Part VI. Scientific Computing in Python

Part VI Scientific Computing in Python Compact Course @ GRS, June 03-07, 2013 80

More on Maths Module math Constants pi and e Functions that operate on int and float All return values float ceil (x) floor (x) exp (x) fabs ( x) # same as globally defined abs () ldexp (x, i) # x * 2** i log (x [, base ]) log10 (x) # == log (x, 10) modf ( x) # ( fractional, integer part ) pow (x, y) # x**y sqrt (x) Compact Course @ GRS, June 03-07, 2013 81

Module math (2) Trigonometric functions assume radians cos (x); cosh (x); acos (x) sin (x);... tan (x);... degrees ( x) # rad -> deg radians ( x) # deg -> rad Compact Course @ GRS, June 03-07, 2013 82

Module math (2) Trigonometric functions assume radians cos (x); cosh (x); acos (x) sin (x);... tan (x);... degrees ( x) # rad -> deg radians ( x) # deg -> rad inf/nan float (" inf ") float ("-inf ") float (" nan ") Compact Course @ GRS, June 03-07, 2013 82

Module math (2) Trigonometric functions assume radians cos (x); cosh (x); acos (x) sin (x);... tan (x);... degrees ( x) # rad -> deg radians ( x) # deg -> rad inf/nan float (" inf ") float ("-inf ") float (" nan ") Use module cmath for complex numbers Compact Course @ GRS, June 03-07, 2013 82

Now to Real Maths... Standard sequence types (list, tuple,... ) Can be used as arrays Can contain different types of objects Very flexible, but slow Loops are not very efficient either For efficient scientific computing, other datatypes and methods required Compact Course @ GRS, June 03-07, 2013 83

Now to Real Maths... Standard sequence types (list, tuple,... ) Can be used as arrays Can contain different types of objects Very flexible, but slow Loops are not very efficient either For efficient scientific computing, other datatypes and methods required Modules NumPy Matplotlib SciPy Compact Course @ GRS, June 03-07, 2013 83

NumPy Compact Course @ GRS, June 03-07, 2013 84

Module numpy Homogeneous arrays NumPy provides arbitrary-dimensional homogeneous arrays Example from numpy import * a = array ([[1,2,3],[4,5,6]]) print a type (a) a. shape print a [0,2] a [0,2] = -1 b = a*2 print b Compact Course @ GRS, June 03-07, 2013 85

Array creation Create from (nested) sequence type Direct access with method [] a = array ([1,2,3,4,5,6,7,8]) a [1] a = array ([[1,2,3,4],[5,6,7,8]]) a [1,1] a = array ([[[1,2],[3,4]],[[5,6],[7,8]]]) a [1,1,1] Compact Course @ GRS, June 03-07, 2013 86

Array creation Create from (nested) sequence type Direct access with method [] a = array ([1,2,3,4,5,6,7,8]) a [1] a = array ([[1,2,3,4],[5,6,7,8]]) a [1,1] a = array ([[[1,2],[3,4]],[[5,6],[7,8]]]) a [1,1,1] Properties of arrays a. ndim # number of dimensions a. shape # dimensions a. size # number of elements a. dtype # data type a. itemsize # number of bytes Compact Course @ GRS, June 03-07, 2013 86

Data Types Exact, C/C++-motivated type of array elements can be specified Otherwise, defaults are used Some types (different storage requirements): int_, int8, int16, int32, int64, float_, float8, float16, float32, float64, complex_, complex64, bool_, character, object_ Standard python type names result in default behaviour array ([[1,2,3],[4,5,6]], dtype = int ) array ([[1,2,3],[4,5,6]], dtype = complex ) array ([[1,2,3],[4,5,6]], dtype = int8 ) array ([[1,2,3],[4,5,1000]], dtype = int8 ) # wrong array ([[1,2,3],[4,5, "hi"]], dtype = object ) Exception: object_ stores pointers to objects Compact Course @ GRS, June 03-07, 2013 87

Create Arrays (Some) default matrices (optional parameter: dtype) arange ([a,] b [, stride ]) # as range, 1D zeros ( (3,4) ) ones ( (1,3,4) ) empty ( (3,4) ) # uninitialized ( fast ) linspace (a, b [, n]) # n equidistant in [a, b] logspace (a, b [, n]) # 10** a to 10** b identity (n) # 2d fromfunction ( lambda i,j: i+j, (3,4), dtype = int ) def f(i,j): return i+j fromfunction (f, (3,4), dtype = int ) Compact Course @ GRS, June 03-07, 2013 88

Manipulate Arrays Reshaping arrays a = arange (12) b = a. reshape ((3,4)) a. resize ((3,4)) # in - place! a. transpose () a. flatten () # Example use - case : a = arange (144) a. resize ((12,12)) Compact Course @ GRS, June 03-07, 2013 89

Create Arrays (2) Create/Copy from existing data a = arange (12); a. resize ((3,4)) copy (a) diag (a); tril (a); triu (a) empty_ like ( a) # copy shape zeros_like (a) ones_like (a) a = loadtxt (" matrix. txt ") # fromfile () if binary # plenty of options : comments, delim., usecols,... Matrix output a. tolist () savetxt (" matrix. txt ", a) # tofile () if binary Compact Course @ GRS, June 03-07, 2013 90

Array Access and Manipulation Typical slicing operations can be used Separate dimensions by comma a = arange (20); a. resize ((4,5)) a [1] a [1:2,:] a [:,::2] a [::2,::2] a [::2,::2] = [[0, -2, -4],[ -10, -12, -14]] a [1::2,1::2] = -1*a [1::2,1::2] Selective access a[ a > 3] a[ a > 3] = -1 Compact Course @ GRS, June 03-07, 2013 91

Array Access Iterating over entries for row in a: print row b = arange (30); b. resize ((2,3,4)) for row in b: for col in row : print col for entry in a. flat : print entry Compact Course @ GRS, June 03-07, 2013 92

Computing with Arrays Fast built-in methods working on arrays a = arange (12); a. resize ((3,4)) 3*a a **2 a+a^2 sin (a) sqrt (a) prod (a) sum (a) it = transpose ( a) x = array ([1,2,3]) y = array ([10,20,30]) inner (x, y) dot (it, x) cross (x,y) Compact Course @ GRS, June 03-07, 2013 93

Computing with Arrays There is much more... var () cov () std () mean () median () min () max () svd () tensordot ()... Matrices (with mat) are subclasses of ndarray, but strictly two-dimensional, with additional attributes m = mat (a) m. T # transpose m. I # inverse m. A # as 2d array m. H # conjugate transpose Compact Course @ GRS, June 03-07, 2013 94

Submodules Module numpy.random Draw from plenty of different distributions More powerful than module random Work on and return arrays from numpy. random import * binomial (10, 0.5) # 10 trials, success 50% binomial (10, 0.5, 15) randint (0, 10, 15) # [0,10), int rand () # [0,1) rand (3,4) # (3 x4) array Compact Course @ GRS, June 03-07, 2013 95

Submodules (2) Module numpy.linalg Core linear algebra tools norm (a); norm (x) inv (a) solve (a, b) # LAPACK LU decomp. det (a) eig (a) cholesky ( a) Compact Course @ GRS, June 03-07, 2013 96

Submodules (2) Module numpy.linalg Core linear algebra tools norm (a); norm (x) inv (a) solve (a, b) # LAPACK LU decomp. det (a) eig (a) cholesky ( a) Module numpy.fft Fourier transforms Compact Course @ GRS, June 03-07, 2013 96

Submodules (2) Module numpy.linalg Core linear algebra tools norm (a); norm (x) inv (a) solve (a, b) # LAPACK LU decomp. det (a) eig (a) cholesky ( a) Module numpy.fft Fourier transforms There is more... Compact Course @ GRS, June 03-07, 2013 96

Version Mania Current Situation SciPy Matplotlib NumPy pylab IPython python Compact Course @ GRS, June 03-07, 2013 97

Version Mania Problems: Numpy, scipy, pylab, ipython and matplotlib often used simultaneously The packages depend on each other (matplotlib uese numpy arrays, e.g.) Depending on OS (version), different packages may have to be installed (i.e. the module name in import command may be different!). Vision: All in one (new) module PyLab! exists as unofficial package Attention Name: again pylab! Compact Course @ GRS, June 03-07, 2013 98

Matplotlib Compact Course @ GRS, June 03-07, 2013 99

Matplotlib What is it? Object-oriented library for plotting 2D Designed to be similar to the matlab plotting functionality Designed to plot scientific data, built on numpy datastructures Compact Course @ GRS, June 03-07, 2013 100

Several Ways to do the Same python/ipython interactive > ipython import scipy, matplotlib. pylab x = scipy. randn (10000) matplotlib. pylab. hist (x, 100) > ipython import numpy. random, matplotlib. pylab x = numpy. random. randn (10000) matplotlib. pylab. hist (x, 100) ipython in pylab mode > ipython - pylab x = randn (10000) hist (x, 100) Compact Course @ GRS, June 03-07, 2013 101

Example - First Plot partially taken from http://matplotlib.sourceforge.net/users/screenshots.html from pylab import * x = arange (0.0, 2* pi, 0.01) y = sin (x) plot (x, y, linewidth =4) plot (x,y) xlabel ( Label for x axis ) ylabel ( Label for y axis ) title ( Simple plot of sin ) grid ( True ) show () Compact Course @ GRS, June 03-07, 2013 102

Example Using Subplots from pylab import * def f(t): s1 = cos (2* pi*t) e1 = exp (-t) return multiply (s1,e1) t1 = arange (0.0, 5.0, 0.1) t2 = arange (0.0, 5.0, 0.02) t3 = arange (0.0, 2.0, 0.01) show () # gives error but helps ; -) subplot (2,1,1) # rows, columns, which to show plot (t1, f(t1), go, t2, f(t2), k-- ) subplot (2,1,2) plot (t3, cos (2* pi*t3), r. ) Compact Course @ GRS, June 03-07, 2013 103

Example Using Subplots # previous slide continued subplot (2,1,1) grid ( True ) title ( A tale of 2 subplots ) ylabel ( Damped oscillation ) subplot (2,1,2) grid ( True ) xlabel ( time (s) ) ylabel ( Undamped ) Compact Course @ GRS, June 03-07, 2013 104

Example - Histogram # start ipython - pylab or use imports : # from matplotlib. mlab import * # from matplotlib. pyplot import * from numpy import * mu, sigma = 100, 15 x = mu + sigma * random. randn (10000) n, bins, patches = hist (x, 50, normed =1, \ facecolor = green, alpha =0.75) # add a best fit line y = normpdf ( bins, mu, sigma ) plot (bins, y, r--, linewidth =1) axis ([40, 160, 0, 0.03]) plt. show () Compact Course @ GRS, June 03-07, 2013 105

SciPy Compact Course @ GRS, June 03-07, 2013 106

More than NumPy? SciPy depends on NumPy Built to work on NumPy arrays Providing functionality for mathematics, science and engineering Still under development NumPy is mostly about (N-dimensional) arrays SciPy comprises a large number of tools using these arrays SciPy includes the NumPy functionality (only one import necessary) A lot more libraries for scientific computing are available, some of them using NumPy and SciPy Here, just a short overview will be given www.scipy.org for more material (incl. the content of the following slides) Compact Course @ GRS, June 03-07, 2013 107

SciPy Organisation - Subpackages cluster Clustering algorithms constants Physical and mathematical constants fftpack Fast Fourier Transform routines integrate Integration and ordinary differential equation solvers interpolate Interpolation and smoothing splines io Input and Output linalg Linear algebra maxentropy Maximum entropy methods ndimage N-dimensional image processing odr Orthogonal distance regression optimize Optimization and root-finding routines signal Signal processing sparse Sparse matrices and associated routines spatial Spatial data structures and algorithms special Special functions stats Statistical distributions and functions weave C/C++ integration Compact Course @ GRS, June 03-07, 2013 108

Special Functions Airy functions Elliptic functions Bessel functions (+ Zeros, Integrals, Derivatives, Spherical, Ricatti-) Struve functions A large number of statistical functions Gamma functions Legendre functions Orthogonal polynomials (Legendre, Chebyshev, Jacobi,...) Hypergeometric functios parabolic cylinder functions Mathieu functions Spheroidal wave functions Kelvin functions... Compact Course @ GRS, June 03-07, 2013 109

Example: Interpolation - Linear import numpy as np import matplotlib. pyplot as plt from scipy import interpolate x = np. arange (0,10) y = np.exp (-x /3.0) f = interpolate. interp1d (x, y) xnew = np. arange (0,9,0.1) plt. plot (x,y, o,xnew,f( xnew ), - ) plt. title ( Linear interpolation ) plt. show () Compact Course @ GRS, June 03-07, 2013 110

Example: Interpolation - Cubic Spline import numpy as np import matplotlib. pyplot as plt from scipy import interpolate x = np. arange (0, 2.25* np.pi, np.pi /4) y = np.sin (x) spline = interpolate. splrep (x,y,s =0) xnew = np. arange (0,2.02* np.pi,np.pi /50) ynew = interpolate. splev ( xnew, spline ) plt. plot (x,y, o,xnew, ynew ) plt. legend ([ Linear, Cubic Spline ]) plt. axis ([ -0.05,6.33, -1.05,1.05]) plt. title ( Cubic - spline interpolation ) plt. show () Compact Course @ GRS, June 03-07, 2013 111