python day4

8/13/2019 Python Day4

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Scientific Computing in Computer Science, Technische Universitat Munchen

Part VI

Scientific Computing in Python

Tobias Neckel: Scripting with Python... and beyond

Compact Course @ GRS, June 03 - 07, 2013 80


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More on Maths

Modulemath

Constants piand e

Functions that operate onintand float

All return values float

c e i l ( x )f l o o r ( x )

e x p ( x )

f a b s ( x ) # s am e as g lo ba ll y d ef in ed ab s ()

l de xp ( x , i ) # x * 2** i

l og ( x [ , b as e ])

l o g 1 0 ( x ) # == log (x , 10)modf ( x ) # ( f r a ct i on a l , i n te g er p ar t )

p ow ( x , y ) # x ** y

s q r t ( x )




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Module math (2)

Trigonometric functions assume radians

c os ( x ); c os h ( x ); a co s ( x)

s in ( x ); . ..

t an ( x ); . ..

d e g r e e s ( x ) # rad -> d eg r a d i a n s ( x ) # deg -> r ad




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Module math (2)


c os ( x ); c os h ( x ); a co s ( x)

s in ( x ); . ..

t an ( x ); . ..


inf/nan

f l o a t (" i n f " )

f l o a t (" - i n f " )

f l o a t (" n a n " )




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Module math (2)


c os ( x ); c os h ( x ); a co s ( x)

s in ( x ); . ..

t an ( x ); . ..


inf/nan

f l o a t (" i n f " )

f l o a t (" - i n f " )

f l o a t (" n a n " )

Use module cmathfor complex numbers




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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




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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




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NumPy




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Module numpy

Homogeneous arrays

NumPy provides arbitrary-dimensional homogeneous arrays

Example

from numpy import *

a = a rr ay ( [ [1 , 2 , 3 ] , [4 , 5 , 6 ]] )print a

t y p e ( a )

a . s h a p e

print a [ 0 , 2 ]

a [0 ,2] = -1

b = a * 2print b




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Array creation

Create from (nested) sequence type

Direct access with method []

a = a rr ay ( [1 , 2 , 3 ,4 , 5 , 6 ,7 , 8] )

a[1]

a = a r ra y ( [ [1 , 2 , 3 , 4] , [ 5 , 6 , 7 , 8 ]] )

a [ 1 , 1 ]

a = a r ra y ( [ [ [1 , 2] , [3 , 4] ] , [ [5 , 6] , [7 , 8] ]] )

a [ 1 , 1 , 1 ]




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Array creation

Create from (nested) sequence type

Direct access with method []

a = a rr ay ( [1 , 2 , 3 ,4 , 5 , 6 ,7 , 8] )

a[1]

a = a r ra y ( [ [1 , 2 , 3 , 4] , [ 5 , 6 , 7 , 8 ]] )

a [ 1 , 1 ]

a = a r ra y ( [ [ [1 , 2] , [3 , 4] ] , [ [5 , 6] , [7 , 8] ]] )a [ 1 , 1 , 1 ]

Properties of arrays

a . n d i m # n um be r of d im en si on s

a . s h a p e # d i m en s io n s

a . s i z e # n um be r of e le me nt s

a . d t y p e # data type

a . i t e m s i z e # n umb er o f byt es




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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

a r ra y ( [ [1 , 2 , 3 ] , [4 , 5 , 6] ] , d ty p e = i nt )

a r ra y ( [ [1 , 2 , 3 ] , [4 , 5 , 6] ] , d ty p e = c o m pl e x )

a r ra y ( [ [1 , 2 , 3 ] , [4 , 5 , 6] ] , d ty p e = i n t8 )a r ra y ( [ [1 , 2 , 3] , [ 4 , 5 , 1 00 0] ] , d t yp e = i n t8 ) # w ro ng

a r r a y ( [ [ 1 , 2 , 3 ] , [ 4 , 5 , " h i " ]] , d t yp e = o b j ec t )

Exception: object_stores pointers to objects




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Create Arrays

(Some) default matrices (optional parameter: dtype)

a ra ng e ( [a , ] b [ , s tr id e ] ) # as range , 1 D

z er os ( (3 , 4) )

o ne s ( (1 , 3 ,4) )

e mp ty ( (3 , 4) ) # u ni n it ia l iz e d ( f as t )

l in sp ac e (a , b [ , n ]) # n e qu id is ta nt in [a , b]

l og sp ac e (a , b [ , n ]) # 10** a to 10** b

i d e n t i t y ( n ) # 2 d

f r o m f u n c t i o n ( lambda i , j : i +j , (3 , 4) , d ty pe = i nt )

def f ( i , j ) :return i + j

f r o mf u n ct i o n ( f , ( 3 ,4 ) , d t yp e = i n t )




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Manipulate Arrays

Reshaping arrays

a = a ra ng e ( 12 )

b = a . r e sh ap e ( (3 , 4 ))

a . r e s i z e ( ( 3 , 4 ) ) # in - p la ce !

a . t r a n s p o s e ( )a . f l a t t e n ( )

# E xa mp le use - c as e :

a = a ra ng e ( 14 4)

a . r e s i z e ( ( 1 2 , 1 2 ) )




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Create Arrays (2)

Create/Copy from existing data

a = a ra ng e ( 1 2) ; a . r es iz e ( (3 , 4 ))

c o p y ( a )

d ia g ( a ); t ri l ( a ); t ri u ( a)

e m p t y _ l i k e ( a ) # c op y s ha pe

z e r o s _ l i k e ( a )o n e s _ l i k e ( a )

a = l oa dt xt ( " m a t r i x . t x t " ) # f ro mf il e () if b in ar y

# p le nt y of o pt io ns : c om me nt s , d el im . , u se co ls , . ..

Matrix output

a . t o l i s t ( )

s a v e t x t (" m a t r i x . t x t " , a ) # t of il e () if b i na ry



S i ifi C i i C S i T h i h U i i M h


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Array Access and Manipulation

Typical slicing operations can be used

Separate dimensions by comma

a = a ra ng e ( 2 0) ; a . r es iz e ( (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



S i tifi C ti i C t S i T h i h U i it t M h


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Array Access

Iterating over entries

for row in a :

print row

b = a ra ng e ( 3 0) ; b . r es iz e ( (2 , 3 , 4) )for row in b :

for col in row:

print col

for entry in a . f l a t :

print entry



Scientific Computing in Computer Science Technische Universitat Munchen


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Computing with Arrays

Fast built-in methods working on arrays

a = a ra ng e ( 1 2) ; a . r es iz e ( (3 , 4 ))

3* a

a**2

a + a ^ 2

s i n ( a )

s q r t ( a )p r o d ( a )

s u m ( a )

it = t ra ns po se ( a )

x = a rr ay ( [1 , 2 , 3])y = a rr ay ( [ 10 , 2 0 , 30 ])

i nn er ( x , y )

d ot ( it , x )

c r o s s ( x , y )





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Computing with Arrays

There is much more. . .

var () cov () std ()

mean () median ()

min () max ()

svd()

t e n s o r d o t ( )

...

Matrices (withmat) are subclasses of ndarray, but strictlytwo-dimensional, with additional attributes

m = mat ( a )

m . T # t r an sp os em . I # i nv er se

m . A # as 2 d ar ray

m . H # c o nj ug at e t ra ns po s e





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Submodules

Modulenumpy.random

Draw from plenty of different distributions

More powerful than module random

Work on and return arrays

from numpy.random import *

b i no m ia l ( 10 , 0 .5 ) # 10 t ri al s , s uc ce ss 5 0%

b in om ia l ( 10 , 0.5 , 1 5)

r an di nt ( 0 , 10 , 1 5) # [0 , 10) , i nt

rand() # [0 , 1)r a n d ( 3 , 4 ) # (3 x4 ) arr ay





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Submodules (2)

Modulenumpy.linalg

Core linear algebra tools

n or m ( a ) ; n or m ( x )

i n v ( a )

s ol ve ( a , b ) # L AP AC K LU d ec om p .d e t ( a )

e i g ( a )

c h o l e s k y ( a )





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p g p ,

Submodules (2)

Modulenumpy.linalg


n or m ( a ) ; n or m ( x )

i n v ( a )


e i g ( a )


Modulenumpy.fft

Fourier transforms





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Submodules (2)

Modulenumpy.linalg


n or m ( a ) ; n or m ( x )

i n v ( a )


e i g ( a )


Modulenumpy.fft

Fourier transforms

There is more. . .





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Version Mania

Current Situation

python

MatplotlibSciPy

NumPy

IPythonpylab




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Matplotlib

Tobias Neckel: Scripting with Python... and beyondCompact Course @ GRS, June 03 - 07, 2013 99



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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




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Several Ways to do the Same

python/ipython interactive

> i py th on

import s c ip y , m a t pl o tl i b . p y l ab

x = s ci py . r a nd n ( 1 00 00 )

m atplo tlib . pylab . hist (x , 100)

> i py th onimport n u m py . r an d om , m a t p l o tl i b . p y l ab

x = n um p y . r a nd o m . r a nd n ( 1 0 0 00 )

m atplo tlib . pylab . hist (x , 100)

ipython in pylab mode

> i py th on - p yl ab

x = r an dn ( 1 00 00 )

h is t ( x , 1 00 )




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Example - First Plot

partially taken fromhttp://matplotlib.sourceforge.net/users/screenshots.html

from pylab import *

x = a ra ng e (0.0 , 2* pi , 0 .0 1)

y = sin (x )p lo t (x , y , l in ew i dt h = 4)

p l o t ( x , y )

x l a b e l ( L a b e l f o r x a x i s )

y l a b e l ( L a b e l f o r y a x i s )

title( S i m p l e p l o t o f s i n )g r i d ( T r u e )

show()




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Example Using Subplots

from pylab import *def f ( t ) :

s1 = cos ( 2* p i *t )

e1 = exp ( -t )

return m u l t i p l y ( s 1 , e 1 )

t1 = a ra ng e (0.0 , 5.0 , 0 .1)t2 = a ra ng e (0.0 , 5.0 , 0 .0 2)

t3 = a ra ng e (0.0 , 2.0 , 0 .0 1)

show() # giv es e rro r but hel ps ; -)

s u b p l o t ( 2 , 1 , 1 ) # rows , co lum ns , w hi ch to s ho w p lo t ( t1 , f ( t 1 ) , g o , t2 , f ( t2 ), k - - )

s u b p l o t ( 2 , 1 , 2 )

p lo t ( t3 , c os ( 2* p i * t 3 ) , r . )




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Example Using Subplots

# p re vi ou s s li de c on ti nu ed

s u b p l o t ( 2 , 1 , 1 )

g r i d ( T r u e )

title( A t a l e o f 2 s u b p l o t s )y l a b e l ( D a m p e d o s c i l l a t i o n )

s u b p l o t ( 2 , 1 , 2 )

g r i d ( T r u e )

x l a b e l ( t i m e ( s ) )

y l a b e l ( U n d a m p e d )




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Example - Histogram

# s ta rt i py th on - py la b or us e i mp or ts :# f ro m m at p lo tl i b . m la b i mp or t *

# f ro m m at p lo tl i b . p yp lo t i mp or t *

from numpy import *

mu , sigma = 100 , 15

x = mu + s ig ma * r an do m . ra nd n ( 10 00 0)n , bins , p at ch es = hist (x , 50 , no rm ed =1 , \

f a c e c o l o r = g r e e n , a l ph a = 0 . 7 5)

# add a best fit line

y = n orm pd f ( bins , mu , s igm a )

p lo t ( bi ns , y , r - - , l i ne w id t h = 1 )

a xi s ([40 , 160 , 0 , 0 .0 3] )

p l t . s h o w ( )




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SciPy




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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 importnecessary)

A lot more libraries for scientific computing are available, some ofthem using NumPy and SciPy

Here, just a short overview will be given

www.scipy.orgfor more material (incl. the content of thefollowing slides)




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SciPy Organisation - Subpackages

cluster Clustering algorithmsconstants Physical and mathematical constantsfftpack Fast Fourier Transform routinesintegrate Integration and ordinary differential equation solversinterpolate Interpolation and smoothing splinesio Input and Outputlinalg Linear algebra

maxentropy Maximum entropy methodsndimage N-dimensional image processingodr Orthogonal distance regressionoptimize Optimization and root-finding routinessignal Signal processing

sparse Sparse matrices and associated routinesspatial Spatial data structures and algorithmsspecial Special functionsstats Statistical distributions and functionsweave C/C++ integration




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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

...




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Example: Interpolation - Linear

import n ump y as np

import m a tp l ot l i b . p y p lo t a s p lt

from scipy import interpolate

x = np . a r an ge ( 0 , 10 )

y = np . e xp ( - x / 3. 0)

f = i nt e rp ol a te . i nt er p1 d ( x , y )

x ne w = n p . ar an ge ( 0 ,9 , 0 .1 )

p l t . p l o t ( x , y , o , x n e w , f ( x n e w ) , - )

p l t . t i t l e ( L i n e a r i n t e r p o l a t i o n )

p l t . s h o w ( )



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