# Posts tagged ‘maths’

Finding the highest common factor is useful in mathematics and cryptography.

It would be computationally expensive to calculate all the common factors of two numbers and then compare to find the highest factor in common. Instead (as usual) there is a mathematical theory that can be used to speed up this process, in this case it’s the Euclidean algorithm.

The Euclidean algorithm subtracts the smaller number from the larger number, then using this new number and the smaller number repeats this process until the numbers are equal and hence the subtraction goes to zero. This is the highest common factor.

**Example:**

Find the highest common factor of 252 and 105

252-105=147

147-105=42

105-42=63

63-42=21

42-21=21

21-21=0

Below is the python code to preform this

def hcf(no1,no2): while no1!=no2: if no1>no2: no1-=no2 elif no2>no1: no2-=no1 return no1

This method is quite quick even and scales well for larger numbers. If you know of a faster method or can see any improvements to make then please let me know.

## Python: Factors of a number

Project Euler often requires the factors of a number to be found, here is the function I have written for that purpose.

def factors(n): fact=[1,n] check=2 rootn=sqrt(n) while check<rootn: if n%check==0: fact.append(check) fact.append(n/check) check+=1 if rootn==check: fact.append(check) fact.sort() return fact

As always any help in speeding up my function would be appreciated. I think it could be improved using a sieve method or by checking if its divisible by 2 first then using check+=2 starting at 3 to improve the speed of the loop by a factor of 2.

Below is a the commented version to help explain what’s going on if it’s not clear.

def factors(n): # 1 and n are automatically factors of n fact=[1,n] # starting at 2 as we have already dealt with 1 check=2 # calculate the square root of n and use this as the # limit when checking if a number is divisible as # factors above sqrt(n) will already be calculated as # the inverse of a lower factor IE. finding factors of # 100 only need go up to 10 (sqrt(100)=10) as factors # such as 25 can be found when 5 is found to be a # factor 100/5=25 rootn=sqrt(n) while check<rootn: if n%check==0: fact.append(check) fact.append(n/check) check+=1 # this line checks the sqrt of the number to see if # it is a factor putting it here prevents it appearing # twice in the above while loop and putting it outside # the loop should save some time. if rootn==check: fact.append(check) # return an array of factors sorted into numerial order. fact.sort() return fact

So a friend of mine introduced me to **Evolutionary Algorithms** a while back and I got some lecture notes passed onto me explaining the basics and a simple example in pseudo-code.

After a bit of work I’ve written up the queens example as an evolutionary algorithm in python. Here goes:

**The queens problem**

The following is the quote from wikipedia regarding the problem.

The eight queens puzzle is the problem of putting eight chess queens on an 8×8 chessboard such that none of them is able to capture any other using the standard chess queen’s moves. The queens must be placed in such a way that no two queens would be able to attack each other. Thus, a solution requires that no two queens share the same row, column, or diagonal.

The above image is one of the many solutions to the problem. Hint: if you are looking for many solutions once you have found one, all of its 4 rotations (if unique) are solutions too.

Check out the wikipedia article for more information

**Code**

This is the current code with medium amount of comments, I have also provided a heavily commented version for people with less python experience (scroll to bottom).

# -*- coding: utf-8 -*- ################################################################## ################################################################## # QUEENS EXAMPLE # - Program Type: Demonstration of Evolutionary alogorithms # - Authour: Matthew Rollings # - Date: 14/09/09 ################################################################## ################################################################## # Import randint from the random module from random import randint # Creates a array I just use for convenience board=[1,2,3,4,5,6,7,8] ################################################################## # FUNCTIONS # Function to check amount of attacking queens def check(array): # Collisions represent number of attacking queens, starts at zero obv. collisions=0 for i in range(1,9): if i not in array: print "DUPLICATE NUMBER ERROR - KILLING STUPID GENE" # Never happen return 0 # Total Collisions # For each queen in the array for i in range(0, 8): col=0 # For every other queen in the array for j in range(0, 8): # avoids checking self vs self if i!=j: # if queen i is on a diagonal from queen j, then the # difference in magnitude x-cord will equal the difference in # the magnitude of the y-coord if abs(board[i]-board[j])==abs(array[j]-array[i]): # Sets a variable to tell the next part that a collision was detected col=1 # If a collision was detected add one to collisions if col==1: collisions+=1 # Return 8-colllisions, so that 0 is bad (8 attacking queens) and 8 is good (no attacking queens) return 8-collisions # The reproduce function, takes two arguements, the two parents def reproduce(array1, array2): # FIRST BABY (mother first then father) # Takes first half of mothers gene baby=array1[0:4] # Can't just add two halves as I will get duplicate numbers for i in array2: # add fathers genes going in order of numbers left if i not in baby: baby.append(i) # Add a little variation by giving a percentage chance of mutation if randint(0,100)>20: baby=mutate(baby) # Add the baby to the population and add its corresponding fitness population.append(baby) fitness.append(check(baby)) # SECOND BABY (arrays just swapped around, Father first then mother) baby=array2[0:4] for i in array1: if i not in baby: baby.append(i) if randint(0,100)>20: baby=mutate(baby) population.append(baby) fitness.append(check(baby)) # Mutate the array given to the function def mutate(array1): #Chooses two random places (WARNING CAN BE SAME POSITION) a=randint(0,7) b=randint(0,7) # Swaps the two over (temporary var must be used (c)) c=array1[a] array1[a]=array1[b] array1[b]=c return array1 # Prints the population and corresponding fitness to the screen # Only used for debugging def printpop(): # Prints the group for i in range(0, len(population)): print population[i],fitness[i] ################################################################## # VARIABLES # The size of the population (how many genes, more = faster convergance but more cpu + memory, fewer = opp.) popsize=40 # The starting variation of the group because I havn't created a better method # for starting off. I just add arrays of 1,2,3,4,5,6,7,8 to the population and # give them the number of mutations between the following two values to make # psudo random starting group variation=[4,14] # How many of the genes to kill each time in percentage, each dead gene will be replaced by a new child die=0.40 # Kill limit is calculated from the above percentage kill_limit=die*popsize ################################################################## # MAIN population=[] fitness=[] for i in range(0,popsize): population.append([1,2,3,4,5,6,7,8]) a=0 while a<randint(variation[0],variation[1]): a+=1 population[i]=mutate(population[i]) fitness.append(check(population[i])) maxi=0 generations=1 while maxi!=8: # Picks top # in group to be parents, kills rest killed=0 # starting at the lowest fitness (0 and increasing untill kill limit reached) x=0 while killed<kill_limit: for i in range(0,popsize): # Try is here to catch any errors try: if fitness[i]==x: # This bit removes the crappy gene from the population and from fitness population.pop(i) fitness.pop(i) # increases the kill counter killed+=1 if killed==kill_limit: break except: break # increments fitness x+=1 babies=0 cpop=len(population)-1 #current population while babies<killed: # produces two babies from two random parents (should prob give fittest parents preference) reproduce(population[randint(0,cpop)],population[randint(0,cpop)]) babies+=2 generations+=1 # Looks for highest fitness in the group and checks if any have won! maxi=0 for i in range(0,popsize): if fitness[i]>maxi: maxi=fitness[i] if maxi==8: print population[i] break print "Took",generations,"generations"

**Quick Analysis**

I did a very quick analysis keeping the kill off percentage at 0.4 and varying the population size to see the improvement on generations required before a solution is found. I’ve also added a link (scroll to bottom) to the .ods file I used to calculate this (and an excel .xls file for the windows chaps). Below is the graph I calculated for different population sizes and it looks like we would expect, decreasing as the population increases. I only did 10 averages for each point and there were some anomalies, which are to be expected with this method of programming as it essentially relies on random numbers so it is a bit noisey:

**Links**

Note the following might be more desirable to copy and pasting the above as they use tabs rather than spaces.

queens.py – The normal version

queens_commented.py – The heavily commented version.

evolutionary.ods – Analysis in ods format (open office)

evolutionary.xls – Analysis in excel format

If you have any corrections or suggestions please contact me.

During my undergraduate degree I wrote a program in fortran 95 to calculate pi using random numbers. My aim is to rewrite it efficiently in python. I know its a terrible way to calculate pi, and there are much better ways to do it but its fun!

First I’ll explain the maths so you can visualise what’s going on. As we should know _pi_ is the ratio of circle’s radius to its circumference, which is conveniently the same as the ratio of a circle’s area to the square of its radius (wiki…)

So what we are going to be doing is picking lots of random coordinates in an x-y grid and calculating if they are within the circle or the square.

We will assign the radius to be 1, because that makes it easy to work with. By default a random number in python ( random() ) will return a floating point number between 0 and 1. To test if a point is within a circle we simply use Pythagoras.

So if the sqrt(a**2+b**2)<=1 then the point lies inside the circle’s radius. In the diagram above we see that point A lies within the circle, and point B lies outside the circle.

We can really don’t need to use the whole circle as it has symmetry, so we can just take a quartre, which makes the generating of random numbers easier as you only need to use a random number for x and y between 0 and 1, rather than -1 and 1. It will look like the diagram below.

Now for a confusing bit of maths. We are calculating the ratio of the area of a circle to the area of a square.

# Area of circle A=pi*r**2 # where r = 1 A = pi # Area of square A = l ** 2 # in this case (see diagram) our square's length is twice the radius l=2*r A=(1+1)**2 = 4 #Therefore our ratio will be pi : 4. # Which means we must multiply our result by four to get pi.

**Final version (efficient for using)**

from random import * from math import sqrt inside=0 n=1000 for i in range(0,n): x=random() y=random() if sqrt(x*x+y*y)<=1: inside+=1 pi=4*inside/n print pi

Below we can see the values it creates

n calc error 1 4.00000000 0.73686317 10 3.60000000 0.45840735 100 3.24000000 0.09840735 1000 3.06400000 -0.07759265 10000 3.16160000 0.02000735 100000 3.14140000 -0.00019265 1000000 3.14293600 0.00134335 10000000 3.14117920 -0.00041345

So we can see that the program quickly solves pi to about two decimal places, but it is a terribly inefficient method and will struggle to get much more accuracy than this.

Resources to check out:

This blog post – Solves pi via taylor series expansion

Super pi – Program that calculate pi often used for benchmarking

## Python: palindrome checking function

I created a reasonable palindrome checking function in python.

**Method 1**

def ispalindrome(num): n=str(num) while len(n)>1: print n if n[0]!=n[-1]: return 0 n=n[1:-1] return 1

I thought that it would be faster avoiding a string conversion, and to somehow use the modulus (modulo) function. However when I came to write it, I found it quite difficult to code, and I’m sure there must be a better way.

**Method 2**

def ispalindrome2(num): l=1 while num/10**l>=1.0: l+=1 r=0 d=[] for i in range(1,l+1): p=num%10**i-r r+=p p=p/10**(i-1) d.append(p) for i in range(0,l/2): if d[i]!=d[-i-1]: return 0 return 1

Doing the speed tests show that Method 1 is over 5.2 times faster than Method 2.

Method 1 0.355437994003 Seconds elapsed Method 2 1.85815691948 Seconds elapsed

**Update: Mike of mikemeat sent me his method using slices (Method 3) and I adapted it slight (Method 4), then shortly after I realised it could be even more efficient by not needing to differentiate between even and odd strings (Method 5).**

**Method 3**

def ispalindrome3(x): z = str(x) if len(z)%2 == 0 and z[:len(z)/2]==z[-len(z)/2:][::-1]: return 1 if len(z)%2 != 0 and z[:(len(z)- 1)/2]==z[(-len(z) + 1)/2:][::-1]: return 1 else: return 0

**Method 4**

def ispalindrome4(x): z = str(x) if z[:len( z)/2]==z[len( z)/2+len( z)%2:][::-1]: return 1 return 0

**Method 5**

def ispalindrome5(x): z = str(x) l=len(z)/2 if z[:l]==z[-l:][::-1]: return 1 return 0

Method 1 0.357168912888 Seconds elapsed Method 2 1.83943104744 Seconds elapsed Method 3 0.179126977921 Seconds elapsed Method 4 0.179482936859 Seconds elapsed Method 5 0.149376153946 Seconds elapsed