computational mathematics with python · computationalmathematicswithpython unit5(cont):...

Computational Mathematics with PythonUnit 5 (cont): More on arrays, more on functions

Numerical Analysis, Lund UniversityLecturer: Claus Führer

2015

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Solving a Linear SystemIf A is a matrix and b is a vector you solve the linear equation

A · x = b

using solve which has the syntax x = solve(A,b).

ExampleWe want to solve {

x1 + 2x2 = 13x1 + 4x2 = 4

from scipy. linalg import solve

A = array([[1., 2.],[3., 4.]])

b = array([1., 4.])x = solve(A,b)dot(A, x) # should be almost b

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Slices

Slices are similar to that of lists and vectors except that there arenow two dimensions.

M[i,:] a vector filled by the row i of MM[:,j] a vector filled by the column j of M

M[2:4,:] slice 2 : 4 on the rows onlyM[2:4,1:4] slice on rows and columns

Omitting a dimensionIf you omit an index or a slice, Scipy assumes you are taking rowsonly.

M[3] is the third row of MM[1:3] is a matrix with the second and third rows of M .

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Altering a Matrix

You may alter a matrix using slices or direct access.I M [ 2,3] = 2.I M [ 2, : ] = < a vector>I M [ 1 : 3, : ] = < a matrix>I M [ 1 : 4,2:5 ] = < a matrix>

The matrices and vectors above must have the right size to “fit” inthe matrix M .

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Some technical terms

Rank The rank of an array is the number of indices used to identifyan element. A matrix has rank 2, a vector has rank 1, a scalarhas rank 0.Don’t confound this term with the rank used in LinearAlgebra.

Shape The shape is a tuple with the dimensions of the array: A 2× 3matrix is represented in Python by an array with shape (2, 3).

Length The number of elements in an array. An array with shape(2, 3) has length 6. An array with shape (2, ) (representing avector) has length 2.Don’t confound this term with the term length used in LinearAlgebra.

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Rank of matrix slices

When slicing the rank of the result object is as follows:

access rank kindindex,index 0 scalarslice,index 1 vectorslice,slice 2 matrix

Example

0 1 2 3

4 5 6 7

8 9 10 11

0 1 2 3

4 5 6 7

8 9 10 11

0 1 2 3

4 5 6 7

8 9 10 11

0 1 2 3

4 5 6 7

8 9 10 11

0 1 2 3

4 5 6 7

8 9 10 11

access shape rank kindM [ : 2,1: - 1 ] (2, 2) 2 matrixM [ 1, : ] 4 1 vectorM [ 1,1] ∅ 0 scalarM [ 1 : 2, : ] (1, 4) 2 matrixM [ 1 : 2,1:2 ] (1, 1) 2 matrix

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Reshaping

From a given tensor (vector or matrix) one may obtain anothertensor by reshaping.

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Reshaping Example

A = arange (6)

A. reshape (1,6)

A. reshape (6,1)

A. reshape (2,3)

A. reshape (3,2)

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2

3 4 5

0 1

2 3

4 5

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Reshaping Trick

Note that Python can guess one of the new dimensions. Just givea negative integer for the dimension to be guessed:A = arange (12) # a vector of length 12

A. reshape (1,-1) # row matrixA. reshape (-1,1) # column matrix

A. reshape (3,-1) # 3,4 matrixA. reshape (-1,4) # same

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Building Matrices

I Piling vectorsI Stacking vectorsI Stacking column matrices

The universal method to build matrices is concatenate. Thisfunction is called by several convenient functions

I hstack to stack matrices horizontallyI vstack to stack matrices verticallyI column_stack to stack vectors in columns

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Stacking Vectors

v1 = array([1,2])v2 = array([3,4])

vstack ([v1 ,v2])

column_stack ([v1 ,v2])

1 2

3 4

1 3

2 4

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sum, max, min

You may perform a number ofoperations on arrays, either onthe whole array, orcolumn-wise or row-wise. Themost common are

I maxI minI sum

Example

1 2 3 4

5 6 7 8

A.sum ()

36

1 2 3 4

5 6 7 8

A.sum(axis=0)

The result is a vector

6 8 10 12

1 2 3 4

5 6 7 8

A.sum(axis=1)

The result is a vector

10 26

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Boolean Arrays

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Boolean Arrays

One may use Boolean arrays to create a “template” for modifyinganother array:

B = array([[True , False],[False , True]]) # the template array

M = array([[2, 3],[1, 4]]) # the other array

M[B] = 0 # using the templateM # [[0, 3], [1, 0]]M[B] = 10 , 20M # [[10 , 3], [1, 20]]

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Creating Boolean Arrays

It might be just as tedious to create the boolean array by handthan to change the array directly. There are however manymethods to create Boolean arrays.Any logical operator will create a Boolean array instead of aBoolean.M = array([[2, 3], [1, 4]])M > 2 # array ([[ False , True], [False , True ]])M == 0 # array ([[ False , False], [False , False ]])N = array([[2, 3], [0, 0]])M == N # array ([[ True , True], [False , False ]])...

This allows the elegant syntax:M[M>2] = 0# all the elements > 2 are replaced by 0

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Comparing Arrays

Note that because array comparison create Boolean arrays, onecannot compare arrays directly.The solution is to use the methods all and any:A = array([[1,2],[3,4]])B = array([[1,2],[3,3]])A == B # creates array ([[ True , True], [True , False ]])(A == B).all () # False(A != B).any () # True

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Boolean Operations

For the same reason as before you cannot use and, or nor not onBoolean arrays! Use the following replacement operators instead:

logic operator replacement for Bool arraysA and B A & BA or B A | Bnot A - A

a = array([True , True , False , False])b = array([True , False , True , False])

a and b # error!a & b # array ([True , False , False , False ])a | b # array ([True , Trues , True , False ])-a # array ([ False , False , True , True ])

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Universal Functions

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Universal Functions

DefinitionA universal function (or ufunc) is a function that operates onarrays in an element-by-element fashion. That is, a ufunc is a“vectorized” wrapper for a function that takes a fixed number ofscalar inputs and produces a fixed number of scalar outputs.

Examples:from scipy import *sincosexp

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Vectorized Functions

Non-universal functions can be wrapped to behave like universalfuntions.This is done by the command vectorize.# A non - universal functiondef heaviside (x):

if x >= 0:return 1.

else:return 0.

heaviside ( linspace (-1,1,100)) # returns an error

vheaviside = vectorize ( heaviside ) # a new functionvheaviside ( linspace (-1,1,100)) # does the job

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Vectorized Functions (Cont.)

xvals= linspace (-1,1,100)plot(xvals , vectorize ( heaviside )(xvals))axis([-1.5,1.5,-0.5,1.5])

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More on Functions

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Unpacking Arguments

Positional arguments remind us of listsKeyword arguments remind us of dictionaries

data = [[1,2], [3,4]]style = {’linewidth ’:3, ’marker ’:’o’, ’color ’:’green ’}

Star operators unpack these to form a valid parameter listplot(*data , ** style)

∗ unpacks a list to positional arguments∗∗ unpacks dictionaries to keyword arguments

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Passing (tunneling) argumentsAlso in the definition of functions you might find these constructs.This is often used to pass arguments through a functiondef outer(f, x, *args , ** keywords ):

return f(x, *args , ** keywords )

def inner(x, y, z, u):print(y, z)print(u)return x ** 2

A callL=[1, 2]D={’u’:15}outer(inner , 3, *L, ** D)

Equivalently:outer(inner , 3, 1, 2, u=15)

Note, the function outer cannot know how many arguments itneeds to provide a full parameter list to the “inner” function f.

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Return

The return statement returns a single object!

def my_func (x):return 1,2,3,4,5,6

What is the object that is returned here? Which type does it have?(see Unit 3)

Statements after the return statement are ignored:

def my_func (x):return 1,2,3z = 25 # ignored

def my_func (x):if x > 0:

return 1else:

return -1z = 25 # ignored

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No Return

A function without a return statement returns None:

def my_func (x):z = 2*x

a = my_func (10.)

type(a) # <type ’NoneType ’>a == None # true

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Functions are objects

Functions are objects, they can be deleted, reassigned, copied ...

def square (x):""" Return the square of ‘x‘"""return x ** 2

square (4) # 16sq = square # now sq is the same as squaresq(4) # 16del square # ‘square ‘ doesn ’t exist anymoreprint ( newton (sq , .2)) # passing as argument

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Partial ApplicationPartial application = closures

In mathematics we often “freeze” a parameter of a function:

fω(x) = f (x, ω) = sin(ωx)

In Python there are many possibilities to do this, here is one ...def make_sine (omega):

def f(x):return sin(omega*x)

return f

fomega = make_sin (omega)

In order to understand this solution, recallI the scope of a variable and references to variables out of scopeI functions are objects

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Anonymous Functions: the lambda keyword

With λ-functions one has a handy tool for making one-linefunction definitions:f = lambda x: 3.*x ** 2 + 2.*x + 0.5f(3) # returns 33.5g = lambda x,y: 3.*x-2.*yg(1,1) # returns 1.0

Example for a common application, compute∫ 1

0 x2 + 5 dx:

import scipy. integrate as sisi.quad( lambda x: x ** 2+5,0,1)

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Partial Applications and λ

fω(x) = f (x, ω) = sin(ωx)

.... now simply becomes

omega = 3.fomega = lambda x: sin(omega*x)fomega (1.) # returns 0.14112 ...

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