Everybody in this country should learn how to program a computer... because it teaches you how to think. Steve Jobs, 1955-2011.
A physicist, a biologist and a mathematician were at a cafe when across the street two people entered a house. Moments later three people came out. The physicist said, "Hmm, that must be a measurement error." The biologist wondered, "It must be reproduction!" And the mathematician said, "If someone goes into the house, it will be empty again."
$$ y(t) = v_0t- \frac{1}{2}gt^2 $$
where
Task: given \( v_0 \), \( g \) and \( t \), compute \( y \).
A sequence of instructions to the computer, written in a programming language, somewhat like English, but very much simpler - and very much stricter.
This course teaches the Python language.
Evaluate \( y(t) = v_0t- \frac{1}{2}gt^2 \) for \( v_0=5 \), \( g=9.81 \) and \( t=0.6 \):
$$ y = 5\cdot 0.6 - \frac{1}{2}\cdot 9.81 \cdot 0.6^2$$
The complete Python program:
print 5*0.6 - 0.5*9.81*0.6**2
Step 1. Write the program in a text editor, here the line
print 5*0.6 - 0.5*9.81*0.6**2
Step 2.
Save the program to a file (say) ball1.py.
(.py denotes Python.)
Step 3. Move to a terminal window and go to the folder containing the program file.
Step 4. Run the program:
Terminal> python ball1.py
The program prints out 1.2342 in the terminal window.
You cannot calculate this integral by hand:
$$ \int_{-\infty}^1 e^{-x^2}dx\tp$$
A little program can compute this and "all" other integrals:
from numpy import *
def integrate(f, a, b, n=100):
"""
Integrate f from a to b,
using the Trapezoidal rule with n intervals.
"""
x = linspace(a, b, n+1) # Coordinates of the intervals
h = x[1] - x[0] # Interval spacing
I = h*(sum(f(x)) - 0.5*(f(a) + f(b)))
return I
# Define my special integrand
def my_function(x):
return exp(-x**2)
minus_infinity = -20 # Approximation of minus infinity
I = integrate(my_function, minus_infinity, 1, n=1000)
print 'Value of integral:', I
The program computes an approximation with error \( 10^{-12} \) within 0.1 s (\( n=10^6 \))!
Look at
print 5*0.6 - 0.5*9.81*0.6**2 write 5*0,6 - 0,5*9,81*0,6^2
Would you consider these two lines to be equal?
write is not a legal Python word in this context, comma has another
meaning than in math, and the hat is not exponentiation
People only become computer programmers if they're obsessive about details, crave power over machines, and can bear to be told day after day exactly how stupid they are. G. J. E. Rawlins
From mathematics you are used to variables, e.g.,
$$ v_0=5,\quad g=9.81,\quad t=0.6,\quad y = v_0t -\frac{1}{2}gt^2$$
We can use variables in a program too, and this makes the last program easier to read and understand:
v0 = 5
g = 9.81
t = 0.6
y = v0*t - 0.5*g*t**2
print y
This program spans several lines of text and use variables, otherwise the program performs the same calculations and gives the same output as the previous program
_ and the digits 0-9, but cannot start with a digita is different from A)
initial_velocity = 5
accel_of_gravity = 9.81
TIME = 0.6
VerticalPositionOfBall = initial_velocity*TIME - \
0.5*accel_of_gravity*TIME**2
print VerticalPositionOfBall
(Note: the backslash allows an instruction to be continued on the next line)
Good variable names make a program easier to understand!
Certain words have a special meaning in Python and cannot be used as variable names. These are: and,
as,
assert,
break,
class,
continue,
def,
del,
elif,
else,
except,
exec,
finally,
for,
from,
global,
if,
import,
in,
is,
lambda,
not,
or,
pass,
print,
raise,
return,
try,
with,
while, and
yield.
# program for computing the height of a ball
# in vertical motion
v0 = 5 # initial velocity
g = 9.81 # acceleration of gravity
t = 0.6 # time
y = v0*t - 0.5*g*t**2 # vertical position
print y
# on a line is a comment and ignored by Pythona = 5 # set a to 5
Normal rule: Python programs, including comments, can only contain characters from the English alphabet.
hilsen = 'Kjære Åsmund!' # er æ og Å lov i en streng?
print hilsen
leads to an error:
SyntaxError: Non-ASCII character ...
Remedy: put this special comment line as the first line in your program
# -*- coding: utf-8 -*-
Or stick to English everywhere in a program
Output from calculations often contain text and numbers, e.g.,
At t=0.6 s, y is 1.23 m.
We want to control the formatting of numbers: no of decimals, style: 0.6 vs 6E-01 or 6.0e-01. So-called printf formatting is useful for this purpose:
t = 0.6; y = 1.2342
print 'At t=%g s, y is %.2f m.' % (t, y)
The printf format has "slots" where the variables listed at the end are put: %g \( \leftarrow \) t, %.2f \( \leftarrow \) y
%g most compact formatting of a real number
%f decimal notation (-34.674)
%10.3f decimal notation, 3 decimals, field width 10
%.3f decimal notation, 3 decimals, minimum width
%e or %E scientific notation (1.42e-02 or 1.42E-02)
%9.2e scientific notation, 2 decimals, field width 9
%d integer
%5d integer in a field of width 5 characters
%s string (text)
%-20s string, field width 20, left-adjusted
%% the percentage sign % itself
(See the the book for more explanation and overview)
Triple-quoted strings (""") can be used for multi-line output, and here we combine such a string with printf formatting:
v0 = 5
g = 9.81
t = 0.6
y = v0*t - 0.5*g*t**2
print """
At t=%f s, a ball with
initial velocity v0=%.3E m/s
is located at the height %.2f m.
""" % (t, v0, y)
Running the program:
Terminal> python ball_print2.py
At t=0.600000 s, a ball with
initial velocity v0=5.000E+00 m/s
is located at the height 1.23 m.
Computer science meaning of terms is often different from the human language meaning
a = 1 # 1st statement (assignment statement)
b = 2 # 2nd statement (assignment statement)
c = a + b # 3rd statement (assignment statement)
print c # 4th statement (print statement)
Normal rule: one statement per line, but multiple statements per line is possible with a semicolon in between the statements:
a = 1; b = 2; c = a + b; print c
myvar = 10
myvar = 3*myvar # = 30
Programs must have correct syntax, i.e., correct use of the computer language grammar rules, and no misprints!
myvar = 5.2
prinnt Myvar
prinnt Myvar
^
SyntaxError: invalid syntax
Only the first encountered error is reported and the program is stopped (correct the error and continue with next error)
Programming demands significantly higher standard of accuracy. Things don't simply have to make sense to another human being, they must make sense to a computer. Donald Knuth, computer scientist, 1938-
Blanks may or may not be important in Python programs. These statements are equivalent (blanks do not matter):
v0=3
v0 = 3
v0= 3
v0 = 3
Here blanks do matter:
counter = 1
while counter <= 4:
counter = counter + 1 # correct (4 leading blanks)
while counter <= 4:
counter = counter + 1 # invalid syntax
(more about this in Ch. 2)
v0 = 3; g = 9.81; t = 0.6
position = v0*t - 0.5*g*t*t
velocity = v0 - g*t
print 'position:', position, 'velocity:', velocity
v0, g, and tposition and velocity
mkdir folder; cd folder; lsmkdir folder; cd folder; dir
Given \( C \) as a temperature in Celsius degrees, compute the corresponding Fahrenheit degrees \( F \):
$$ F = \frac{9}{5}C + 32 $$
Program:
C = 21
F = (9/5)*C + 32
print F
Execution:
Terminal> python c2f_v1.py
53
9/5 times 21 plus 32 is 69.8, not 53.
9.0 or 9. is float9 is an integer9.0/5 yields 1.8, 9/5. yields 1.8, 9/5 yields 1Corrected program (with correct output 69.8):
C = 21
F = (9.0/5)*C + 32
print F
Variables refer to objects:
a = 5 # a refers to an integer (int) object
b = 9 # b refers to an integer (int) object
c = 9.0 # c refers to a real number (float) object
d = b/a # d refers to an int/int => int object
e = c/a # e refers to float/int => float object
s = 'b/a=%g' % (b/a) # s is a string/text (str) object
We can convert between object types:
a = 3 # a is int
b = float(a) # b is float 3.0
c = 3.9 # c is float
d = int(c) # d is int 3
d = round(c) # d is float 4.0
d = int(round(c)) # d is int 4
d = str(c) # d is str '3.9'
e = '-4.2' # e is str
f = float(e) # f is float -4.2
5/9 + 2*a**4/2r1 = 5/9 (=0)r2 = a**4r3 = 2*r2r4 = r3/2r5 = r1 + r4(5/9) + (2*(a**4))/2math module
math module
Compute \( \sqrt{2} \) using the sqrt function in the math module:
import math
r = math.sqrt(2)
# or
from math import sqrt
r = sqrt(2)
# or
from math import * # import everything in math
r = sqrt(2)
math
Evaluate
$$ Q = \sin x\cos x + 4\ln x $$
for \( x=1.2 \).
from math import sin, cos, log
x = 1.2
Q = sin(x)*cos(x) + 4*log(x) # log is ln (base e)
print Q
Let us compute \( 1/49\cdot 49 \) and \( 1/51\cdot 51 \):
v1 = 1/49.0*49
v2 = 1/51.0*51
print '%.16f %.16f' % (v1, v2)
Output with 16 decimals becomes
0.9999999999999999 1.0000000000000000
What is printed?
a = 1; b = 2;
computed = a + b
expected = 3
correct = computed == expected
print 'Correct:', correct
Change to a = 0.1 and b = 0.2 (expected = 0.3). What is now printed?
Why? How can the comparison be performed?
>>> a = 0.1; b = 0.2; expected = 0.3
>>> a + b == expected
False
>>> print '%.17f\n%.17f\n%.17f\n%.17f' % (0.1, 0.2, 0.1 + 0.2, 0.3)
0.10000000000000001
0.20000000000000001
0.30000000000000004
0.29999999999999999
>>> a = 0.1; b = 0.2; expected = 0.3
>>> computed = a + b
>>> diff = abs(expected - computed)
>>> tol = 1E-15
>>> diff < tol
The \( \sinh x \) function is defined as
$$\sinh (x) = \frac{1}{2}\left(e^{x} - e^{-x}\right) $$
We can evaluate this function in three ways:
math.sinhmath.expmath.e
from math import sinh, exp, e, pi
x = 2*pi
r1 = sinh(x)
r2 = 0.5*(exp(x) - exp(-x))
r3 = 0.5*(e**x - e**(-x))
print '%.16f %.16f %.16f' % (r1,r2,r3)
Output: r1 is \( 267.744894041016\underline{4369} \), r2 is
\( 267.744894041016\underline{4369} \), r3 is
\( 267.744894041016\underline{3232} \) (!)
python, ipython, or idle in the terminal window>>>
(IPython has a different prompt)
Terminal> python
Python 2.7.6 (r25:409, Feb 27 2014, 19:35:40)
...
>>> C = 41
>>> F = (9.0/5)*C + 32
>>> print F
105.8
>>> F
105.8
Previous commands can be recalled and edited
2 + 3j in Python
>>> a = -2
>>> b = 0.5
>>> s = complex(a, b) # make complex from variables
>>> s
(-2+0.5j)
>>> s*w # complex*complex
(-10.5-3.75j)
>>> s/w # complex/complex
(-0.25641025641025639+0.28205128205128205j)
>>> s.real
-2.0
>>> s.imag
0.5
See the book for additional info
>>> from sympy import *
>>> t, v0, g = symbols('t v0 g')
>>> y = v0*t - Rational(1,2)*g*t**2
>>> dydt = diff(y, t) # 1st derivative
>>> dydt
-g*t + v0
>>> print 'acceleration:', diff(y, t, t) # 2nd derivative
acceleration: -g
>>> y2 = integrate(dydt, t)
>>> y2
-g*t**2/2 + t*v0
>>> y = v0*t - Rational(1,2)*g*t**2
>>> roots = solve(y, t) # solve y=0 wrt t
>>> roots
[0, 2*v0/g]
>>> x, y = symbols('x y')
>>> f = -sin(x)*sin(y) + cos(x)*cos(y)
>>> simplify(f)
cos(x + y)
>>> expand(sin(x+y), trig=True) # requires a trigonometric hint
sin(x)*cos(y) + sin(y)*cos(x)
int, float, str
Mathematical functions like \( \sin x \) and \( \ln x \) must be imported from the math module:
from math import sin, log
x = 5
r = sin(3*log(10*x))
Use printf syntax for full control of output of text and numbers!
Important terms: object, variable, algorithm, statement, assignment, implementation, verification, debugging
- You think you know when you can learn,
are more sure when you can write,
even more when you can teach,
but certain when you can program- Within a computer, natural language is unnatural
- To understand a program you must become both the machine and the program
Alan Perlis, computer scientist, 1922-1990.
We throw a ball with velocity \( v_0 \), at an angle \( \theta \) with the horizontal, from the point \( (x=0,y=y_0) \). The trajectory of the ball is a parabola (we neglect air resistance):
$$ y = x\tan\theta - \frac{1}{2v_0}\frac{gx^2}{\cos^2\theta} + y_0 $$
math
Program:
g = 9.81 # m/s**2
v0 = 15 # km/h
theta = 60 # degrees
x = 0.5 # m
y0 = 1 # m
print """v0 = %.1f km/h
theta = %d degrees
y0 = %.1f m
x = %.1f m""" % (v0, theta, y0, x)
# convert v0 to m/s and theta to radians:
v0 = v0/3.6
from math import pi, tan, cos
theta = theta*pi/180
y = x*tan(theta) - 1/(2*v0)*g*x**2/((cos(theta))**2) + y0
print 'y = %.1f m' % y