Computational Mathematics with Python

Numerical Analysis, Lund University, 2011 1 Computational Mathematics with Python Chapter 1: Basics Numerical Analysis, Lund University Claus Führer, Jan Erik Solem, Olivier Verdier, Tony Stillfjord Spring 2011

Numerical Analysis, Lund University, 2011 2 Introduction and Motivation Python vs Other Languages Examples Concepts Basic Types Variables Lists For Loop Practical Information

Numerical Analysis, Lund University, 2011 3 Introduction and Motivation Python vs Other Languages Examples Concepts Basic Types Variables Lists For Loop Practical Information

Numerical Analysis, Lund University, 2011 4 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. Used by YouTube, Google, NASA, Los Alamos, NSA among others

Numerical Analysis, Lund University, 2011 5 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.

Numerical Analysis, Lund University, 2011 6 Examples Python may be used in interactive mode:»> x = 3»> y = 5»> print ( x + y) 8 Here we solve [ ] 1 2 x = 3 4»> M = array ([[1., 2.], [3., 4.]])»> V = array ([2., 1.])»> x = solve (M, V)»> print (x) [-3. 2.5] [ ] 2 1

Numerical Analysis, Lund University, 2011 7 More examples Computing e iπ and 2 100 :»> print ( exp (1j*pi )) # should return -1 : -) (-1+1. 22464679915e - 16j )»> print (2** 100 ) 1267650600228229401496703205376L Computing ζ(x) = k=1 1 k x. For x = 2 we know that ζ(2) = π2 6 : # for x = 2:»> print ( scipy. special. zeta (2., 1)) 1. 64493406685»> print (pi**2/6) 1. 6449340668482264

Numerical Analysis, Lund University, 2011 8 Introduction and Motivation Python vs Other Languages Examples Concepts Basic Types Variables Lists For Loop Practical Information

Numerical Analysis, Lund University, 2011 9 Numbers A number may be an integer, a real number or a complex number. The usual operations are + and - addition and substraction * and / multiplication and division ** power 2**(2+2) # 16 1j**2 # -1

Numerical Analysis, Lund University, 2011 10 Strings Strings are lists of characters, enclosed by simple or double quotes: valid string " string with double quotes "

Numerical Analysis, Lund University, 2011 10 Strings Strings are lists of characters, enclosed by simple or double quotes: valid string " string with double quotes " You may also use triple quotes for strings including multiple lines: """ This is a long, long string """

Numerical Analysis, Lund University, 2011 11 Concept: Variable Variables A variable is a reference to an object. An object may have several references. One uses the assignment operator = to assign a value to a variable. Example x = [3, 4] # a list object is created y = x # this object now has two labels : x and y del x # we delete one of the labels del y # both labels are removed : the object is deleted

Numerical Analysis, Lund University, 2011 12 Concept: Lists Lists A python list is an ordered list of objects, enclosed in square brackets. One accesses elements of a list using zero-based indices inside square brackets.

Numerical Analysis, Lund University, 2011 13 List Examples Example L1 = [1, 2] L1[0] # 1 L1[1] # 2 L1[ 2] # raises IndexError L2 = [ a, 1, [3, 4]] L2[0] # a L2[2][0] # 3 L2[-1] # last element : [3,4] L2[- 2] # second to last : 1

Numerical Analysis, Lund University, 2011 14 List Utilities range(n) creates a list with n elements, starting with zero: print ( range (5)) [0, 1, 2, 3, 4]

Numerical Analysis, Lund University, 2011 14 List Utilities range(n) creates a list with n elements, starting with zero: print ( range (5)) [0, 1, 2, 3, 4] len(l) gives the length of a list: len ([ a, 1, 2, 34]) # returns 4

Numerical Analysis, Lund University, 2011 14 List Utilities range(n) creates a list with n elements, starting with zero: print ( range (5)) [0, 1, 2, 3, 4] len(l) gives the length of a list: len ([ a, 1, 2, 34]) # returns 4 Use append to append an element to a list: L = [ a, b, c ] L[-1] # c L. append ( d ) L # L is now [ a, b, c, d ] L[-1] # d

Numerical Analysis, Lund University, 2011 15 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 [< expr > for < variable > in < list >] Example L = [2, 3, 10, 1, 5] L2 = [x*2 for x in L] # [4, 6, 20, 2, 10] L3 = [ x* 2 for x in L if 4 < x <= 10] # [ 20, 10]

Numerical Analysis, Lund University, 2011 16 Comprehensive Lists in Maths Mathematical Notation This is very close to the mathematical notation for sets. Compare: and L2 = [ 2* x for x in L] L 2 = {2x; x L} One big difference though is that lists are ordered while sets aren t.

Numerical Analysis, Lund University, 2011 17 Operations on Lists Adding two lists concatenates (sammanfoga) them: L1 = [1, 2] L2 = [3, 4] L = L1 + L2 # [1, 2, 3, 4]

Numerical Analysis, Lund University, 2011 17 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. } {{ } n times L = [1, 2] 3 * L # [1, 2, 1, 2, 1, 2] (To multiply each element by c, we use arrays instead of lists.)

Numerical Analysis, Lund University, 2011 18 Concept: for loop for loop A for loop allows to loop through a list using an index variable. This variable is successively equal to all the elements in the list.

Numerical Analysis, Lund University, 2011 18 Concept: for loop for loop A for loop allows 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: print ( s * 2) # output : 2 4 20

Numerical Analysis, Lund University, 2011 19 Indentation The part to be repeated in the for loop has to be properly indented: for elt in my_list : do_ something () something_ else () etc print (" loop finished ") # outside the for block

Numerical Analysis, Lund University, 2011 20 Repeating a Task One typical use of the for loop is to repeat a certain task a fixed number of time: n = 30 for i in range (n): do_ something # this gets executed n times

Numerical Analysis, Lund University, 2011 21 Introduction and Motivation Python vs Other Languages Examples Concepts Basic Types Variables Lists For Loop Practical Information

Numerical Analysis, Lund University, 2011 22 Python Shell Start a python session by typing scipython in 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, write exit() When you want to run python at home please follow the installation instruction on http://www.maths.lth.se/na/python/install (see also the course compendium)

Numerical Analysis, Lund University, 2011 23 Executing Scripts You often want to execute the contents of a file. We recommand to use Geany 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 named file.py just write execfile( file.py ) in ipython.

Numerical Analysis, Lund University, 2011 24 Getting Help Some tips on how to use ipython: To get help 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 for completion in general