Numerical Methods. Centre for Mathematical Sciences Lund University. Spring 2015

Transcription

1 Numerical Methods Claus Führer Alexandros Sopasakis Centre for Mathematical Sciences Lund University Spring 2015

2 Preface These notes serve as a skeleton for the course. They document together with the assignments the course outline and course content. All references in the notes refer to the textbook by Sauer if not otherwise stated. The notes are a guide to read the textbook. They are no textbook. Please report missprints or your comments to the teacher. 1

3 Python 1: Python basics for MATLAB users This course comes with many programming examples in Python. You are encouraged to do your computer assignments in Python also. Here we collect some things to know: Install Python 3.4 as described at nahelp Use Spyder as workbench and IPython in its console. 2

4 Python 1-1: Elementary datatypes in Python integers: 1; -1; floats: complex: strings: Boolean: 1.;1.0;1.e+5;.3e j;3j;0.2-1.j Anna ;"Anna s book" True; False Note, the semicolon, ;, or a newline separate Python commands. 3

5 Python 1-2: Operations with elementary datatypes Here some operations which differ from MATLAB powers 2.**3 string concatenation 3* Carl ; matte + huset integer division: 2/3 returns a float, 2//3 returns an integer. 4

6 Python 1-3: Methods of elementary datatypes Datatypes have special methods (functions which operate on these datatypes): a=3.+5j a. conjugate () s = victor s. upper () Find out other string methods by writing the s.<tab> <TAB> denotes pressing the TAB-key which does command completion in this case. Do the same also for a variable of type complex. 5

7 Python 1-4: Container datatypes Container datatypes are those which collect objects of various kinds. Lists: Dictionaries: a=[1,2., Anna,[3,4]] style={ line : dotted, points :300} Tuples: (see later in this course) Arrays: v=array([1.,2.,3.]) Note, array requires that you import the package SciPy first, from scipy import * This automatically also imports NumPy. 6

8 Python 1-5: Lists and indexing A list is an ordered collection of items. Indexing is zero based. a=[1,2., Anna,[3,4]] a[0] # returns 1 a[3] # returns [3,4] a[3][0] # returns 3 Negative indexes count from the last element: a[-1] # returns [3,4] a[-2] # returns Anna Slices return (sub-)lists. Note the hotel-night convention of slice indexing: a[1:3] # returns [2., Anna ] a[1:1] # returns an empty list : [] a[-3:] # returns [2., Anna,[3,4]] 7

9 Python 1-6: Play with list indexing Try out the following commands and find an explanation for the results: a=[1,2., Anna,[3,4]] a[1:1]=[-5,-6,-7] a[:2]=[] a[3]. upper () b=[ Carl, 26] a[-1:-1]=b 8

10 Python 1-7: List operations and methods Concatenating (see strings): a=[3,4] b=a+a # returns : [3,4,3,4] c=3*a # returns : [3,4,3,4,3,4] d=a-a # returns an error e=a/a # returns an error Appending: a=[3,4] a. append (5) # replaces a by [3,4,5] b=[6,7] a. append (b) # replaces a by [3,4,5,[6,7]] a=[3,4] b=[5,6] a. append (b) Compare the two command sequences a=[3,4] b=[5,6] a+b 9

11 Python 1-8: Dictionaries Access to dictionary items is by keyword: style ={ line : dotted, points : 300 } style [ points ] # returns : 300 style [ scale ]= linear # adds new items A key/value pair is called an item. These can be extracted to lists: style ={ line : dotted, points : 300 } style. values () # returns : [ dotted, 300 ] style. keys () # returns : [ line, points ] style. items () # returns : [( line, dotted ), ( points, 300 )] 10

12 Python 1-9: Arrays You need to import SciPy(or at least NumPy) to access arrays. This can be done automatically in your IPython startup script or (even better) in a Spyder template see our help pages. from scipy import * Arrays are constructed from lists and operations are elementwise: v= array ([1.,2.,3.]) w= array ([4.,5.,6.]) 2*v # returns : array ([2.,4.,6.]) v+w # returns : array ([5.,7.,9.]) v*w # returns : array ([4.,10.,18.]) dot (v,w) # returns : 38 11

13 Python 1-10: Arrays - Matrices, vectors and methods Vectors are constructed from simple lists, matrices from a list of lists: v= array ([1,2,3]) A= array ([[1,2,3],[4,5,6]]) v[2] # returns : 3 A[1,2] # returns : 6 Some methods: A. shape # returns (2,3) A. shape [ 0] # returns number of rows 2 A. T # returns A transpose Note, a vector and a column of a matrix are different mathematical objects! 12

14 Python 1-11: Block commands Block commands are characterized by a keyword, a colon and a sequence of equally intented commands: for i in [1,2,3]: print (i**2} print (2*i) print ( finished ) Use four blanks or one TAB for indentation. Don t mix. a=2 b=3 if a > b: print ( a wins } else : print ( b wins ) print ( test finished ) 13

15 Python 1-12: An example A list can be generated with the range command. range(200) can be used to generate a list starting of integers starting at 0 and ending at 199. The point x with x = f(x) is called a fixed point of f. We try to find the fixed point of sin(x) ax 30 by a simple iteration. If it converged we found the fixed point: x=0.5 a=0.5 for i in range ( 200 ): x= sin (x)-a*x -30 print ( After 200 iterations we got the result { res }. format ( res =x)) # test if abs (( sin (x) -a*x -30) - x) < 1.e-8: print ( We found the fixed point. ) else : print ( Still far away ) Note, again SciPy has to be imported (for the abs). Note also the string formatting in the first print command. 14