Computational Mathematics with Python

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Boolean Arrays Classes

Computational Mathematics with Python

Basics

Olivier Verdier and Claus Führer

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Introduction and Motivation Concepts Practical Information

1 Introduction and Motivation Python vs Other Languages Examples and Demo

2 Concepts Basic Types Variables Lists For Loop 3 Practical Information

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Python vs Other Languages Examples and Demo

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Python vs Other Languages

Examples and Demo

Why Python?

Python is. . .

Free and open source

It is a scripting language, meaning that it is interpreted

It is modern: object oriented, exception handling, dynamic typing etc. Plenty of libraries, in particular scientific ones: linear algebra;

visualisation tools: plotting, image analysis; differential equations solving; symbolic computations; statistics ; etc.

Many possible usages: Scientific computing (of course :-)), scripting, web sites, text parsing, etc.

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Examples and Demo

Python vs language XX

Java, C++ Object oriented compiled languages. Very limited and extremely verbose. Low level compared to python. Few scientific libraries.

C, FORTRAN Very low level compiled language. Useful in some CPU critical situations.

php, ruby Other interpreted languages. PHP is web oriented. Ruby is as flexible as python but has no scientific library.

MATLAB Tool for matrix computation that evolved for scientific computing. The scientific library is huge but it is not a programming language. Extremely expensive.

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Examples and Demo

Examples

Python may be used in interactivemode:

>>>x = 3 >>>y = 5 >>>p r i n t x + y 8 Here we solve " 1 2 3 4 # · x = " 2 1 # >>>M = a r r a y( [ [1 . , 2 .], [3 . , 4 .] ] ) >>>V = a r r a y( [2 . , 1 .] )

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Examples and Demo

More examples

Computing eiπ and 2100:

>>>p r i n t exp(1j*pi) # s h o u l d r e t u r n - 1 : -) ( -1+1 . 2 2 4 6 4 6 7 9 9 1 5 e-16j) >>>p r i n t 2* *100 1 2 6 7 6 5 0 6 0 0 2 2 8 2 2 9 4 0 1 4 9 6 7 0 3 2 0 5 3 7 6 L Computing ζ(x) =P∞ k=1 1

kx. For x = 2 we know that ζ(2) = π

2 6 : # for x = 2 : >>>p r i n t s c i p y . s p e c i a l . z e t a(2 . , 1) 1 . 6 4 4 9 3 4 0 6 6 8 5 >>>p r i n t pi* *2/6 1 . 6 4 4 9 3 4 0 6 6 8 4 8 2 2 6 4

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Examples and Demo

Demo

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Introduction and Motivation Concepts Practical Information Basic Types Variables Lists For Loop

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Numbers

A number may be an integer, a real numberor acomplex number. The usual operations are

+and-addition and substraction

*and/multiplication and division

** power 2* * (2+2) # 16

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Strings

Strings are “lists” of characters, enclosed by simple or double quotes:

’ v a l i d s t r i n g ’

" s t r i n g w i t h d o u b l e q u o t e s "

You may also use triple quotesfor strings including multiple lines:

""" T h i s is a long ,

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Strings

Strings are “lists” of characters, enclosed by simple or double quotes:

’ v a l i d s t r i n g ’

" s t r i n g w i t h d o u b l e q u o t e s "

You may also use triple quotesfor strings including multiple lines:

""" T h i s is a long ,

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Concept: Variable

Variables

A variable is a reference to an object. An object may have several references. One uses theassignment operator =to assign a value to a variable.

Example

x = [3 , 4] # a l i s t o b j e c t is c r e a t e d

y = x # t h i s o b j e c t now has two l a b e l s : x and y

del x # we d e l e t e one of the l a b e l s

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Concept: Lists

Lists

A python list is an orderedlist of objects, enclosed in square brackets. One accesses elements of a list using zero-basedindices inside square brackets.

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List Examples

Example L1 = [1 , 2] L1[0] # 1 L1[1] # 2 L1[2] # r a i s e s I n d e x E r r o r L2 = [’ a ’, 1 , [3 , 4] ] L2[0] # ’ a ’ L2[2] [0] # 3 L2[ -1] # l a s t e l e m e n t : [ 3 , 4 ] L2[ -2] # s e c o n d to l a s t : 1

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List Utilities

range(n)creates a list with n elements, starting with zero: p r i n t r a n g e(5)

[0 , 1 , 2 , 3 , 4]

len(L) gives thelength of a list:

len( [’ a ’, 1 , 2 , 34] ) # r e t u r n s 4

Useappend to append an element to a list: L = [’ a ’, ’ b ’, ’ c ’]

L[ -1] # ’ c ’

L .a p p e n d(’ d ’)

L # L is now [ ’ a ’ , ’ b ’ , ’ c ’ , ’ d ’]

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List Utilities

[0 , 1 , 2 , 3 , 4]

len( [’ a ’, 1 , 2 , 34] ) # r e t u r n s 4

L[ -1] # ’ c ’

L .a p p e n d(’ d ’)

L # L is now [ ’ a ’ , ’ b ’ , ’ c ’ , ’ d ’]

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List Utilities

[0 , 1 , 2 , 3 , 4]

len( [’ a ’, 1 , 2 , 34] ) # r e t u r n s 4

L[ -1] # ’ c ’

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Comprehensive lists

A convenient way to build up lists is to use the comprehensive lists construct, possibly with a conditional inside.

Definition

The syntax of a comprehensive list is [ <e x p r> for <x> in <l i s t> ]

Example

L = [2 , 3 , 10 , 1 , 5]

L2 = [x*2 for x in L] # [ 4 , 6 , 20 , 2 , 10 ]

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Comprehensive Lists in Maths

Mathematical Notation

This is very close to the mathematical notation for sets. Compare: S2 = {2x; x ∈ S }

and

L2 = [2*x for x in L]

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Operations on Lists

Adding two lists concatenates(sammanfoga) them: L1 = [1 , 2]

L2 = [3 , 4]

L = L1 + L2 # [ 1 , 2 , 3 , 4 ]

Logically, multiplying a list with an integer concatenates the list with itself several times: n*L is equivalent to L + L + · · · + L

| {z }

n times

. L = [1 , 2]

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Operations on Lists

Adding two lists concatenates(sammanfoga) them: L1 = [1 , 2]

L2 = [3 , 4]

L = L1 + L2 # [ 1 , 2 , 3 , 4 ]

Logically, multiplying a list with an integer concatenates the list with itself several times: n*L is equivalent to L + L + · · · + L

| {z }

n times

. L = [1 , 2]

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Concept: for loop

for loop

A for loopallows to loop through a list using an index variable. This variable is successively equal to all the elements in the list.

Example

L = [1 , 2 , 10] for s in L:

p r i n t s * 2 ,

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Concept: for loop

for loop

A for loopallows to loop through a list using an index variable. This variable is successively equal to all the elements in the list.

Example

L = [1 , 2 , 10] for s in L:

p r i n t s * 2 ,

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Indentation

The part to be repeated in the for loop has to be properly indented: for elt in m y _ l i s t:

d o _ s o m e t h i n g( ) s o m e t h i n g _ e l s e( ) etc

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Repeating a Task

One typical useof the forloop is to repeat a certain task a fixed number of time:

n = 30

for i in r a n g e(n) :

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Python Shell

Start a python session by typingscipythonin a unix shell Check that it is working with: plot(rand(4));show()

A window should appear with a graph; you should be able to type other commands without having to close the graph window when you want to quit, writeexit()

When you want to run python at home please follow the installation instruction

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Executing Scripts

You often want to execute the contents of a file.

We recommand to useKate on the Linux machines (but any other good editor will do)

Save your files in (for example) in $HOME/course/

Type (once) in ipython: cd course

To execute the contents of a file namedfile.py just write execfile(’file.py’) in ipython.

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Getting Help

Some tips on how to use ipython:

To gethelp on an object just type ? after it and then return Use the arrow keys to reuse the last executed commands

We will see later that you may use the tabulation key forcompletion in general