplotlib (optional) • ipython (the enhancements come handy) Sparsity Structure Visualization • spy() from matplotlib • example plots:

2.2 Sparse Matrices in SciPy
author Robert Cimrman

2.2.1 Introduction
(dense) matrix is: • mathematical object • data structure for storing a 2D array of values important features: • memory allocated once for all items – usually a contiguous chunk, think NumPy ndarray • fast access to individual items (*) Why Sparse Matrices? • the memory, that grows like n**2 • small example (double precision matrix):
>>> >>> >>> >>> >>> >>> >>> import numpy as np import matplotlib.pyplot as plt x = np.linspace(0, 1e6, 10) plt.plot(x, 8.0 * (x**2) / 1e6, lw=5) plt.xlabel(’size n’) plt.ylabel(’memory [MB]’) plt.show()

Sparse Matrices vs. Sparse Matrix Storage Schemes • sparse matrix is a matrix, which is almost empty • storing all the zeros is wasteful -> store only nonzero items • think compression 2.2. Sparse Matrices in SciPy 138 2.2. Sparse Matrices in SciPy 139

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Common Methods • all scipy.sparse classes are subclasses of spmatrix – default implementation of arithmetic operations * always converts to CSR * subclasses override for efﬁciency – shape, data type set/get – nonzero indices – format conversion, interaction with NumPy (toarray(), todense()) – ... • attributes: – mtx.A - same as mtx.toarray() – mtx.T - transpose (same as mtx.transpose()) – mtx.H - Hermitian (conjugate) transpose – mtx.real - real part of complex matrix – mtx.imag - imaginary part of complex matrix – mtx.size - the number of nonzeros (same as self.getnnz()) – mtx.shape - the number of rows and columns (tuple) • data usually stored in NumPy arrays Sparse Matrix Classes

2.2.2 Storage Schemes
• seven sparse matrix types in scipy.sparse: 1. csc_matrix: Compressed Sparse Column format 2. csr_matrix: Compressed Sparse Row format 3. bsr_matrix: Block Sparse Row format 4. lil_matrix: List of Lists format 5. dok_matrix: Dictionary of Keys format 6. coo_matrix: COOrdinate format (aka IJV, triplet format) 7. dia_matrix: DIAgonal format • each suitable for some tasks • many employ sparsetools C++ module by Nathan Bell • assume the following is imported:
>>> import numpy as np >>> import scipy.sparse as sps >>> import matplotlib.pyplot as plt

Diagonal Format (DIA)
• very simple scheme • diagonals in dense NumPy array of shape (n_diag, length) – ﬁxed length -> waste space a bit when far from main diagonal – subclass of _data_matrix (sparse matrix classes with .data attribute) • offset for each diagonal – 0 is the main diagonal – negative offset = below – positive offset = above • fast matrix * vector (sparsetools) • fast and easy item-wise operations – manipulate data array directly (fast NumPy machinery) • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) – (data, offsets) tuple • no slicing, no individual item access 2.2. Sparse Matrices in SciPy 141

• warning for NumPy users: – the multiplication with ‘*’ is the matrix multiplication (dot product) – not part of NumPy! * passing a sparse matrix object to NumPy functions expecting ndarray/matrix does not work

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• use: – rather specialized – solving PDEs by ﬁnite differences – with an iterative solver Examples • create some DIA matrices:
>>> data = np.array([[1, 2, 3, 4]]).repeat(3, axis=0) >>> data array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]) >>> offsets = np.array([0, -1, 2]) >>> mtx = sps.dia_matrix((data, offsets), shape=(4, 4)) >>> mtx <4x4 sparse matrix of type ’<type ’numpy.int32’>’ with 9 stored elements (3 diagonals) in DIAgonal format> >>> mtx.todense() matrix([[1, 0, 3, 0], [1, 2, 0, 4], [0, 2, 3, 0], [0, 0, 3, 4]]) >>> data = np.arange(12).reshape((3, 4)) + 1 >>> data array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> mtx = sps.dia_matrix((data, offsets), shape=(4, 4)) >>> mtx.data array([[ 1, 2, 3, 4], [ 5, 6, 7, 8], [ 9, 10, 11, 12]]) >>> mtx.offsets array([ 0, -1, 2]) >>> print mtx (0, 0) 1 (1, 1) 2 (2, 2) 3 (3, 3) 4 (1, 0) 5 (2, 1) 6 (3, 2) 7 (0, 2) 11 (1, 3) 12 >>> mtx.todense() matrix([[ 1, 0, 11, 0], [ 5, 2, 0, 12], [ 0, 6, 3, 0], [ 0, 0, 7, 4]])

-2: -3:

. 6 3 . . . 7 4 ---------8

• matrix-vector multiplication
>>> vec = np.ones((4,)) >>> vec array([ 1., 1., 1., 1.]) >>> mtx * vec array([ 12., 19., 9., 11.]) >>> mtx.toarray() * vec array([[ 1., 0., 11., 0.], [ 5., 2., 0., 12.], [ 0., 6., 3., 0.], [ 0., 0., 7., 4.]])

List of Lists Format (LIL)
• row-based linked list – each row is a Python list (sorted) of column indices of non-zero elements – rows stored in a NumPy array (dtype=np.object) – non-zero values data stored analogously • efﬁcient for constructing sparse matrices incrementally • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) • ﬂexible slicing, changing sparsity structure is efﬁcient • slow arithmetics, slow column slicing due to being row-based • use: – when sparsity pattern is not known apriori or changes – example: reading a sparse matrix from a text ﬁle Examples • create an empty LIL matrix:
>>> mtx = sps.lil_matrix((4, 5))

• prepare random data:
>>> from >>> data >>> data array([[ [ numpy.random import rand = np.round(rand(2, 3)) 0., 0., 0., 0., 1.], 1.]])

• explanation with a scheme:
offset: row 2: 1: 0: -1: 9 --10-----1 . 11 . 5 2 . 12

• assign the data using fancy indexing:

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>>> mtx[:2, [1, 2, 3]] = data >>> mtx <4x5 sparse matrix of type ’<type ’numpy.float64’>’ with 2 stored elements in LInked List format> >>> print mtx (0, 3) 1.0 (1, 3) 1.0 >>> mtx.todense() matrix([[ 0., 0., 0., 1., 0.], [ 0., 0., 0., 1., 0.], [ 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0.]]) >>> mtx.toarray() array([[ 0., 0., 0., 1., 0.], [ 0., 0., 0., 1., 0.], [ 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0.]])

• ﬂexible slicing, changing sparsity structure is efﬁcient • can be efﬁciently converted to a coo_matrix once constructed • slow arithmetics (for loops with dict.iteritems()) • use: – when sparsity pattern is not known apriori or changes Examples • create a DOK matrix element by element:
>>> mtx = sps.dok_matrix((5, 5), dtype=np.float64) >>> mtx <5x5 sparse matrix of type ’<type ’numpy.float64’>’ with 0 stored elements in Dictionary Of Keys format> >>> for ir in range(5): >>> for ic in range(5): >>> mtx[ir, ic] = 1.0 * (ir != ic) >>> mtx <5x5 sparse matrix of type ’<type ’numpy.float64’>’ with 25 stored elements in Dictionary Of Keys format> >>> mtx.todense() matrix([[ 0., 1., 1., 1., 1.], [ 1., 0., 1., 1., 1.], [ 1., 1., 0., 1., 1.], [ 1., 1., 1., 0., 1.], [ 1., 1., 1., 1., 0.]])

• more slicing and indexing:
>>> mtx = sps.lil_matrix([[0, 1, 2, 0], [3, 0, 1, 0], [1, 0, 0, 1]]) >>> mtx.todense() matrix([[0, 1, 2, 0], [3, 0, 1, 0], [1, 0, 0, 1]]) >>> print mtx (0, 1) 1 (0, 2) 2 (1, 0) 3 (1, 2) 1 (2, 0) 1 (2, 3) 1 >>> mtx[:2, :] <2x4 sparse matrix of type ’<type ’numpy.int32’>’ with 4 stored elements in LInked List format> >>> mtx[:2, :].todense() matrix([[0, 1, 2, 0], [3, 0, 1, 0]]) >>> mtx[1:2, [0,2]].todense() matrix([[3, 1]]) >>> mtx.todense() matrix([[0, 1, 2, 0], [3, 0, 1, 0], [1, 0, 0, 1]])

• something is wrong, let’s start over:
>>> mtx = sps.dok_matrix((5, 5), dtype=np.float64) >>> for ir in range(5): >>> for ic in range(5): >>> if (ir != ic): mtx[ir, ic] = 1.0 >>> mtx <5x5 sparse matrix of type ’<type ’numpy.float64’>’ with 20 stored elements in Dictionary Of Keys format>

• slicing and indexing:
>>> mtx[1, 1] 0.0 >>> mtx[1, 1:3] <1x2 sparse matrix of type ’<type ’numpy.float64’>’ with 1 stored elements in Dictionary Of Keys format> >>> mtx[1, 1:3].todense() matrix([[ 0., 1.]]) >>> mtx[[2,1], 1:3].todense() -----------------------------------------------------------Traceback (most recent call last): <snip> NotImplementedError: fancy indexing supported over one axis only

Dictionary of Keys Format (DOK)
• subclass of Python dict – keys are (row, column) index tuples (no duplicate entries allowed) – values are corresponding non-zero values • efﬁcient for constructing sparse matrices incrementally • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) • efﬁcient O(1) access to individual elements

Coordinate Format (COO)
• also known as the ‘ijv’ or ‘triplet’ format – three NumPy arrays: row, col, data – data[i] is value at (row[i], col[i]) position

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– permits duplicate entries – subclass of _data_matrix (sparse matrix classes with .data attribute) • fast format for constructing sparse matrices • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) – (data, ij) tuple • very fast conversion to and from CSR/CSC formats • fast matrix * vector (sparsetools) • fast and easy item-wise operations – manipulate data array directly (fast NumPy machinery) • no slicing, no arithmetics (directly) • use: – facilitates fast conversion among sparse formats – when converting to other format (usually CSR or CSC), duplicate entries are summed together * facilitates efﬁcient construction of ﬁnite element matrices Examples • create empty COO matrix:
>>> mtx = sps.coo_matrix((3, 4), dtype=np.int8) >>> mtx.todense() matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], dtype=int8)

[0, 0, 0, 0], [0, 0, 0, 1]])

• no slicing...:
>>> mtx[2, 3] -----------------------------------------------------------Traceback (most recent call last): File "<ipython console>", line 1, in <module> TypeError: ’coo_matrix’ object is unsubscriptable

Compressed Sparse Row Format (CSR)
• row oriented – three NumPy arrays: indices, indptr, data * indices is array of column indices * data is array of corresponding nonzero values * indptr points to row starts in indices and data * length is n_row + 1, last item = number of values = length of both indices and data * nonzero values of the i-th row are data[indptr[i]:indptr[i+1]] with column indices indices[indptr[i]:indptr[i+1]] * item (i, j) can be accessed as data[indptr[i]+k], where k is position of j in indices[indptr[i]:indptr[i+1]] – subclass of _cs_matrix (common CSR/CSC functionality) * subclass of _data_matrix (sparse matrix classes with .data attribute) • fast matrix vector products and other arithmetics (sparsetools) • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) – (data, ij) tuple – (data, indices, indptr) tuple • efﬁcient row slicing, row-oriented operations • slow column slicing, expensive changes to the sparsity structure • use: – actual computations (most linear solvers support this format) Examples • create empty CSR matrix:
>>> mtx = sps.csr_matrix((3, 4), dtype=np.int8) >>> mtx.todense() matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], dtype=int8)

• create using (data, ij) tuple:
>>> row = np.array([0, 3, 1, 0]) >>> col = np.array([0, 3, 1, 2]) >>> data = np.array([4, 5, 7, 9]) >>> mtx = sps.coo_matrix((data, (row, col)), shape=(4, 4)) >>> mtx <4x4 sparse matrix of type ’<type ’numpy.int32’>’ with 4 stored elements in COOrdinate format> >>> mtx.todense() matrix([[4, 0, 9, 0], [0, 7, 0, 0], [0, 0, 0, 0], [0, 0, 0, 5]])

• duplicates entries are summed together:
>>> row = np.array([0, 0, 1, 3, 1, 0, 0]) >>> col = np.array([0, 2, 1, 3, 1, 0, 0]) >>> data = np.array([1, 1, 1, 1, 1, 1, 1]) >>> mtx = sps.coo_matrix((data, (row, col)), shape=(4, 4)) >>> mtx.todense() matrix([[3, 0, 1, 0], [0, 2, 0, 0],

• create using (data, ij) tuple: 2.2. Sparse Matrices in SciPy 146 2.2. Sparse Matrices in SciPy 147

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>>> row = np.array([0, 0, 1, 2, 2, 2]) >>> col = np.array([0, 2, 2, 0, 1, 2]) >>> data = np.array([1, 2, 3, 4, 5, 6]) >>> mtx = sps.csr_matrix((data, (row, col)), shape=(3, 3)) >>> mtx <3x3 sparse matrix of type ’<type ’numpy.int32’>’ with 6 stored elements in Compressed Sparse Row format> >>> mtx.todense() matrix([[1, 0, 2], [0, 0, 3], [4, 5, 6]]) >>> mtx.data array([1, 2, 3, 4, 5, 6]) >>> mtx.indices array([0, 2, 2, 0, 1, 2]) >>> mtx.indptr array([0, 2, 3, 6])

• efﬁcient column slicing, column-oriented operations • slow row slicing, expensive changes to the sparsity structure • use: – actual computations (most linear solvers support this format) Examples • create empty CSC matrix:
>>> mtx = sps.csc_matrix((3, 4), dtype=np.int8) >>> mtx.todense() matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], dtype=int8)

• create using (data, ij) tuple:
>>> row = np.array([0, 0, 1, 2, 2, 2]) >>> col = np.array([0, 2, 2, 0, 1, 2]) >>> data = np.array([1, 2, 3, 4, 5, 6]) >>> mtx = sps.csc_matrix((data, (row, col)), shape=(3, 3)) >>> mtx <3x3 sparse matrix of type ’<type ’numpy.int32’>’ with 6 stored elements in Compressed Sparse Column format> >>> mtx.todense() matrix([[1, 0, 2], [0, 0, 3], [4, 5, 6]]) >>> mtx.data array([1, 4, 5, 2, 3, 6]) >>> mtx.indices array([0, 2, 2, 0, 1, 2]) >>> mtx.indptr array([0, 2, 3, 6])

• create using (data, indices, indptr) tuple:
>>> data = np.array([1, 2, 3, 4, 5, 6]) >>> indices = np.array([0, 2, 2, 0, 1, 2]) >>> indptr = np.array([0, 2, 3, 6]) >>> mtx = sps.csr_matrix((data, indices, indptr), shape=(3, 3)) >>> mtx.todense() matrix([[1, 0, 2], [0, 0, 3], [4, 5, 6]])

Compressed Sparse Column Format (CSC)
• column oriented – three NumPy arrays: indices, indptr, data * indices is array of row indices * data is array of corresponding nonzero values * indptr points to column starts in indices and data * length is n_col + 1, last item = number of values = length of both indices and data * nonzero values of the i-th column are data[indptr[i]:indptr[i+1]] with row indices indices[indptr[i]:indptr[i+1]] * item (i, j) can be accessed as data[indptr[j]+k], where k is position of i in indices[indptr[j]:indptr[j+1]] – subclass of _cs_matrix (common CSR/CSC functionality) * subclass of _data_matrix (sparse matrix classes with .data attribute) • fast matrix vector products and other arithmetics (sparsetools) • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) – (data, ij) tuple – (data, indices, indptr) tuple

• create using (data, indices, indptr) tuple:
>>> data = np.array([1, 4, 5, 2, 3, 6]) >>> indices = np.array([0, 2, 2, 0, 1, 2]) >>> indptr = np.array([0, 2, 3, 6]) >>> mtx = sps.csc_matrix((data, indices, indptr), shape=(3, 3)) >>> mtx.todense() matrix([[1, 0, 2], [0, 0, 3], [4, 5, 6]])

Block Compressed Row Format (BSR)
• basically a CSR with dense sub-matrices of ﬁxed shape instead of scalar items – block size (R, C) must evenly divide the shape of the matrix (M, N) – three NumPy arrays: indices, indptr, data * indices is array of column indices for each block * data is array of corresponding nonzero values of shape (nnz, R, C) * ... – subclass of _cs_matrix (common CSR/CSC functionality)

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* subclass of _data_matrix (sparse matrix classes with .data attribute) • fast matrix vector products and other arithmetics (sparsetools) • constructor accepts: – dense matrix (array) – sparse matrix – shape tuple (create empty matrix) – (data, ij) tuple – (data, indices, indptr) tuple • many arithmetic operations considerably more efﬁcient than CSR for sparse matrices with dense submatrices • use: – like CSR – vector-valued ﬁnite element discretizations Examples • create empty BSR matrix with (1, 1) block size (like CSR...):
>>> mtx = sps.bsr_matrix((3, 4), dtype=np.int8) >>> mtx <3x4 sparse matrix of type ’<type ’numpy.int8’>’ with 0 stored elements (blocksize = 1x1) in Block Sparse Row format> >>> mtx.todense() matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], dtype=int8)

[[3]], [[4]], [[5]], [[6]]]) >>> mtx.indices array([0, 2, 2, 0, 1, 2]) >>> mtx.indptr array([0, 2, 3, 6])

• create using (data, indices, indptr) tuple with (2, 2) block size:
>>> indptr = np.array([0, 2, 3, 6]) >>> indices = np.array([0, 2, 2, 0, 1, 2]) >>> data = np.array([1, 2, 3, 4, 5, 6]).repeat(4).reshape(6, 2, 2) >>> mtx = sps.bsr_matrix((data, indices, indptr), shape=(6, 6)) >>> mtx.todense() matrix([[1, 1, 0, 0, 2, 2], [1, 1, 0, 0, 2, 2], [0, 0, 0, 0, 3, 3], [0, 0, 0, 0, 3, 3], [4, 4, 5, 5, 6, 6], [4, 4, 5, 5, 6, 6]]) >>> data array([[[1, 1], [1, 1]], [[2, 2], [2, 2]], [[3, 3], [3, 3]], [[4, 4], [4, 4]], [[5, 5], [5, 5]], [[6, 6], [6, 6]]])

• create empty BSR matrix with (3, 2) block size:
>>> mtx = sps.bsr_matrix((3, 4), blocksize=(3, 2), dtype=np.int8) >>> mtx <3x4 sparse matrix of type ’<type ’numpy.int8’>’ with 0 stored elements (blocksize = 3x2) in Block Sparse Row format> >>> mtx.todense() matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], dtype=int8)

– a bug? • create using (data, ij) tuple with (1, 1) block size (like CSR...):
>>> row = np.array([0, 0, 1, 2, 2, 2]) >>> col = np.array([0, 2, 2, 0, 1, 2]) >>> data = np.array([1, 2, 3, 4, 5, 6]) >>> mtx = sps.bsr_matrix((data, (row, col)), shape=(3, 3)) >>> mtx <3x3 sparse matrix of type ’<type ’numpy.int32’>’ with 6 stored elements (blocksize = 1x1) in Block Sparse Row format> >>> mtx.todense() matrix([[1, 0, 2], [0, 0, 3], [4, 5, 6]]) >>> mtx.data array([[[1]], [[2]],

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Summary Table 2.1: Summary of storage schemes. format DIA LIL DOK COO CSR CSC BSR matrix * vector sparsetools via CSR python sparsetools sparsetools sparsetools sparsetools get item . yes yes . yes yes . fancy get . yes one axis only . yes yes . set item . yes yes . slow slow . fancy set . yes yes . . . . solvers iterative iterative iterative iterative any any specialized note has data array, specialized arithmetics via CSR, incremental construction O(1) item access, incremental construction has data array, facilitates fast conversion has data array, fast row-wise ops has data array, fast column-wise ops has data array, specialized

– check-out the new scikits.suitesparse by Nathaniel Smith

Examples
• import the whole module, and see its docstring:
>>> import scipy.sparse.linalg.dsolve as dsl >>> help(dsl)

• both superlu and umfpack can be used (if the latter is installed) as follows: – prepare a linear system:
>>> import numpy as np >>> import scipy.sparse as sps >>> mtx = sps.spdiags([[1, 2, 3, 4, 5], [6, 5, 8, 9, 10]], [0, 1], 5, 5) >>> mtx.todense() matrix([[ 1, 5, 0, 0, 0], [ 0, 2, 8, 0, 0], [ 0, 0, 3, 9, 0], [ 0, 0, 0, 4, 10], [ 0, 0, 0, 0, 5]]) >>> rhs = np.array([1, 2, 3, 4, 5])

2.2.3 Linear System Solvers
• sparse matrix/eigenvalue problem solvers live in scipy.sparse.linalg • the submodules: – dsolve: direct factorization methods for solving linear systems – isolve: iterative methods for solving linear systems – eigen: sparse eigenvalue problem solvers • all solvers are accessible from:
>>> import scipy.sparse.linalg as spla >>> spla.__all__ [’LinearOperator’, ’Tester’, ’arpack’, ’aslinearoperator’, ’bicg’, ’bicgstab’, ’cg’, ’cgs’, ’csc_matrix’, ’csr_matrix’, ’dsolve’, ’eigen’, ’eigen_symmetric’, ’factorized’, ’gmres’, ’interface’, ’isolve’, ’iterative’, ’lgmres’, ’linsolve’, ’lobpcg’, ’lsqr’, ’minres’, ’np’, ’qmr’, ’speigs’, ’spilu’, ’splu’, ’spsolve’, ’svd’, ’test’, ’umfpack’, ’use_solver’, ’utils’, ’warnings’]

– solve as single precision real:
>>> >>> >>> >>> mtx1 = mtx.astype(np.float32) x = dsl.spsolve(mtx1, rhs, use_umfpack=False) print x print "Error: ", mtx1 * x - b

– solve as double precision real:
>>> >>> >>> >>> mtx2 = mtx.astype(np.float64) x = dsl.spsolve(mtx2, rhs, use_umfpack=True) print x print "Error: ", mtx2 * x - b

– solve as single precision complex:
>>> >>> >>> >>> mtx1 = mtx.astype(np.complex64) x = dsl.spsolve(mtx1, rhs, use_umfpack=False) print x print "Error: ", mtx1 * x - b

– solve as double precision complex:
>>> >>> >>> >>> mtx2 = mtx.astype(np.complex128) x = dsl.spsolve(mtx2, rhs, use_umfpack=True) print x print "Error: ", mtx2 * x - b

Sparse Direct Solvers • default solver: SuperLU 4.0 – included in SciPy – real and complex systems – both single and double precision • optional: umfpack – real and complex systems – double precision only – recommended for performance – wrappers now live in scikits.umfpack 2.2. Sparse Matrices in SciPy 152

""" Construct a 1000x1000 lil_matrix and add some values to it, convert it to CSR format and solve A x = b for x:and solve a linear system with a direct solver. """ import numpy as np import scipy.sparse as sps from matplotlib import pyplot as plt from scipy.sparse.linalg.dsolve import linsolve rand = np.random.rand

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mtx = sps.lil_matrix((1000, 1000), dtype=np.float64) mtx[0, :100] = rand(100) mtx[1, 100:200] = mtx[0, :100] mtx.setdiag(rand(1000)) plt.clf() plt.spy(mtx, marker=’.’, markersize=2) plt.show() mtx = mtx.tocsr() rhs = rand(1000) x = linsolve.spsolve(mtx, rhs) print ’rezidual:’, np.linalg.norm(mtx * x - rhs)

LinearOperator Class
from scipy.sparse.linalg.interface import LinearOperator

• common interface for performing matrix vector products • useful abstraction that enables using dense and sparse matrices within the solvers, as well as matrix-free solutions • has shape and matvec() (+ some optional parameters) • example:
>>> import numpy as np >>> from scipy.sparse.linalg import LinearOperator >>> def mv(v): ... return np.array([2*v[0], 3*v[1]]) ... >>> A = LinearOperator((2, 2), matvec=mv) >>> A <2x2 LinearOperator with unspecified dtype> >>> A.matvec(np.ones(2)) array([ 2., 3.]) >>> A * np.ones(2) array([ 2., 3.])

• examples/direct_solve.py Iterative Solvers • the isolve module contains the following solvers: – bicg (BIConjugate Gradient) – bicgstab (BIConjugate Gradient STABilized) – cg (Conjugate Gradient) - symmetric positive deﬁnite matrices only – cgs (Conjugate Gradient Squared) – gmres (Generalized Minimal RESidual) – minres (MINimum RESidual) – qmr (Quasi-Minimal Residual)

A Few Notes on Preconditioning
• problem speciﬁc • often hard to develop • if not sure, try ILU – available in dsolve as spilu()

Common Parameters
• mandatory: A [{sparse matrix, dense matrix, LinearOperator}] The N-by-N matrix of the linear system. b [{array, matrix}] Right hand side of the linear system. Has shape (N,) or (N,1). • optional: x0 [{array, matrix}] Starting guess for the solution. tol [ﬂoat] Relative tolerance to achieve before terminating. maxiter [integer] Maximum number of iterations. Iteration will stop after maxiter steps even if the speciﬁed tolerance has not been achieved. M [{sparse matrix, dense matrix, LinearOperator}] Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback [function] User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

Eigenvalue Problem Solvers

The eigen module
• arpack * a collection of Fortran77 subroutines designed to solve large scale eigenvalue problems • lobpcg (Locally Optimal Block Preconditioned Conjugate Gradient Method) * works very well in combination with PyAMG * example by Nathan Bell:
""" Compute eigenvectors and eigenvalues using a preconditioned eigensolver In this example Smoothed Aggregation (SA) is used to precondition the LOBPCG eigensolver on a two-dimensional Poisson problem with Dirichlet boundary conditions. """ import scipy from scipy.sparse.linalg import lobpcg from pyamg import smoothed_aggregation_solver from pyamg.gallery import poisson N = 100 K = 9 A = poisson((N,N), format=’csr’)

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# create the AMG hierarchy ml = smoothed_aggregation_solver(A) # initial approximation to the K eigenvectors X = scipy.rand(A.shape[0], K) # preconditioner based on ml M = ml.aspreconditioner() # compute eigenvalues and eigenvectors with LOBPCG W,V = lobpcg(A, X, M=M, tol=1e-8, largest=False)

#plot the eigenvectors import pylab pylab.figure(figsize=(9,9)) for i in range(K): pylab.subplot(3, 3, i+1) pylab.title(’Eigenvector %d’ % i) pylab.pcolor(V[:,i].reshape(N,N)) pylab.axis(’equal’) pylab.axis(’off’) pylab.show()

– examples/pyamg_with_lobpcg.py • example by Nils Wagner: – examples/lobpcg_sakurai.py • output:
$ python examples/lobpcg_sakurai.py Results by LOBPCG for n=2500 [ 0.06250083 0.06250028 0.06250007]

2.2.4 Other Interesting Packages
• PyAMG – algebraic multigrid solvers – http://code.google.com/p/pyamg • Pysparse – own sparse matrix classes – matrix and eigenvalue problem solvers – http://pysparse.sourceforge.net/

Exact eigenvalues [ 0.06250005 0.0625002 0.06250044]

Elapsed time 7.01

2.3 Sympy : Symbolic Mathematics in Python
author Fabian Pedregosa

2.3.1 Objectives
At the end of this session you will be able to: 1. Evaluate expressions with arbitrary precission. 2. Perform algebraic manipulations on symbolic expressions. 3. Perform basic calculus tasks (limits, differentiation and integration) with symbolic expressions.

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4. Solve polynomial and trascendental equations. 5. Solve some differential equations.

Symbols In contrast to other Computer Algebra Systems, in SymPy you have to declare symbolic variables explicitly:
>>> from sympy import * >>> x = Symbol(’x’) >>> y = Symbol(’y’)

2.3.2 What is SymPy?
SymPy is a Python library for symbolic mathematics. It aims become a full featured computer algebra system thatn can compete directly with commercial alternatives (Mathematica, Maple) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python and does not require any external libraries.

Then you can manipulate them:
>>> x+y+x-y 2*x >>> (x+y)**2 (x + y)**2

2.3.3 First Steps with SymPy
Using SymPy as a calculator Sympy deﬁnes three numerical types: Real, Rational and Integer. The Rational class represents a rational number as a pair of two Integers: the numerator and the denominator, so Rational(1,2) represents 1/2, Rational(5,2) 5/2 and so on:
>>> from sympy import * >>> a = Rational(1,2) >>> a 1/2 >>> a*2 1

Symbols can now be manipulated using some of python operators: +, -, , * (arithmetic), &, |, ~ , >>, << (boolean).

2.3.4 Algebraic manipulations
SymPy is capable of performing powerful algebraic manipulations. We’ll take a look into some of the most frequently used: expand and simplify. Expand Use this to expand an algebraic expression. It will try to denest powers and multiplications:
In [23]: expand((x+y)**3) Out[23]: 3*x*y**2 + 3*y*x**2 + x**3 + y**3

SymPy uses mpmath in the background, which makes it possible to perform computations using arbitrary - precission arithmetic. That way, some special constants, like e, pi, oo (Inﬁnity), are treated as symbols and can be evaluated with arbitrary precission:
>>> pi**2 pi**2 >>> pi.evalf() 3.14159265358979 >>> (pi+exp(1)).evalf() 5.85987448204884

Further options can be given in form on keywords:
In [28]: expand(x+y, complex=True) Out[28]: I*im(x) + I*im(y) + re(x) + re(y) In [30]: expand(cos(x+y), trig=True) Out[30]: cos(x)*cos(y) - sin(x)*sin(y)

Simplify Use simplify if you would like to transform an expression into a simpler form:
In [19]: simplify((x+x*y)/x) Out[19]: 1 + y

as you see, evalf evaluates the expression to a ﬂoating-point number. There is also a class representing mathematical inﬁnity, called oo:
>>> oo > 99999 True >>> oo + 1 oo

Simpliﬁcation is a somewhat vague term, and more precises alternatives to simplify exists: powsimp (simpliﬁcation of exponents), trigsimp (for trigonometrical expressions) , logcombine, radsimp, together. Exercises

Exercises 1. Calculate √ 2 with 100 decimals.

1. Calculate the expanded form of (x + y)6 . 2. Symplify the trigonometrical expression sin(x) / cos(x)

2. Calculate 1/2 + 1/3 in rational arithmetic.

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2.3.5 Calculus
Limits Limits are easy to use in sympy, they follow the syntax limit(function, variable, point), so to compute the limit of f(x) as x -> 0, you would issue limit(f, x, 0):
>>> limit(sin(x)/x, x, 0) 1

2. Calulate the derivative of log(x) for x. Integration SymPy has support for indeﬁnite and deﬁnite integration of transcendental elementary and special functions via integrate() facility, which uses powerful extended Risch-Norman algorithm and some heuristics and pattern matching. You can integrate elementary functions:
>>> integrate(6*x**5, x) x**6 >>> integrate(sin(x), x) -cos(x) >>> integrate(log(x), x) -x + x*log(x) >>> integrate(2*x + sinh(x), x) cosh(x) + x**2

you can also calculate the limit at inﬁnity:
>>> limit(x, x, oo) oo >>> limit(1/x, x, oo) 0 >>> limit(x**x, x, 0) 1

Also special functions are handled easily:
>>> integrate(exp(-x**2)*erf(x), x) pi**(1/2)*erf(x)**2/4

Differentiation You can differentiate any SymPy expression using diff(func, var). Examples:
>>> diff(sin(x), x) cos(x) >>> diff(sin(2*x), x) 2*cos(2*x) >>> diff(tan(x), x) 1 + tan(x)**2

It is possible to compute deﬁnite integral:
>>> integrate(x**3, (x, -1, 1)) 0 >>> integrate(sin(x), (x, 0, pi/2)) 1 >>> integrate(cos(x), (x, -pi/2, pi/2)) 2

Also improper integrals are supported as well:
>>> integrate(exp(-x), (x, 0, oo)) 1 >>> integrate(exp(-x**2), (x, -oo, oo)) pi**(1/2)

You can check, that it is correct by:
>>> limit((tan(x+y)-tan(x))/y, y, 0) 1 + tan(x)**2

Higher derivatives can be calculated using the diff(func, var, n) method:
>>> diff(sin(2*x), x, 1) 2*cos(2*x) >>> diff(sin(2*x), x, 2) -4*sin(2*x) >>> diff(sin(2*x), x, 3) -8*cos(2*x)

Exercises

2.3.6 Equation solving
SymPy is able to solve algebraic equations, in one and several variables:
In [7]: solve(x**4 - 1, x) Out[7]: [I, 1, -1, -I]

Series expansion SymPy also knows how to compute the Taylor series of an expression at a point. Use series(expr, var):
>>> 1 >>> 1 + series(cos(x), x) x**2/2 + x**4/24 + O(x**6) series(1/cos(x), x) x**2/2 + 5*x**4/24 + O(x**6)

As you can see it takes as ﬁrst argument an expression that is supposed to be equaled to 0. It is able to solve a large part of polynomial equations, and is also capable of solving multiple equations with respect to multiple variables giving a tuple as second argument:
In [8]: solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y]) Out[8]: {y: 1, x: -3}

It also has (limited) support for trascendental equations:
In [9]: solve(exp(x) + 1, x) Out[9]: [pi*I]

Exercises 1. Calculate lim x− > 0, sin(x)/x 2.3. Sympy : Symbolic Mathematics in Python 160

Another alternative in the case of polynomial equations is factor. factor returns the polynomial factorized into irreducible terms, and is capable of computing the factorization over various domains:

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In [10]: f = x**4 - 3*x**2 + 1 In [11]: factor(f) Out[11]: (1 + x - x**2)*(1 - x - x**2) In [12]: factor(f, modulus=5) Out[12]: (2 + x)**2*(2 - x)**2

In [5]: dsolve(f(x).diff(x, x) + f(x), f(x)) Out[5]: C1*sin(x) + C2*cos(x)

Keyword arguments can be given to this function in order to help if ﬁnd the best possible resolution system. For example, if you know that it is a separable equations, you can use keyword hint=’separable’ to force dsolve to resolve it as a separable equation. In [6]: dsolve(sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x), f(x), hint=’separable’) Out[6]: -log(1 sin(f(x))**2)/2 == C1 + log(1 - sin(x)**2)/2 Exercises 1. Solve the Bernoulli differential equation x*f(x).diff(x) + f(x) - f(x)**2 2. Solve the same equation using hint=’Bernoulli’. What do you observe ?

SymPy is also able to solve boolean equations, that is, to decide if a certain boolean expression is satisﬁable or not. For this, we use the function satisﬁable:
In [13]: satisfiable(x & y) Out[13]: {x: True, y: True}

This tells us that (x & y) is True whenever x and y are both True. If an expression cannot be true, i.e. no values of its arguments can make the expression True, it will return False:
In [14]: satisfiable(x & ~x) Out[14]: False

2.4 3D plotting with Mayavi
author Gaël Varoquaux

Exercises 1. Solve the system of equations x + y = 2, 2*x + y = 0 2. Are there boolean values x, y that make (~x | y) & (~y | x) true?

2.3.7 Linear Algebra
Matrices Matrices are created as instances from the Matrix class:
>>> >>> [1, [0, from sympy import Matrix Matrix([[1,0], [0,1]]) 0] 1]

2.4.1 A simple example
Warning: Start ipython -wthread
import numpy as np x, y = np.mgrid[-10:10:100j, -10:10:100j] r = np.sqrt(x**2 + y**2) z = np.sin(r)/r from enthought.mayavi import mlab mlab.surf(z, warp_scale=’auto’) mlab.outline() mlab.axes()

unlike a numpy array, you can also put Symbols in it:
>>> >>> >>> >>> [1, [y, x = Symbol(’x’) y = Symbol(’y’) A = Matrix([[1,x], [y,1]]) A x] 1]

np.mgrid[-10:10:100j, -10:10:100j] creates an x,y grid, going from -10 to 10, with 100 steps in each directions.

>>> A**2 [1 + x*y, 2*x] [ 2*y, 1 + x*y]

Differential Equations SymPy is capable of solving (some) Ordinary Differential Equations. sympy.ode.dsolve works like this:
In [4]: f(x).diff(x, x) + f(x) Out[4]: 2 d -----(f(x)) + f(x) dx dx

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2.4.2 3D plotting functions
Points
In [1]: import numpy as np In [2]: from enthought.mayavi import mlab In [3]: x, y, z, value = np.random.random((4, 40)) In [4]: mlab.points3d(x, y, z, value) Out[4]: <enthought.mayavi.modules.glyph.Glyph object at 0xc3c795c>

In [10]: r = np.sqrt(x**2 + y**2) In [11]: z = np.sin(r)/r In [12]: mlab.surf(z, warp_scale=’auto’) Out[12]: <enthought.mayavi.modules.surface.Surface object at 0xcdb98fc>

Lines
In [5]: mlab.clf() In [6]: t = np.linspace(0, 20, 200) In [7]: mlab.plot3d(np.sin(t), np.cos(t), 0.1*t, t) Out[7]: <enthought.mayavi.modules.surface.Surface object at 0xcc3e1dc>

Arbitrary regular mesh
In [13]: mlab.clf() In [14]: phi, theta = np.mgrid[0:np.pi:11j, 0:2*np.pi:11j] In [15]: x = np.sin(phi) * np.cos(theta) In [16]: y = np.sin(phi) * np.sin(theta) In [17]: z = np.cos(phi) In [18]: mlab.mesh(x, y, z) In [19]: mlab.mesh(x, y, z, representation=’wireframe’, color=(0, 0, 0)) Out[19]: <enthought.mayavi.modules.surface.Surface object at 0xce1017c>

Elevation surface
In [8]: mlab.clf() In [9]: x, y = np.mgrid[-10:10:100j, -10:10:100j]

Note: A surface is deﬁned by points connected to form triangles or polygones. In mlab.func and mlab.mesh, the connectivity is implicity given by the layout of the arrays. See also mlab.triangular_mesh.

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Our data is often more than points and values: it needs some connectivity information Volumetric data
In [20]: mlab.clf() In [21]: x, y, z = np.mgrid[-5:5:64j, -5:5:64j, -5:5:64j] In [22]: values = x*x*0.5 + y*y + z*z*2.0 In [23]: mlab.contour3d(values) Out[24]: <enthought.mayavi.modules.iso_surface.IsoSurface object at 0xcfe392c>

2.4.3 Figures and decorations
Figure management Here is a list of functions useful to control the current ﬁgure Get the current ﬁgure: Clear the current ﬁgure: Set the current ﬁgure: Save ﬁgure to image ﬁle: Change the view: Changing plot properties In general, many properties of the various objects on the ﬁgure can be changed. If these visualization are created via mlab functions, the easiest way to change them is to use the keyword arguments of these functions, as described in the docstrings. mlab.gcf() mlab.clf() mlab.ﬁgure(1, bgcolor=(1, 1, 1), fgcolor=(0.5, 0.5, 0.5) mlab.saveﬁg(‘foo.png’, size=(300, 300)) mlab.view(azimuth=45, elevation=54, distance=1.)

This function works with a regular orthogonal grid:

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Example docstring: mlab.mesh Plots a surface using grid-spaced data supplied as 2D arrays. Function signatures:
mesh(x, y, z, ...)

In [8]: mlab.mesh(x, y, z, extent=[0, 1, 0, 1, 0, 1], ...: representation=’wireframe’, line_width=1, color=(0.5, 0.5, 0.5)) Out[8]: <enthought.mayavi.modules.surface.Surface object at 0xdd6a71c>

x, y, z are 2D arrays, all of the same shape, giving the positions of the vertices of the surface. The connectivity between these points is implied by the connectivity on the arrays. For simple structures (such as orthogonal grids) prefer the surf function, as it will create more efﬁcient data structures. Keyword arguments: color the color of the vtk object. Overides the colormap, if any, when speciﬁed. This is speciﬁed as a triplet of ﬂoat ranging from 0 to 1, eg (1, 1, 1) for white. colormap type of colormap to use. extent [xmin, xmax, ymin, ymax, zmin, zmax] Default is the x, y, z arrays extents. Use this to change the extent of the object created. ﬁgure Figure to populate. line_width The with of the lines, if any used. Must be a ﬂoat. Default: 2.0 mask boolean mask array to suppress some data points. mask_points If supplied, only one out of ‘mask_points’ data point is displayed. This option is usefull to reduce the number of points displayed on large datasets Must be an integer or None. mode the mode of the glyphs. Must be ‘2darrow’ or ‘2dcircle’ or ‘2dcross’ or ‘2ddash’ or ‘2ddiamond’ or ‘2dhooked_arrow’ or ‘2dsquare’ or ‘2dthick_arrow’ or ‘2dthick_cross’ or ‘2dtriangle’ or ‘2dvertex’ or ‘arrow’ or ‘cone’ or ‘cube’ or ‘cylinder’ or ‘point’ or ‘sphere’. Default: sphere name the name of the vtk object created. representation the representation type used for the surface. Must be ‘surface’ or ‘wireframe’ or ‘points’ or ‘mesh’ or ‘fancymesh’. Default: surface resolution The resolution of the glyph created. For spheres, for instance, this is the number of divisions along theta and phi. Must be an integer. Default: 8 scalars optional scalar data. scale_factor scale factor of the glyphs used to represent the vertices, in fancy_mesh mode. Must be a ﬂoat. Default: 0.05 scale_mode the scaling mode for the glyphs (‘vector’, ‘scalar’, or ‘none’). transparent make the opacity of the actor depend on the scalar. tube_radius radius of the tubes used to represent the lines, in mesh mode. If None, simple lines are used. tube_sides number of sides of the tubes used to represent the lines. Must be an integer. Default: 6 vmax vmax is used to scale the colormap If None, the max of the data will be used vmin vmin is used to scale the colormap If None, the min of the data will be used Example:
In [1]: import numpy as np In [2]: r, theta = np.mgrid[0:10, -np.pi:np.pi:10j] In [3]: x = r*np.cos(theta) In [4]: y = r*np.sin(theta) In [5]: z = np.sin(r)/r In [6]: from enthought.mayavi import mlab In [7]: mlab.mesh(x, y, z, colormap=’gist_earth’, extent=[0, 1, 0, 1, 0, 1]) Out[7]: <enthought.mayavi.modules.surface.Surface object at 0xde6f08c>

Decorations Different items can be added to the ﬁgure to carry extra information, such as a colorbar or a title.
In [9]: mlab.colorbar(Out[7], orientation=’vertical’) Out[9]: <tvtk_classes.scalar_bar_actor.ScalarBarActor object at 0xd897f8c> In [10]: mlab.title(’polar mesh’) Out[10]: <enthought.mayavi.modules.text.Text object at 0xd8ed38c> In [11]: mlab.outline(Out[7]) Out[11]: <enthought.mayavi.modules.outline.Outline object at 0xdd21b6c> In [12]: mlab.axes(Out[7]) Out[12]: <enthought.mayavi.modules.axes.Axes object at 0xd2e4bcc>

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Warning: extent: If we speciﬁed extents for a plotting object, mlab.outline’ and ‘mlab.axes don’t get them by default.

2.4.4 Interaction
The quicket way to create beautiful visualization with Mayavi is probably to interactivly tweak the various settings. Click on the ‘Mayavi’ button in the scene, and you can control properties of objects with dialogs.

Bibliography

[Mallet09] Mallet, C. and Bretar, F. Full-Waveform Topographic Lidar: State-of-the-Art. ISPRS Journal of Photogrammetry and Remote Sensing 64(1), pp.1-16, January 2009 http://dx.doi.org/10.1016/j.isprsjprs.2008.09.007

To ﬁnd out what code can be used to program these changes, click on the red button as you modify those properties, and it will generate the corresponding lines of code.

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Index

D
diff, 160, 162 differentiation, 160 dsolve, 162

E
equations algebraic, 161 differential, 162

I
integration, 161

M
Matrix, 162

P
Python Enhancement Proposals PEP 3118, 125

S
solve, 161

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