Inheritance

The chapter title Object-oriented programming (OO) may mean two different things

Programming with classes (better: object-based programming)

Programming with class hierarchies (class families)

New concept: collect classes in families (hierarchies)

What is a class hierarchy?

A family of closely related classes

A key concept is inheritance: child classes can inherit attributes and methods from parent class(es) - this saves much typing and code duplication

As usual, we shall learn through examples!

OO is a Norwegian invention by Ole-Johan Dahl and Kristen Nygaard in the 1960s - one of the most important inventions in computer science, because OO is used in all big computer systems today!

Warning: OO is difficult and takes time to master

Let ideas mature with time

Study many examples

OO is less important in Python than in C++, Java and C#, so the benefits of OO are less obvious in Python

Our examples here on OO employ numerical methods for $\int_a^b f(x)dx$ , $f'(x)$ , $u'=f(u,t)$ - make sure you understand the simplest of these numerical methods before you study the combination of OO and numerics

Our goal: write general, reusable modules with lots of methods for numerical computing of $\int_a^b f(x)dx$ , $f'(x)$ , $u'=f(u,t)$

A class for straight lines

Problem:

Make a class for evaluating lines $y=c_0 + c_1x$ .

Code:

class Line:
    def __init__(self, c0, c1):
        self.c0, self.c1 = c0, c1

    def __call__(self, x):
        return self.c0 + self.c1*x

    def table(self, L, R, n):
        """Return a table with n points for L <= x <= R."""
        s = ''
        for x in linspace(L, R, n):
            y = self(x)
            s += '%12g %12g\n' % (x, y)
        return s

A class for parabolas

Problem:

Make a class for evaluating parabolas $y=c_0 + c_1x + c_2x^2$ .

Code:

class Parabola:
    def __init__(self, c0, c1, c2):
        self.c0, self.c1, self.c2 = c0, c1, c2

    def __call__(self, x):
        return self.c2*x**2 + self.c1*x + self.c0

    def table(self, L, R, n):
        """Return a table with n points for L <= x <= R."""
        s = ''
        for x in linspace(L, R, n):
            y = self(x)
            s += '%12g %12g\n' % (x, y)
        return s

Observation:

This is almost the same code as class Line, except for the things with c2

Class Parabola as a subclass of Line; principles

Parabola code = Line code + a little extra with the $c_2$ term

Can we utilize class Line code in class Parabola?

This is what inheritance is about!

Writing

class Parabola(Line):
    pass

makes Parabola inherit all methods and attributes from Line, so Parabola has attributes c0 and c1 and three methods

Line is a superclass, Parabola is a subclass
(parent class, base class; child class, derived class)

Class Parabola must add code to Line's constructor (an extra c2 attribute), __call__ (an extra term), but table can be used unaltered

The principle is to reuse as much code in Line as possible and avoid duplicating code

Class Parabola as a subclass of Line; code

A subclass method can call a superclass method in this way:

superclass_name.method(self, arg1, arg2, ...)

Class Parabola as a subclass of Line:

class Parabola(Line):
    def __init__(self, c0, c1, c2):
        Line.__init__(self, c0, c1)  # Line stores c0, c1
        self.c2 = c2

    def __call__(self, x):
        return Line.__call__(self, x) + self.c2*x**2

What is gained?

Class Parabola just adds code to the already existing code in class Line - no duplication of storing c0 and c1, and computing $c_0+c_1x$

Class Parabola also has a table method - it is inherited

__init__ and __call__ are overridden or redefined in the subclass

Class Parabola as a subclass of Line; demo

p = Parabola(1, -2, 2)
p1 = p(2.5)
print p1
print p.table(0, 1, 3)

Output:

8.5
           0            1
         0.5          0.5
           1            1

Exercise 1: Point out the program flow

We can check class type and class relations with `isinstance(obj, type)` and `issubclass(subclassname, superclassname)`

>>> from Line_Parabola import Line, Parabola
>>> l = Line(-1, 1)
>>> isinstance(l, Line)
True
>>> isinstance(l, Parabola)
False

>>> p = Parabola(-1, 0, 10)
>>> isinstance(p, Parabola)
True
>>> isinstance(p, Line)
True

>>> issubclass(Parabola, Line)
True
>>> issubclass(Line, Parabola)
False

>>> p.__class__ == Parabola
True
>>> p.__class__.__name__   # string version of the class name
'Parabola'

Line as a subclass of Parabola

Subclasses are often special cases of a superclass

A line $c_0+c_1x$ is a special case of a parabola $c_0+c_1x+c_2x^2$

Can Line be a subclass of Parabola?

No problem - this is up to the programmer's choice

Many will prefer this relation between a line and a parabola

Code when Line is a subclass of Parabola

class Parabola:
    def __init__(self, c0, c1, c2):
        self.c0, self.c1, self.c2 = c0, c1, c2

    def __call__(self, x):
        return self.c2*x**2 + self.c1*x + self.c0

    def table(self, L, R, n):
        """Return a table with n points for L <= x <= R."""
        s = ''
        for x in linspace(L, R, n):
            y = self(x)
            s += '%12g %12g\n' % (x, y)
        return s

class Line(Parabola):
    def __init__(self, c0, c1):
        Parabola.__init__(self, c0, c1, 0)

Note: __call__ and table can be reused in class Line!

Recall the class for numerical differentiation from Ch. 7

$f'(x) \approx {f(x+h)-f(x)\over h}$

class Derivative:
    def __init__(self, f, h=1E-5):
        self.f = f
        self.h = float(h)

    def __call__(self, x):
        f, h = self.f, self.h      # make short forms
        return (f(x+h) - f(x))/h

def f(x):
    return exp(-x)*cos(tanh(x))

from math import exp, cos, tanh
dfdx = Derivative(f)
print dfdx(2.0)

There are numerous formulas numerical differentiation

$\begin{align*} f'(x) &= \frac{f(x+h)-f(x)}{h} + {\cal O}(h)\\ f'(x) &= \frac{f(x)-f(x-h)}{h} + {\cal O}(h)\\ f'(x) &= \frac{f(x+h)-f(x-h)}{2h} + {\cal O}(h^2)\\ f'(x) &= \frac{4}{3}\frac{f(x+h)-f(x-h)}{2h} -\frac{1}{3}\frac{f(x+2h) - f(x-2h)}{4h} + {\cal O}(h^4)\\ f'(x) &= \frac{3}{2}\frac{f(x+h)-f(x-h)}{2h} -\frac{3}{5}\frac{f(x+2h) - f(x-2h)}{4h} + \nonumber\\ & \frac{1}{10}\frac{f(x+3h) - f(x-3h)}{6h} + {\cal O}(h^6)\\ f'(x) &= \frac{1}{h}\left( -\frac{1}{6}f(x+2h) + f(x+h) - \frac{1}{2}f(x) - \frac{1}{3}f(x-h)\right) + {\cal O}(h^3) \end{align*}$

How can we make a module that offers all these formulas?

It's easy:

class Forward1:
    def __init__(self, f, h=1E-5):
        self.f = f
        self.h = float(h)

    def __call__(self, x):
        f, h = self.f, self.h
        return (f(x+h) - f(x))/h

class Backward1:
    def __init__(self, f, h=1E-5):
        self.f = f
        self.h = float(h)

    def __call__(self, x):
        f, h = self.f, self.h
        return (f(x) - f(x-h))/h

class Central2:
    # same constructor
    # put relevant formula in __call__

What is the problem with this type of code?

All the constructors are identical so we duplicate a lot of code.

A general OO idea: place code common to many classes in a superclass and inherit that code

Here: inhert constructor from superclass,
let subclasses for different differentiation formulas implement their version of __call__

Class hierarchy for numerical differentiation

Superclass:

class Diff:
    def __init__(self, f, h=1E-5):
        self.f = f
        self.h = float(h)

Subclass for simple 1st-order forward formula:

class Forward1(Diff):
    def __call__(self, x):
        f, h = self.f, self.h
        return (f(x+h) - f(x))/h

Subclass for 4-th order central formula:

class Central4(Diff):
    def __call__(self, x):
        f, h = self.f, self.h
        return (4./3)*(f(x+h)   - f(x-h))  /(2*h) - \
               (1./3)*(f(x+2*h) - f(x-2*h))/(4*h)

Use of the differentiation classes

Interactive example: $f(x)=\sin x$ , compute $f'(x)$ for $x=\pi$

>>> from Diff import *
>>> from math import sin
>>> mycos = Central4(sin)
>>> # compute sin'(pi):
>>> mycos(pi)
-1.000000082740371

Central4(sin) calls inherited constructor in superclass, while mycos(pi) calls __call__ in the subclass Central4

Exercise 2: Point out the program flow

A flexible main program for numerical differentiation

Suppose we want to differentiate function expressions from the command line:

Terminal> python df.py 'exp(sin(x))' Central 2 3.1
-1.04155573055

Terminal> python df.py 'f(x)' difftype difforder x
f'(x)

With eval and the Diff class hierarchy this main program can be realized in a few lines (many lines in C# and Java!):

import sys
from Diff import *
from math import *
from scitools.StringFunction import StringFunction

f = StringFunction(sys.argv[1])
difftype = sys.argv[2]
difforder = sys.argv[3]
classname = difftype + difforder
df = eval(classname + '(f)')
x = float(sys.argv[4])
print df(x)

Investigating numerical approximation errors

We can empirically investigate the accuracy of our family of 6 numerical differentiation formulas

Sample function: $f(x)=\exp{(-10x)}$

See the book for a little program that computes the errors:

.   h        Forward1        Central2        Central4
6.25E-02 -2.56418286E+00  6.63876231E-01 -5.32825724E-02
3.12E-02 -1.41170013E+00  1.63556996E-01 -3.21608292E-03
1.56E-02 -7.42100948E-01  4.07398036E-02 -1.99260429E-04
7.81E-03 -3.80648092E-01  1.01756309E-02 -1.24266603E-05
3.91E-03 -1.92794011E-01  2.54332554E-03 -7.76243120E-07
1.95E-03 -9.70235594E-02  6.35795004E-04 -4.85085874E-08

Observations:

Halving $h$ from row to row reduces the errors by a factor of 2, 4 and 16, i.e, the errors go like $h$ , $h^2$ , and $h^4$

Central4 has really superior accuracy compared with Forward1

Alternative implementations (in the book)

Pure Python functions
downside: more arguments to transfer, cannot apply formulas twice to get 2nd-order derivatives etc.

Functional programming
gives the same flexibility as the OO solution

One class and one common math formula
applies math notation instead of programming techniques to generalize code

These techniques are beyond scope in the course, but place OO into a bigger perspective. Might better clarify what OO is - for some.

Formulas for numerical integration

There are numerous formulas for numerical integration and all of them can be put into a common notation:

$\int_a^b f(x)dx \approx \sum_{i=0}^{n-1} w_i f(x_i)$

$w_i$ : weights, $x_i$ : points (specific to a certain formula)

The Trapezoidal rule has $h=(b-a)/(n-1)$ and

$x_i = a+ih, \quad w_0=w_{n-1}={h\over2},\ w_i=h\ (i\neq 0,n-1)$

The Midpoint rule has $h=(b-a)/n$ and

$x_i = a + {h\over 2} + ih,\quad w_i=h$

More formulas

Simpson's rule has

$\begin{align*} x_i &= a+ih,\quad h={b-a\over n-1}\\ w_0 &=w_{n-1}={h\over6}\\ w_i &= {h\over3}\hbox{ for }i\hbox{ even},\quad w_i={2h\over3}\hbox{ for }i\hbox{ odd} \end{align*}$

Other rules have more complicated formulas for $w_i$ and $x_i$

Why should these formulas be implemented in a class hierarchy?

A numerical integration formula can be implemented as a class: $a$ , $b$ and $n$ are attributes and an integrate method evaluates the formula

All such classes are quite similar: the evaluation of $\sum_jw_jf(x_j)$ is the same, only the definition of the points and weights differ among the classes

Recall: code duplication is a bad thing!

The general OO idea: place code common to many classes in a superclass and inherit that code

Here we put $\sum_jw_jf(x_j)$ in a superclass (method integrate)

Subclasses extend the superclass with code specific to a math formula, i.e., $w_i$ and $x_i$ in a class method construct_rule

The superclass for integration

class Integrator:
    def __init__(self, a, b, n):
        self.a, self.b, self.n = a, b, n
        self.points, self.weights = self.construct_method()

    def construct_method(self):
        raise NotImplementedError('no rule in class %s' % \
                                  self.__class__.__name__)

    def integrate(self, f):
        s = 0
        for i in range(len(self.weights)):
            s += self.weights[i]*f(self.points[i])
        return s

    def vectorized_integrate(self, f):
        # f must be vectorized for this to work
        return dot(self.weights, f(self.points))

A subclass: the Trapezoidal rule

class Trapezoidal(Integrator):
    def construct_method(self):
        h = (self.b - self.a)/float(self.n - 1)
        x = linspace(self.a, self.b, self.n)
        w = zeros(len(x))
        w[1:-1] += h
        w[0] = h/2;  w[-1] = h/2
        return x, w

Another subclass: Simpson's rule

Simpson's rule is more tricky to implement because of different formulas for odd and even points

Don't bother with the details of $w_i$ and $x_i$ in Simpson's rule now - focus on the class design!

class Simpson(Integrator):

    def construct_method(self):
        if self.n % 2 != 1:
            print 'n=%d must be odd, 1 is added' % self.n
            self.n += 1

        <code for computing x and w>
        return x, w

About the program flow

Let us integrate $\int_0^2 x^2dx$ using 101 points:

def f(x):
    return x*x

method = Simpson(0, 2, 101)
print method.integrate(f)

Important:

method = Simpson(...): this invokes the superclass constructor, which calls construct_method in class Simpson

method.integrate(f) invokes the inherited integrate method, defined in class Integrator

Exercise 3: Point out the program flow

Applications of the family of integration classes

We can empirically test out the accuracy of different integration methods Midpoint, Trapezoidal, Simpson, GaussLegendre2, ... applied to, e.g.,

$\int\limits_0^1 \left(1 + {1\over m}\right)t^{1\over m} dt= 1$

This integral is "difficult" numerically for $m>1$ .

Key problem: the error in numerical integration formulas is of the form $Cn^{-r}$ , mathematical theory can predict $r$ (the "order"), but we can estimate $r$ empirically too

See the book for computational details

Here we focus on the conclusions

Convergence rates for $m < 1$ (easy case)

Simpson and Gauss-Legendre reduce the error faster than Midpoint and Trapezoidal (plot has ln(error) versus $\ln n$ )

Convergence rates for $m>1$ (problematic case)

Simpson and Gauss-Legendre, which are theoretically "smarter" than Midpoint and Trapezoidal do not show superior behavior!

Summary of object-orientation principles

A subclass inherits everything from the superclass

When to use a subclass/superclass?

if code common to several classes can be placed in a superclass

if the problem has a natural child-parent concept

The program flow jumps between super- and sub-classes

It takes time to master when and how to use OO

Study examples!

Recall the class hierarchy for differentiation

Mathematical principles:

Collection of difference formulas for $f'(x)$ . For example,

$f'(x) \approx {f(x+h)-f(x-h)\over 2h}$

Superclass Diff contains common code (constructor), subclasses implement various difference formulas.

Implementation example (superclass and one subclass).

class Diff:
    def __init__(self, f, h=1E-5):
        self.f = f
        self.h = float(h)

class Central2(Diff):
    def __call__(self, x):
        f, h = self.f, self.h
        return (f(x+h) - f(x-h))/(2*h)

Recall the class hierarchy for integration (1)