2 Getting Started with Numerical Computations in Python

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1 1 Documentation and Resources * Download: o Requirements: Python, IPython, Numpy, Scipy, Matplotlib o Windows: google "windows download (Python,IPython,Numpy,Scipy,Matplotlib" o Debian based: sudo apt-get install python ipython numpy scipy matplotlib * Screencast for IPython o * Documentation for Plotting o o * Documentation for Arrays/Matrices/Linear Algebra o o 2 Getting Started with Numerical Computations in Python from numpy import from numpy. l i n a l g import Listing 1: Numpy examples # defining matrices 6 A = array ( [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ], dtype = f l o a t ) print 3x3 identity matrix print eye ( 3 ) print 3x3 matrix filled with 1 print ones ( ( 3, 3 ) ) 11 print repeat matrix along the first axis print r e p e a t (A, 2, a x i s =1) print array with 3 elements equally spaced from to 5 print l i n s p a c e (, 5, 3 ) print populate matrix 16 G = array ( [ [ s i n ( x)+ cos ( y ) for x in l i n s p a c e (,2 pi, 4 ) ] for y in l i n s p a c e (2 pi, 4 pi, 4 ) ] ) print G print functions operate elementwise on arrays G = s i n (G) 21 # modifying matrices print transpose of a 2 dim array G transposed = G. T 26 print second column of A print G[ :, 1 ] print second column and all rows but the last print G[ : 1, 1 ] x = l i n s p a c e (, 5, 1 ) 31 print whole vector x print x print every second element print x [ : : 2 ] print every second element in reverse 36 print x [ : : 2 ] # l i n a l g procedures print rank(g)=, 1

2 2 GETTING STARTED WITH NUMERICAL COMPUTATIONS IN PYTHON 41 print rank (G) print matrix multiplication if 2D arrays A \ cdot B print dot ( ones ( ( 3, 3 ) ), diag ( [ 1, 2, 3 ] ) ) print inner product print dot ( [ 1, 2, 3 ], [ 4, 5, 6 ] ) # works also on l i s t s 46 print dyadic product v v^t print outer ( [ 1, 2, 3 ], [ 4, 5 ] ) 51 print elementwise multiplication A[i,j]* D[i,j] print G G transposed print inverse of G try : i n v (G) # this wont work since rank(g)=2 and G a 4x4 matrix except : 56 print matrix inversion failed print solve A x=b b = array ( [ 1, 2, 3 ] ) A = diag ( ones ( 3 ) ) + diag ( ones ( 2 ), k=1) 61 x = s o l v e (A, b ) print x + diag ( ones ( 2 ), k= 1) #!/ usr/bin/env python 2 from pylab import import numpy as npy Listing 2: Pylab examples # figure with two overlayed plots x = l i n s p a c e (, 1, 1 ) 7 y = s i n ( x ) z = cos ( x ) f i g u r e ( ) p1=p l o t ( x, y, b- ) p2=p l o t ( x, z, r. ) 12 x l a b e l ( lala ) y l a b e l ( dumdedum ) legend ( ( p1, p2 ), ( foo, bar ) ) t e x t ( 2,. 5, r some text with Latex code $ i \frac{d}{dt} \Psi = H \Psi $ ) arrow (,, 2, 1 ) 17 g r i d ( on ) t i t l e ( foo ) s a v e f i g ( lala.eps ) # figure with subplots 22 f i g u r e ( ) subplot ( 2, 2, 1 ) p1=p l o t ( x, y, b- ) subplot ( 2, 2, 2 ) p2=p l o t ( x, y, r- ) 27 subplot ( 2, 2, 3 ) p3=p l o t ( x, y, g- ) subplot ( 2, 2, 4 ) p4=p l o t ( x, y, k- ) s a v e f i g ( subplots.eps ) 32 # figure with contour plots of z=f (x, y)= xˆt A x # A indefinite matrix A = diag ( [ 3, 2 ] ) x i = npy. r [ 1:1:1 j ] 37 X,Y = meshgrid ( xi, x i ) x = X. r a v e l ( ) # make i t 1D array y = Y. r a v e l ( ) xy = a s a r r a y ( z i p ( x, y ) ). T z = diag ( dot ( xy.t, dot (A, xy ) ) ) # this can definitely be done more elegantly ; ) 2

3 42 z = z. reshape ( 1, 1 ) f i g u r e ( ) imshow ( z ) c o l o r b a r ( ) s a v e f i g ( imshow.eps ) 47 ## Contour plot f i g u r e ( ) contour ( xi, xi, z ) c o l o r b a r ( ) s a v e f i g ( contour.eps ) 52 f i g u r e ( ) ### ContourF plot c o n t o u r f ( xi, xi, z ) c o l o r b a r ( ) s a v e f i g ( contourf.eps ) 57 show ( ) #pops up some windows from numpy import Listing 3: Pitfalls examples # a l l objects are passed as references # for example dict io nar ie s ( hash maps, called map in c++) 6 a ={ foo : bar, 2 : 1 } print a c = a c [ foo ] = lala print c 11 print a # or arrays A = ones ( ( 2, 2 ) ) B = A 16 B[, ] = print A # a l l? No, int, float, etc are immutable in Python a = b = a b = 32 print a # solution 26 C = A. copy ( ) C[,1]= print A 31 # at the moment: integer division by standard ( changes in Python 3) print 2/3 # output : print 2. / 3 #output Listing 4: general stuff from numpy import print output formatting 6 print fixed number of integer digits print an int with at least 5 digits length : %5d % 23 print fill whitespaces with print %5d %43 print floating point 11 print %2.7 f %

4 3 OBJECT ORIENTED PROGRAMMING 16 print %2.7 f % print exponential representation print %e %( ) print insert a string %s % INPUTSTRING # elementary datatypes print dictionary (= hash map) mydict = { 1 :, 2 : lala, myarray : array ( [ 1, 2, 3 ] ) } 21 print access with print mydict [ 1 ], mydict [ 2 ], mydict [ myarray ] 26 print list m y l i s t = [ 1, 2, 3 ] class myclass : a=1 b=2 myobject = myclass ( ) 31 myobject. a = 2 myobject. b = 3 myobject. c = 4 # this also works # looping and i f then else 36 for i in range ( 1 ) : # very slow!!! print i i = while True : # slow!!! 41 print i i = i + 1 i f i ==9: break e l i f i == 8 : 46 print 8 yay! else : print. 51 print normal functions def f (A, x ) : return dot (A, x ) x = l i n s p a c e (, 9, 1 ) A = ones ( ( 1, 1 ) ) 56 print f (A, x ) print lambda functions ( anonymous functions ) f = lambda x : x 2 61 print pass functions as argument of functions def f ( g ) : return g ( 1. 5 ) print f (lambda x : x 2) 3 Object Oriented Programming import numpy Listing 5: Algorithmic differentiation class adouble ( o b j e c t ) : """ derive the class from object """ 6 def i n i t ( s e l f, x, dx = ) : 4

5 """ this function is automatically called when >>> a = adouble (), i.e. runs some initialization. Usually called CONSTRUCTOR """ s e l f. x = x 11 s e l f. dx = dx def s t r ( s e l f ) : """ string representation of adouble : i.e. the result you see when you do: >>> print adouble () """ 16 return [%f,%f] %( s e l f. x, s e l f. dx ) def m u l ( s e l f, rhs ) : """ multiplication between doubles and/ or adboules a*b is equivalent to a. mul (b) 21 """ i f numpy. i s s c a l a r ( rhs ) : return adouble ( s e l f. x rhs, s e l f. dx rhs ) else : return adouble ( s e l f. x rhs. x, s e l f. x rhs. dx + s e l f. dx rhs. x ) 26 def r m u l ( s e l f, l h s ) : """ 2 * adouble () is equivalent to 2. mul ( adouble ()), but integers dont have this functionality. therefore this must be a right multipilcation, i.e. 31 adouble (). mul (2) """ return s e l f l h s i f name == " main " : 36 ### this i s only executed i f this script i s called ###by $python algorithmic differentiation. py a = adouble ( 2, 1 ) b = adouble ( 3 ) print a b 41 print a 2. print 2 a 1..5 foo foo bar some text with Latex code i dtψ=hψ d dumdedum lala Fig. 3.1: Output produced by Listing 2. 5

6 3 OBJECT ORIENTED PROGRAMMING Fig. 3.2: Output produced by Listing (a) (b) (c) Fig. 3.3: Output produced by Listing 2. File contour.eps in a) and contourf.eps in b). In c) imshow has been used, it plots the value of every matrix entry in a different color. 6