Date: Aug 15, 2015
Programming with classes (better: object-based programming)
Programming with class hierarchies (class families)
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!
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 $\int_a^b f(x)dx$, $f'(x)$, $u'=f(u,t)$
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
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
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
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
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
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
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
p = Parabola(1, -2, 2)
print p(2.5)
isinstance(obj, type)
and issubclass(subclassname, superclassname)
¶from Line_Parabola import Line, Parabola
l = Line(-1, 1)
isinstance(l, Line)
isinstance(l, Parabola)
p = Parabola(-1, 0, 10)
isinstance(p, Parabola)
isinstance(p, Line)
issubclass(Parabola, Line)
issubclass(Line, Parabola)
p.__class__ == Parabola
p.__class__.__name__ # string version of the class name
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
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
!
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)
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__
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__
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)
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)
Central4(sin)
calls inherited constructor in superclass, while mycos(pi)
calls __call__
in the subclass Central4
class Diff:
def __init__(self, f, h=1E-5):
self.f = f
self.h = float(h)
class Forward1(Diff):
def __call__(self, x):
f, h = self.f, self.h
return (f(x+h) - f(x))/h
dfdx = Diff(lambda x: x**2)
print dfdx(0.5)
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!):
%matplotlib inline
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)
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
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.
There are numerous formulas for numerical integration and all of them can be put into a common notation:
$w_i$: weights, $x_i$: points (specific to a certain formula)
The Trapezoidal rule has $h=(b-a)/(n-1)$ and
The Midpoint rule has $h=(b-a)/n$ and
Simpson's rule has
Other rules have more complicated formulas for $w_i$ and $x_i$
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
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))
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
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
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
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
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
def f(x):
return x*x
method = Trapezoidal(0, 2, 101)
print method.integrate(f)
We can empirically test out the accuracy of different integration methods Midpoint
, Trapezoidal
, Simpson
, GaussLegendre2
, ... applied to, e.g.,
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
Simpson and Gauss-Legendre reduce the error faster than Midpoint and Trapezoidal (plot has ln(error) versus $\ln n$)
Simpson and Gauss-Legendre, which are theoretically "smarter" than Midpoint and Trapezoidal do not show superior behavior!
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!
Mathematical principles:
Collection of difference formulas for $f'(x)$. For example,
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)
Mathematical principles:
General integration formula for numerical integration:
Superclass Integrator
contains common code (constructor, $\sum_j w_if(x_i)$), subclasses implement definition of $w_i$ and $x_i$.
Implementation example (superclass and one subclass):
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 integrate(self, f):
s = 0
for i in range(len(self.weights)):
s += self.weights[i]*f(self.points[i])
return s
class Trapezoidal(Integrator):
def construct_method(self):
x = linspace(self.a, self.b, self.n)
h = (self.b - self.a)/float(self.n - 1)
w = zeros(len(x)) + h
w[0] /= 2; w[-1] /= 2 # adjust end weights
return x, w
Write a table of $x\in [a,b]$ and $f(x)$ to file:
outfile = open(filename, 'w')
from numpy import linspace
for x in linspace(a, b, n):
outfile.write('%12g %12g\n' % (x, f(x)))
outfile.close()
We want flexible input:
Read a
, b
, n
, filename
and a formula for f
from...
the command line
interactive commands like a=0
, b=2
, filename=mydat.dat
questions and answers in the terminal window
a graphical user interface
a file of the form
a = 0
b = 2
filename = mydat.dat
from ReadInput import *
# define all input parameters as name-value pairs in a dict:
p = dict(formula='x+1', a=0, b=1, n=2, filename='tmp.dat')
# read from some input medium:
inp = ReadCommandLine(p)
# or
inp = PromptUser(p) # questions in the terminal window
# or
inp = ReadInputFile(p) # read file or interactive commands
# or
inp = GUI(p) # read from a GUI
# load input data into separate variables (alphabetic order)
a, b, filename, formula, n = inp.get_all()
# go!
A superclass ReadInput
stores the dict and provides methods for getting input into program variables (get
, get_all
)
Subclasses read from different input sources
ReadCommandLine
, PromptUser
, ReadInputFile
, GUI
See the book or ReadInput.py
for implementation details
For now the ideas and principles are more important than code details!