unit4: functions numericalanalysis,lunduniversity 2015 · computationalmathematicswithpython unit4:...

Computational Mathematics with PythonUnit 4: Functions

Numerical Analysis, Lund UniversityLecturers: Claus Führer, Alexandros Sopasakis

2015

Numerical Analysis, Lund University, 2015 1

Functions


Functions in MathematicsIn Mathematics:A function is written as a map, that uniquelyassigns an element y from the range R to every element x fromthe domain D.

f : x 7→ y.

f is the functionx is its argumenty is its (return) value

There can be several arguments of different type:

Considerf (g, a, b) =

∫ b

ag(x)dx

The arguments are not interchangable. Position matters.


Functions in Python

Definition of a function:def f(x1 , x2 , x3):

..... # some computationsy = ... #return y

Evaluation (call) of a function:f(17 , 18 , -2)f([1, 2, 3], {’Tol ’: 1.e-10}, ’ro’)...

Wording:x1, x2, x3 are called function parameters (needed for the definition)17, 18, -2 are called arguments (needed for the evaluation)


Passing Arguments

Consider:def subtract (x1 , x2):

return x1 - x2

Passing arguments by position:subtract (1, 2) # returns -1

Position matters.

Passing arguments by keyword:subtract (x2=2, x1=1) # returns -1

Position doesn’t matter.


Scope of variables I

Variables defined inside the function are said to belong to thefunction’s scope.They are unknown outside the function.

Definition:def mult2(x):

c = 2.return c*x

Evaluation:mult2(20.) # returns 40c # returns NameError : ’c’

not defined


Scope of variables II

Compare:

a is a parameter of thefunctiondef multiply (x, a):

return a*x

a is referenced from outsidethe function’s scope:a = 3def multiply (x):

return a*x

Discuss the (dis-)advantages of these versions!


Scope of variables III

Variables with the same name belonging to different scopes aredifferent variables:

a = 3def multiply (x):

a = 17return a*x

multiply (2) # returns 34a # returns (still) 3


Local variables

Similarly, immutable variables are copied when calling a function.You cannot change an input variable by changing a local variablelike this:a = 3def f(x):

x = 2return x

print (f(a)) # 2print (a) # 3

However, this can be done for mutable variables like lists,dictionaries, etc.As a principle, never change input variables within a function.


Default Argumentsdefault (eng) = standard (sv)

The standard way ...def mult(x, c):

return c*y

a typical cally=mult(2., 13.)# returns 26y=mult(c=13., x=2.)# returns 26

... now with defaultsdef mult(x, c=1.0):

return c*y

possible callsy=mult(2., 13.)# returns 26y=mult(2.)# returns 2

In the definition of the function mandatory parameters mustprecede optional parameters (those with default values).Why?


Arguments, Parameters – Summary

Make sure that you understood the differenceI between arguments and parametersI between function definition and function evaluation (call)I between positional arguments and keyword arguments


Docstrings

All functions (and everything else) should be documented carefully:

A doctring is the leading comment in a function (or class):

def newton (f, x0):"""Newton ’s method for computing a zero of a functionon input:f ( function ) given function f(x)x0 (float) initial guesson return :y (float) the approximated zero of f"""...

help(newton) in Python or newton? in IPython displays thedocstring as a help text.


Computational Mathematics with PythonUnit 5: Linear Algebra

Numerical Analysis, Lund UniversityLecturers: Claus Führer, Alexandros Sopasakis

2015


Vectors


The array type

Lists are almost like vectors but the operations on list are not thelinear algebra operation.

DefinitionAn array represents a vector or a matrix in linear algebra. It isoften initialised from a list or another vector. Operations + , * ,/ , - are all elementwise. dot is used for the scalar product.


Vector usage

Examplevec = array([1., 3.]) # a vector in the plane2*vec # array ([2., 6.])vec * vec # array ([1., 9.])vec/2 # array ([0.5, 1.5])norm(vec) # normdot(vec , vec) # scalar product


Vectors are similar to lists

I Access vectors via their indicesv = array([1., 2., 3.])v[0] # 1.

I The length of a vector is still obtained by the function len.len(v) # 3

I Parts of vectors using slicesv[:2] # array ([1., 2.])

I Replace parts of vectors using slicesv[:2] = [10 , 20]v # array ([10., 20., 3.])


Vectors are not lists!Operations are not the same:

I Operations + and * are differentI More operations are defined: - , /I Many functions act elementwise on vectors: exp, sin, sqrt,

etc.I Scalar product with dot

I Vectors have a fixed size: no append methodI Only one type throughout the whole vector

(usually float, complex, int or bool)I Vector slices are views:

v = array([1., 2., 3.])v1 = v[:2] # v is array ([1.,2.])v1[0] = 0. # if v1 is changed ...v # ... v is changed too: array ([0., 2., 3.])


More examplesv1 = array([1., 2., 3.]) # don ’t forget the dots!v2 = array([2, 0, 1.]) # one dot is enoughv1 + v2; v1 / v2; v1 - v2; v1 * v2

3*v13*v1 + 2*v2

dot(v1 , v2) # scalar productcos(v1) # cosine , elementwise

# accessv1[0] # 1.v1[0] = 10

# slicesv1[:2] # array ([10., 2.])v1[:2] = [0, 1] # now v1 == array ([0.,1.,3.])v1[:2] = [1,2,3] # error!


Creating Vectors (linspace)

The linspace method is a convenient way to create equallyspaced arrays.xs = linspace (0, 10 , 200) # 200 points between 0 and

10xs[0] # the first point is 0xs[-1] # the last is 10

So for example the plot of the sine function between 0 and 10 willbe obtain by:plot(xs , sin(xs))


Creating Vectors (zeros, ones, ...)

Some handy methods to quickly create vectors:zeros zeros(n) creates a vector of size n filled with zerosones ones(n) is the same filled with onesrand rand(n) creates a vector with uniformly distributed random

numbers between 0 and 1empty empty(n) creates an “empty” vector of size n

(try it!)


Concatenating Vectors

Since the + operation is redefined we need a means toconcatenate vectors. This is where the command hstack comes tohelp. hstack([v1, v2,..., vn]) concatenates the vectors v1,v2, . . . , vn.

Symplectic permutationWe have a vector of size 2n. We want to permute the first halfwith the second half of the vector with sign change:

(x1, x2, . . . , xn , xn+1, . . . , x2n) 7→ (−xn+1,−xn+2, . . . ,−x2n , x1, . . . , xn)

def symp(v):n = len(v) // 2 # use the integer division //return hstack ([-v[-n:], v[:n]])


Vectorized Functions

Note that not all functions may be applied on vectors. For instancethis one:def const(x):

return 1

We will see later how to automatically vectorize a function so thatit works componentwise on vectors.


Matrices


Matrices as Lists of Lists

DefinitionMatrices are represented by arrays of lists of rows, which are listsas well.

# the identity matrix in 2Did = array([[1., 0.], [0., 1.]])# Python allows this:id = array([[1., 0.],

[0., 1.]])# which is more readable


Accessing Matrix Entries

Matrix coefficients are accessed with two indices:M = array([[1., 2.],[3.,4.]])M[0,0] # first row , first column : 1.M[-1,0] # last row , first column : 3.


Creating Matrices

Some convenient methods to create matrices are:eye eye(n) is the identity matrix of size n

zeros zeros([n,m]) fills an n ×m matrix with zerosrand rand(n,m) is the same with random values

empty empty([n,m]) same with “empty” values


Shape

The shape of a matrix is the tuple of its dimensions. The shape ofan n ×m matrix is (n,m). It is given by the method shape:M = eye(3)M.shape # (3, 3)

V = array([1., 2., 1., 4.])V.shape # (4,) <- tuple with one element

Tip:zeros(A.shape) returns a matrix of the same shape as Acontaining only zeros.rand(*A.shape) does the same but with random values

Recall the difference between the argumentsA.shape and * A.shape


Transpose

The transpose of a matrix Aĳ is a matrix B such that

Bĳ = Aji

By transposing a matrix you switch the two shape elements.A = ...A.shape # 3,4

B = A.T # A transposeB.shape # 4,3

Note: B is just a “view” of A. Altering B changes A.


Matrix vector multiplication

The mathematical concept of reduction:∑j

aĳxj

is translated in Python in the function dot:angle = pi/3M = array([[cos(angle), -sin(angle)],

[sin(angle), cos(angle)]])V = array([1., 0.])Y = dot(M, V) # the product M.V

Elementwise vs. matrix multiplicationThe multiplication operator * is always elementwise. It hasnothing to do with the dot operation. A * V is a legal operationwhich will be explained later on.


Dot product

vector vector s =∑

i xiyi s = dot(x,y)

matrix vector yi =∑

j Aĳxj y = dot(A,x)

matrix matrix Cĳ =∑

k AikBkj C = dot(A,B)

vector matrix yj =∑

i xiAĳ y = dot(x,A)


unit4: functions numericalanalysis,lunduniversity 2015 · computationalmathematicswithpython unit4:...

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