CS计算机代考程序代写 python algorithm Digression: Scoring Matrices

What happens if you type: int(“2.87”)?

L6 Maths – 5

REVISION: Definite Loops

for loops alter the flow of program
execution, so they are referred to as
control structures.
>>> for odd in [1, 3, 5, 7]:

print(odd*odd)
1
9
25
49

Loop variable odd first has the value 1, then 3, then 5
and finally 7

L6 Maths – 6

REVISION: module and function

• Module:
– allows you to logically organize your Python code
– grouping related code that makes the code easier to

understand and use
– Simply, a file consisting of Python code which can

define functions, classes, variables and may also
include runnable code

• Function:
– a block of organized, reusable code that is used to

perform a single, related action.
– provide better modularity for your application and a

high degree of code reusing

L6 Maths – 7

Using the Math Library

• Besides (**, *, /, %, //, +, -, abs), we have lots of other
math functions available in a math library.
– ** is exponentiation, e.g. 2**3 == 8

• A library is a module with some useful
definitions/functions.

• In this course, importing library modules is not
encouraged as objective of the course is to focus on
programming skills.

L6 Maths – 8

Using the Math Library

• The formula for computing the roots of a quadratic
equation of the form: ax2 + bx + c is

• The only part of this that we don’t know how to do
yet is the square root function, but that is in the
math library

L6 Maths – 9

Using the Math Library

• To use a library, we need to make sure this line is in
our program:

import math # not maths

• Importing a library makes whatever functions are
defined within it available to the program (from that
point onwards).

L6 Maths – 10

Using the Math Library

• To access the sqrt library routine, we need to access it
as: math.sqrt(x)

• Using this dot notation tells Python to use the sqrt
function found in the math library module.

• To calculate the root, you can do
discRoot = math.sqrt(b*b – 4*a*c)

L6 Maths – 11

Using the Math Library

• We can also import math module as
import math as m # now math is called by m

m.sqrt(9)

• If we do not want to import all functions of module
# Import sqrt function only

from math import sqrt

sqrt(9) # No need a reference

L6 Maths – 12

Algorithm: Roots of quadratic equation

• Print an introduction
• Input the coefficients (a, b and c)
• Calculate roots:

• Output the roots

L6 Maths – 13

Using the Math Library
# quadratic.py
# A program that computes the real roots of a quadratic equation.
# Illustrates use of the math library.
# Author: Unit Coordinator

import math # Makes the math library available.

def main():
print(“This program finds the real solutions to a quadratic equation”)
print()

a = float(input(“Please enter coefficient a: “))
b = float(input(“Please enter coefficient b: “))
c = float(input(“Please enter coefficient c: “))

discRoot = math.sqrt(b * b – 4 * a * c)
root1 = (-b + discRoot) / (2 * a)
root2 = (-b – discRoot) / (2 * a)

print()
print(“The solutions are:”, root1, root2 )
return # optional to write it if function is returning nothing

L6 Maths – 14

Using the Math Library

Running the program

>>> main()

This program finds the real solutions to a quadratic

Please enter coefficient a: 3

Please enter coefficient b: 4

Please enter coefficient c: -1

The solutions are: 0.215250437022 -1.54858377035

L6 Maths – 15

Using the Math Library
Try again with 1, 2, 3

>>> main()

This program finds the real solutions to a quadratic

Please enter coefficient a: 1

Please enter coefficient b: 2

Please enter coefficient c: 3

Traceback (most recent call last):

File “quadratic_roots.py”, line 21, in main()

File “quadratic_roots.py”, line 15, in main

discRoot = math.sqrt(b * b – 4 * a * c)

ValueError: math domain error

>>>
Runtime error

L6 Maths – 16

Math Library

• If a = 1, b = 2, c = 3, then we are trying to take
the square root of a negative number!

• Using the sqrt function is more efficient than
using **
– How could you use ** to calculate a square root?

L6 Maths – 17

Math Library

Python Mathematics English
pi π An approximation of pi
e e An approximation of e

sqrt(x) √ The square root of x
sin(x) sin x The sine of x
cos(x) cos x The cosine of x
tan(x) tan x The tangent of x
asin(x) arcsin x The inverse of sine x
acos(x) arccos x The inverse of cosine x
atan(x) arctan x The inverse of tangent x

Don’t forget to put math in front (depending on the method of
importing module, e.g. math.sqrt(x), math.pi

L6 Maths – 18

Math Library

Python Mathematics English
log(x) ln x The natural (base e) logarithm of x

log10(x) The common (base 10) logarithm of x
exp(x) The exponential of x
ceil(x) The smallest whole number >= x
floor(x) The largest whole number <= x 10log x xe x   x   L6 Maths - 19 Accumulating Results: Factorial • Say you are waiting in a line with five other people. How many ways are there to arrange the six people? • 720 -- which is the factorial of 6 (abbreviated 6!) • Factorial is defined as: n! = n(n-1)(n-2)…(1) • So, 6! = 6*5*4*3*2*1 = 720 L6 Maths - 20 Accumulating Results: Factorial • How we could write a program to do this? • Input number to take factorial of, n Compute factorial of n, fact Output fact L6 Maths - 21 Accumulating Results: Factorial • How did we calculate 6!? • 6*5 = 30 • Take that 30, and 30 * 4 = 120 • Take that 120, and 120 * 3 = 360 • Take that 360, and 360 * 2 = 720 • Take that 720, and 720 * 1 = 720 L6 Maths - 22 Algorithm: Factorial The general form of an accumulator algorithm looks like this: – Initialize the accumulator variable – Perform computation (e.g. in case of factorial multiply by the next smaller number) – Update accumulator variable – Loop until final result is reached (e.g. in case of factorial the next smaller number is 1) – Output accumulator variable Computational Thinking: Pattern recognition ‒ pattern can be used repeatedly for range of problems L6 Maths - 23 Accumulating Results: Factorial • It looks like we’ll need a loop! factorial = 1 for fact in [6, 5, 4, 3, 2, 1]: factorial = fact * factorial • Let’s trace through it to verify that this works! L6 Maths - 24 Accumulating Results: Factorial • Why did we need to initialize factorial to 1? • There are a couple reasons… – Each time through the loop, the previous value of factorial is used to calculate the next value of factorial. By doing the initialization, you know factorial will have a value the first time through. – If you use factorial without assigning it a value, what does Python do? L6 Maths - 25 Improving Factorial • What does range(n) return? 0, 1, 2, 3, …, n-1 • range has another optional parameter: – range(start, n) returns start, start + 1, …, n-1 – E.g. range(1, 11) returns: 1,2,3,4,5,6,7,8,9,10 • But wait! There’s more! range(start, n, step) returns: start, start+step, …stopping before n • list() to make a list

L6 Maths – 26

Range()
• Let’s try some examples!

>>> list(range(10))

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

>>> list(range(5,10))

[5, 6, 7, 8, 9]

>>> list(range(5,10,2))

[5, 7, 9]

L6 Maths – 27

range() Forwards or Backwards

• We can do the range for our loop a couple of different
ways.

– We can count up from 2 to n:
range(2, n+1)
(Why did we have to use n+1?)

– We can count down from n to 2:
range(n, 1, -1)

L6 Maths – 28

Back at the Factorial Program
Our completed factorial program:
# Program to compute the factorial of a number
# Illustrates for loop with an accumulator
# Author: Unit coordinator

def factorial_find():
n = int(input(“Please enter an integer: “))
factorial = 1
for fact in range(n,1,-1):

factorial = fact * factorial
print(“The factorial of”, n, “is”, factorial)
return

factorial_find()

L6 Maths – 29

Lecture Summary

• We learned how to use the Python math library

• We discussed an example of Quadratic equation

• We discussed an example of accumulator: factorial

• We discussed range()function

Lecture 6�Math module
Objectives of this Lecture
REVISION: The Software Development Process
REVISION: Numerical data types
REVISION: Definite Loops
REVISION: module and function
Using the Math Library
Using the Math Library
Using the Math Library
Using the Math Library
Using the Math Library
Algorithm: Roots of quadratic equation
Using the Math Library
Using the Math Library
Using the Math Library
Math Library
Math Library
Math Library
Accumulating Results: Factorial
Accumulating Results: Factorial
Accumulating Results: Factorial
Algorithm: Factorial
Accumulating Results: Factorial
Accumulating Results: Factorial
Improving Factorial
Range()
range() Forwards or Backwards
Back at the Factorial Program
Lecture Summary