mathslib package

mathslib.numtheory module

Various Number Theory functions

Author: Igor van Loo

mathslib.numtheory.phi(n)[source]

Implementation of Eulers Totient Function counts the positive integers up to a given integer n that are relatively prime to n

Parameters

n – An integer

Returns

An integer, numbers, a, less than n, such that gcd(a, n) = 1

```print(phi(20)) #8
print(phi(100)) #40
```
mathslib.numtheory.phi_sieve(n)[source]

A sieve for Eulers Totient Function which counts the positive integers up to a given integer n that are relatively prime to n

Parameters

n – An integer

Returns

An array, where array[x] = phi(x)

```print(phi_sieve(10)) #[0, 1, 1, 2, 2, 4, 2, 6, 4, 6, 4]
print(phi_sieve(20)[11:]) #[10, 4, 12, 6, 8, 8, 16, 6, 18, 8]
```
mathslib.numtheory.phi_sum(n)[source]

Computes the Totient Summatory Function

The algorithm is based on Overview of Project Euler Problem 351 specifically, this is an implementation of Algorithm 6, which you may view after Solving Problem 351

Parameters

n – An integer

Returns

sum of phi(x) where x goes from 1 to n

```print(phi_sum(10**4)) #30397486
print(sum(phi(i) for i in range(1, 10**4 + 1))) #30397486
print(sum(phi_sieve(10**4))) #30397486
```
mathslib.numtheory.mobius(n)[source]

Implementation of the Mobius function of n

Parameters

n – An integer

Returns

An integer

```print(Mobius(10)) #1 - 10 = 2*5, therefore μ(10) = (-1)*(-1) = 1
print(Mobius(9)) #0 - Divisble by 3*3
print(Mobius(7)) #-1 - 7 is prime therefore μ(7) = -1
```

Note

• returns 0 if n is divisible by p^2, where p is a prime

• returns (-1)^k, where k is number of distinct prime factors

mathslib.numtheory.mobius_k_sieve(limit, k=2)[source]

A sieve for the Generalized Mobius function, the mathematics of this function can be read in the following PDF Note that when k = 2 we get the normal mobius function shown above.

Parameters
• limit – An integer

• k – Optional integer, default value is 2 which gives a regular mobius sieve

Returns

An array where array[x] = μ(k, x)

```print(mobius_k_sieve(10)) #[0, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1]
print(mobius_k_sieve(10, 3)) #[0, 1, -1, -1, -1, -1, 1, -1, 0, -1, 1]
```
mathslib.numtheory.count_k_free(n, k)[source]

A function that counts the integers ≤ n which are k-power free. count_k_free(n, 2) would count the number of squarefree integers. The mechanics of this function come from this PDF

Parameters
• n – An integer

• k – An integer

Returns

The number of integers ≤ n which are k-power free

```print(count_k_free(2**50, 2)) #684465067343069
```
mathslib.numtheory.pythagorean_triples(limit, non_primitive=True)[source]

Generates all Pythagorean Triplets up to the limit

Parameters
• limit – An integer, will generate all Pythagorean Triplets such that no side is longer than the limit

• non_primitive – Optional boolean value, If True, returns all triplets, if False returns only primitive triplets

Returns

A list containing all desired triplets

```print(pythagorean_triples(20)) #[[3, 4, 5], [6, 8, 10], [9, 12, 15], [12, 16, 20], [5, 12, 13], [15, 8, 17]]
print(pythagorean_triples(20, False)) #[[3, 4, 5], [5, 12, 13], [15, 8, 17]]
print(len(pythagorean_triples(100, False))) #16
```
mathslib.numtheory.count_primitive_pythagorean_triples(n)[source]

Function counts the number of primitive Pythagorean Triplets with hypotenuse less than n. Algorithm is adapted from the following paper Pythagorean triangles with legs less than n

Parameters

n – An integer

Returns

Number of primitive Pythagorean triplets with hypotenuse less than n

```print(count_primitive_pythagorean_triples(10**10)) #1591549475
print(count_primitive_pythagorean_triples(10**8)) #15915493
```

Note

Due to some precision errors the answer can somtimes be a few numbers off for example the correct answer for n = 10^8 is actually 15915492, one less than what my function gives.

mathslib.numtheory.legendre_factorial(x)[source]

Implementation of Legendres’ Formula

Parameters

x – An integer

Returns

A dictionary containing the prime factorisation of x!

```print(legendre_factorial(6)) #{2: 4, 3: 2, 5: 1}
```
mathslib.numtheory.k_smooth_numbers(max_prime, limit)[source]

Find all k ≤ max_prime smooth numbers up to the limit

Parameters
• max_prime – The maximum prime allowed

• limit – limit up till which to find max_prime smooth numbers

Returns

A list containing all k ≤ max_prime smooth numbers less that limit

```print(len(k_smooth_numbers(5, 10**8))) #1105
```
mathslib.numtheory.k_powerful(k, limit, count=True)[source]

Find all k-powerful numbers less than or equal to upper_bound. Inspired by Rosetta

Parameters
• k – k, representing k-powerful

• limit – limit up till which to k-powerful numbers

• count – Optional, Boolean

Returns

if count is True, it counts the number of k-powerful numbers, otherwise it will return them as a list

```print(kpowerful(2, 10^2)) #14
print(kpowerful(2, 10^2, False)) #[1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100]
```
mathslib.numtheory.legendre_symbol(a, p)[source]

Finds the legendre symbol of a/p

Parameters
• a – An integer

• p – An odd prime

Results

The legendre symbol of a/p

```print(legendre_symbol(3, 3)) #0
print(legendre_symbol(10, 31)) #1
print(legendre_symbol(2, 11)) #-1
```

Note

• returns 1 if a is a quadratic residue modulo p and p does not divide a

• returns -1 if a is a non-quadratic residue modulo p

• returns 0 if p divides a

mathslib.numtheory.tonelli_shanks(a, p)[source]

Implementation of Tonelli Shanks algorithm

Full credit for this alogrithm goes to Eli Bendersky

Parameters
• a – An integer

• p – An integer

Returns

solution to x^2 = a (mod p)

```print(tonelli_shanks(5, 41)) #28
```

Note

This function solves the congruence of the form x^2 = a (mod p) and returns x. Note that p - x is also a root.

0 is returned if no square root exists for these a and p.

mathslib.numtheory.chinese_remainder_theorem(a1, a2, n1, n2)[source]

Simple Chinese Remiander Theorem to solve x = a1 mod n1, x = a2 mod n2

Parameters
• a1 – An integer

• a2 – An integer

• n1 – An integer

• n2 – An integer

Returns

Unique solution to x = a1 mod n1, x = a2 mod n2

```#We solve x = 2 mod 3 = 3 mod 5 = 2 mod 7
#First we solve x = 2 mod = 3 mod 5
print(chinese_remainder_theorem(2, 3, 3, 5)) #8
#Then we solve x = 8 mod 15 = 2 mod 7
print(chinese_remainder_theorem(8, 2, 15, 7)) #23
```
mathslib.numtheory.generalised_CRT(a1, a2, n1, n2)[source]

A generalised Chinese Remiander Theorem which solves for non-coprime moduli

Parameters
• a1 – An integer

• a2 – An integer

• n1 – An integer

• n2 – An integer

Returns

Unique solution to x = a1 mod n1, x = a2 mod n2

```print(generalised_CRT(2, 3, 3, 5)) #8, note that we can use ChineseRemainderTheorem function for this case
print(generalised_CRT(2, 4, 4, 6)) #10
print(generalised_CRT(3, 4, 4, 6)) #'No solution'
```
mathslib.numtheory.frobenius_number(*integers)[source]

Generates the Frobenius Number for given integers.

The below algorithm is based on Faster Algorithms for Frobenius Numbers specifically, this is an implementation of their Breadth-First Method which you may find on page 9

Param

integers

Returns

Frobenius number

```print(frobenius_number(3, 5)) #7
print(frobenius_number(6, 9, 20)) #43
print(frobenius_number(1000, 1476, 3764, 4864, 4871, 7773)) #47350
```
mathslib.numtheory.continued_fraction(x)[source]

Finds the continued Fraction of the sqrt(x)

Parameters

x – An integer

Returns

A list containing the continued fraction of x

```print(continued_fraction(19)) #[4, 2, 1, 3, 1, 2, 8]
```

Note

The continued fraction may repeat forever, the code stops once it repeats, otherwise once we find the 100th number in the continued fraction

mathslib.numtheory.overall_fraction(cf)[source]
Parameters

cf – A list, this list represents the continued fraction of a number

Returns numerator

An integer, the numerator of the fraction

Returns denominator

An integer, the denominator of the fraction

```print(overall_fraction([4, 2, 6, 7])) #(415, 93)
```

mathslib.primes module

Prime related functions

Author: Igor van Loo

mathslib.primes.prime_sieve(limit, segment=False, values=True)[source]

A prime sieve I made with a few different options

Parameters
• limit – The limit up till which the function will generate primes

• segment – Optional boolean value, if segment == True, it will perform a segmented sieve

• values – Optional boolean value, if values == False, it will return an array such that array[x] = True if x is prime

Returns

All primes < limit

```print(prime_sieve(50)) #[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
print(prime_sieve(10, values = False)) #[False, False, True, True, False, True, False, True, False, False, False]

print([i for (i, isprime) in enumerate(prime_sieve(10, values = False)) if isprime]) #[2, 3, 5, 7]
```
mathslib.primes.is_prime(x)[source]

A simple is prime function, checks if a number is prime

Parameters

x – An integer

Returns

True if x is prime, False otherwise

```print(is_prime(10)) #False
print(is_prime(160517089)) #True
```
mathslib.primes.prime_factors(n)[source]

Finds all the prime factors of n

Parameters

n – An integer

Returns

A dictionary containing the prime divisors as keys and value as the corresponding exponent of the prime

```print(prime_factors(123123)) #{3: 1, 7: 1, 11: 1, 13: 1, 41: 1}
print(prime_factors(1123619623)) #{7: 1, 160517089: 1}
```
mathslib.primes.spf_sieve(N)[source]

A smallest prime factor sieve.

Parameters

N – An integer

Returns

An array such that array[x] = smallest prime factor of x

```print(spf_sieve(10)) #[0, 1, 2, 3, 2, 5, 2, 7, 2, 3, 2]
```
mathslib.primes.primepi(x)[source]

Primepi function is commonly known as Prime Counting Function

This function computes primepi(x) based on the Meissel-Lehmer Method

The following article by Adithya Ganesh helped me a lot in understanding and implementing this function

Parameters

x – An integer

Returns

primepi(x)

```print(primepi(10**6)) #78498
print(primepi(10**7)) #664579
print(primepi(10**8)) #5761455
print(primepi(10**9)) #50847534
```
mathslib.primes.primepi_sieve(limit)[source]

Primepi function is commonly known as Prime Counting Function

This function generates an array such that array[x] = primepi(x)

Parameters

limit – An integer,

Returns

An array length limit such that array[x] = primepi(x)

```print(primepi_sieve(10)) #[0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4]
```
mathslib.primes.sum_of_primes(n)[source]

Ultra fast sum of Primes made by Lucy The HedgeHog on Project Euler

You may view it here once you’ve completed problem 10

Parameters

n – An integer

Returns

The sum of all primes up till n

```print(sum_of_primes(2*10**6)) #142913828922
print(sum_of_primes(10**10)) #2220822432581729238
```
mathslib.primes.fermat_primality_test(n, tests=2)[source]
Parameters
• n – The integer to be tested

• tests – Optional, Number of tests to make. The default is 2

Returns

True if n is probably prime

```print(fermat_primality_test(17969800575241)) #True and it is actually True
print(fermat_primality_test(101101)) #True but it is wrong 101101 is not prime
print(fermat_primality_test(101101, 6)) #False, after using 6 tests we see that it is not prime
```

Note

This function will always guess a prime correctly due to Fermats Theorem, but may guess a composite to be a prime. Therefore, it is very useful when we test large numbers, otherwise it is dangerous to use, and hence if n < 10^5 the program will automatically use the is_prime function

You can test more terms to make it more accurate

mathslib.primes.miller_primality_test(n, millerrabin=False, numoftests=2)[source]

The Miller Primality Test with the option to use the Miller-Rabin test

Parameters
• n – The integer to be tested

• millerrabin – Optional, default False. Decides with the use Miller-Rabin or Miller primality test

• numoftests – Optional, default is 2. If millerrabin is True, it uses numoftests bases

Returns

True if n is probably prime, False if n is not prime

```print(miller_primality_test(17969800575241)) #True
print(miller_primality_test(101101)) #False
print(len([x for x in range(1, 10**6) if miller_primality_test(x, True, 1)])) #78544, 46 more primes than needed
print(len([x for x in range(1, 10**6) if miller_primality_test(x, True, 2)])) #78498, correct now
```

Note

Automatically uses the Miller Primality Test to get an exact an answer if n < 3317044064679887385961981 and swaps to using the first 17 primes, but no longer guarantees a correct answer. You may optionally use a regular miller-rabin test with a specified number of tests

mathslib.divisors module

Divisor related functions

Author: Igor van Loo

mathslib.divisors.divisors(n, proper=False)[source]

Finds all the divisors of n using the prime factorisation of n and recursion to find all divisors. Blog by numericalrecipes is an excellent article explaining the algorithm and even faster versions.

Parameters
• x – Integer

• proper – Optional boolean value, If true it will output all proper divisors of n

Returns

A list which contains all divisors of n

```print(divisors(15)) #[1, 3, 5, 15]
print(divisors(15, proper = True)) #[1, 3, 5]
```
mathslib.divisors.divisor(x, n)[source]

Implementation of Divisor function sigma(x, n)

Parameters
• x – An integer, denotes the power till which the divisors will be summed

• n – An integer, denotes the number to find the divisors of

Returns

An integer

```print(divisor(0, 9)) #3, 9 has 3 divisors [1, 3, 9]
print(divisor(1, 9)) #13 = 1 + 3 + 9
print(divisor(2, 9)) #91 = 1*1 + 3*3 + 9*9
```
mathslib.divisors.divisor_sieve(x, N)[source]

Implementation of Divisor function sigma(x, n) sieve. It returns an array such that array[x] = sigma(x, n)

Parameters
• x – An integer

• n – An integer, denotes the length of the array

Returns

An array such that array[x] = sigma(x, n)

```print(divisors_sieve(0, 10)) #[0, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4]
print(divisors_sieve(1, 10)) #[0, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
print(divisors_sieve(2, 10)) #[0, 1, 5, 10, 21, 26, 50, 50, 85, 91, 130]
```

Note

The case x = 0 is especially useful as it gives an array such that array[x] = number of divisors of x

mathslib.linalg module

Various Linear Algebra Functions

Author: Igor van Loo

mathslib.linalg.gauss_jordan_elimination(matrix, augmentedpart=None)[source]
Parameters
• matrix – Matrix to perform Algoithm on

• augmentedpart – Optional argument, will attach the augmented part onto the matrix and then perform the algorithm

Returns

True if algorithm was successful, false otherwise

```matrix = [[2, 1, -1],
[-3, -1, 2],
[-2, 1, 2]]
print(gauss_jordan_elimination(matrix)) #True
```

Note

This function simply performs the Gauss Jordan Algorithm, it is used with with solve and inverse to solve Ax = b, and the find the inverse of a matrix

mathslib.linalg.solve(M, b)[source]

Finds the solution, x, to the equation Mx = b

Parameters
• M – A Matrix

• b – A Vector

Returns

The Vector x, or an error message

```matrix = [[2, 1, -1],
[-3, -1, 2],
[-2, 1, 2]]
b = [[8],
[-11],
[-3]]
print(solve(matrix, b)) #[[2.0], [3.0], [-1.0]]
```
mathslib.linalg.inverse(matrix)[source]

Finds the inverse of the given matrix by performing Gauss Jordan Elimination

Parameters

matrix – Matrix to be inverted

Returns

Inverted matrix, or an error message

```matrix = [[1, -1, 0],
[-8, 9, -1],
[-9, 0, 10]]
print(inverse(matrix)) #[[90.0, 10.0, 1.0], [89.0, 10.0, 1.0], [81.0, 9.0, 1.0]]
```
mathslib.linalg.determinant(matrix)[source]

Finds the determinant of a matrix

Parameters

matrix – Matrix

Returns

det(Matrix)

```matrix = [[7, -1, 0],
[-8, 9, -1],
[-9, 0, 10]]
print(determinant(matrix)) #541.0
```

Performs matrix addition/subtraction on matrix A

Parameters
• A – First Matrix

• B – Second Matrix

• subtract – If True, will perform matrix subtraction

Returns

A + B

```matrix = [[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]
print(matrix_addition(matrix, matrix)) #[[2, 0, 0], [0, 2, 0], [0, 0, 2]]
print(matrix_addition(matrix, matrix, True)) #[[0, 0, 0], [0, 0, 0], [0, 0, 0]]
```
mathslib.linalg.identity(l, val=1)[source]

Generats an Square Identity Matrix

Parameters
• l – Size of matrix

• val – diagonal values, default is 1

Returns

val * identity matrix of size l

```print(identity(2)) #[[1, 0], [0, 1]]
print(identity(2, 5)) #[[5, 0], [0, 5]]
```
mathslib.linalg.concatenate(A, B)[source]

Concatenates 2 matrices, A and B

Parameters
• A – First Matrix

• B – Second Matrix

Returns

A concatenated with B

```A = [[1, 0],
[0, 1]]
print(concatenate(A, A)) #[[1, 0, 1, 0], [0, 1, 0, 1]]
```

Note

This is a helper function for inverse and solve

mathslib.linalg.argmax(alist)[source]

Finds the argmax of a list

Parameters

alist – A list

Returns

the arg max of the list

```print(argmax([1, 3, 2])) #1
```
mathslib.linalg.fillmatrix(size, val=0)[source]

Fills a matrix with a value

Parameters
• size – A tuple, denoted (width, height) of matrix

• val – Value to fill matrix with, default is 0

Returns

A matrix

```print(fillmatrix((2, 2))) #[[0, 0], [0, 0]]
print(fillmatrix((2, 3))) #[[0, 0, 0], [0, 0, 0]]
```
mathslib.linalg.matrix_mul(A, B)[source]

Matrix multplication of 2 matrices, A and B

Parameters
• A – First Matrix

• B – Second Matrix

Returns

Matrix AB

```A = [[2, 0],
[0, 2]]
B = [[1, 2],
[3, 1]]
print(matrix_mul(A, B)) #[[2, 4], [6, 2]]
```
mathslib.linalg.matrix_pow(A, n)[source]

Matrix exponentiation. Uses the Exponentiation by squaring method

Parameters
• A – Matrix

• n – exponenent

Returns

Matrix A^n

```A = [[1, 1],
[1, 0]]
print(matrix_pow(A, 10)) #[[89, 55], [55, 34]]
```

Note

Seem familiar? These are fibonacci numbers! This is nearly identical to my fibonacci generation function as it uses the same method, however the fibonacci is slightly more optimized due to it’s properties

mathslib.fib module

Fibonacci related functions

Author: Igor van Loo

mathslib.fib.fibonacci(n, m=None)[source]

Finds the n-th Fibonacci using matrix exponentiation by squaring. The method is outlined here Specifically, this is an implementation of the third algorithm.

Also includes an option to calculate with a given modulus

Parameters
• n – An integer

• m – An integer, default is None, if specificed will find F(n) (mod m)

Returns

The n-th Fibonacci number (modulus m if specified)

```print(fibonacci(100)) #354224848179261915075
print(fibonacci(100, 10**7 + 9)) #5475613
```
mathslib.fib.fib_till(limit)[source]

Finds all Fibonacci number up till a limit

Parameters

limit – An integer

Returns

A list containing all the fibonacci numbers < limit

```print(fib_till(100)) #[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
print(sum(fib_till(1000))) #2583
```
mathslib.fib.zeckendorf_representation(x)[source]

Finds the Zeckendorf Representation of x

Parameters

x – An integer

Returns

A list containing the zeckendorf representation of x

```print(zeckendorf_representation(64)) #[55, 8, 1]
```

mathslib.algorithms module

Various known algorithms. They are graph theory and optimization related

Author: Igor van Loo

mathslib.algorithms.prims_algorithm(matrix)[source]

Implementation of Prim’s algorithm It finds a Minimum Spanning Tree (MST) for a weighted undirected graph.

Parameters

matrix – Takes a Adjacency matrix as input

Returns Weight

The sum of the minimum spanning tree

The corresponding Adjacency matrix of the MST

Example from Project Euler Problem 107

```matrix = [[0, 16, 12, 21, 0, 0, 0],
[16, 0, 0, 17, 20, 0, 0],
[12, 0, 0, 28, 0, 31, 0],
[21, 17, 28, 0, 18, 19, 23],
[0, 20, 0, 18, 0, 0, 11],
[0, 0, 31, 19, 0, 0, 27],
[0, 0, 0, 23, 11, 27, 0]]

print(prims_algorithm(matrix)) #(93,
#[[0, 16, 12, 0, 0, 0, 0],
#[16, 0, 0, 17, 0, 0, 0],
#[12, 0, 0, 0, 0, 0, 0],
#[0, 17, 0, 0, 18, 19, 0],
#[0, 0, 0, 18, 0, 0, 11],
#[0, 0, 0, 19, 0, 0, 0],
#[0, 0, 0, 0, 11, 0, 0]])
```
mathslib.algorithms.dijkstras_algorithm(graph, start_node=0, INFINITY=10000000000)[source]

Implementation of Dijkstra’s algorithm It finds the the shortest paths between nodes in a graph

Parameters
• graph – Takes an adjacency list as input

• start_node – Optional tuple, default is node 0.

• INFINITY – Optional integer, default is 10^10. It is used to set the “Infinty” value

Returns

Shortest path between start_node and all other nodes

```g = [[[1, 7], [2, 9], [5, 14]],
[[0, 7], [2, 10], [3, 15]],
[[0, 9], [1, 10], [3, 11], [5, 2]],
[[1, 15], [2, 11], [4, 6]],
[[3, 6], [5, 9]],
[[0, 14], [2, 2], [4, 9]]
]

print(dijkstras_algorithm(g)) #[0, 7, 9, 20, 20, 11]
```

Note

A quick comment on this adjacency list. The way it works is for example g[i] contains all the nodes node i is connected to. For example, using the above graph g[0] = [[1, 7], [2, 9], [5, 14]] means node 0 is connected to nodes 1, 2, and 5 and the weight between the edges are 7, 9, and 14 respectively

mathslib.algorithms.floyd_warshall_algorithm(graph, INFINITY=10000000000)[source]

Implementation of the Floyd-Warshall algorithm It finds the the shortest paths between every node in the graph to every node in the graph

Parameters
• graph – Takes an adjacency list as input

• INFINITY – Optional integer, default is 10^10. It is used to set the “Infinty” value

Returns

Shortest path between every node to every node

```g = [[[1, 7], [2, 9], [5, 14]],
[[0, 7], [2, 10], [3, 15]],
[[0, 9], [1, 10], [3, 11], [5, 2]],
[[1, 15], [2, 11], [4, 6]],
[[3, 6], [5, 9]],
[[0, 14], [2, 2], [4, 9]]
]

print(floyd_warshall_algorithm(g)) #[[0, 7, 9, 20, 20, 11],
# [7, 0, 10, 15, 21, 12],
# [9, 10, 0, 11, 11, 2],
# [20, 15, 11, 0, 6, 13],
# [20, 21, 11, 6, 0, 9],
# [11, 12, 2, 13, 9, 0]]
```

Note

This process is like applying Dijkstras Algorithm on every node

mathslib.algorithms.knap_sack(values, weights, n, W, no_values=True)[source]

Implementation of dynamic programming solution to the 0-1 Knapsack Problem

Parameters
• values – A list of values

• weights – A list with weight of corresponding values

• n – Number of items

• W – Desired weight

• no_values – Optional boolean value

Returns

• If no_values == True - It returns the optimal sum of weights

• If no_values == False - it returns the entire array used to build up the solution

```values = [60, 100, 120]
weights = [10, 20, 30]
W = 50
n = len(values)

print(knap_sack(values, weights, n, W)) #220
```
mathslib.algorithms.knap_sack_values(values, weights, n, W)[source]

Extension to KnapSack function It finds the actual values used to obtain the optimal sum

Parameters
• values – A list of values

• weights – A list with weight of corresponding values

• n – Number of items

• W – Desired weight

Returns

A set with the optimal values which form the solution to the knapsack problem

```values = [60, 100, 120]
weights = [10, 20, 30]
W = 50
n = len(values)

print(knap_sack_values(values, weights, n, W)) #{20, 30}
```
mathslib.algorithms.BFS(g, start_node=0, end_node=False)[source]

Parameters
• g – An Adjacency List

• start_node – Optional, pick your start node. Default is 1st node

• end_node – Optional, pick your end node. Default is last node

Returns

A list of nodes which create a path from your start_node to end_node if it exists

```G = [[4, 1], [0, 5], [6, 3], [2, 7],
[0, 8], [1, 6], [2, 5, 10], [3, 11],
[4, 9], [8, 13], [6], [7, 15],
[13], [9, 12, 14], [13, 15], [11, 14] ]

print(BFS(G)) #[0, 4, 8, 9, 13, 14, 15]
```

Note

A quick comment on adjacency lists. The way it works is for example G[i] contains all the nodes node i is connected to. For example using the above graph G[0] = [4, 1] means node 0 is connected to nodes 1, and 4.

mathslib.algorithms.DFS(g, start_node=0, end_node=False)[source]

Implementation of Depth First Search

Parameters
• g

• start_node – Optional, pick your start node. Default is 1st node

• end_node – Optional, pick your end node. Default is last node

Returns

A list of nodes which create a path from your start_node to end_node if it exists

```G = [[4, 1], [0, 5], [6, 3], [2, 7],
[0, 8], [1, 6], [2, 5, 10], [3, 11],
[4, 9], [8, 13], [6], [7, 15],
[13], [9, 12, 14], [13, 15], [11, 14] ]

print(DFS(G)) #[0, 1, 5, 6, 2, 3, 7, 11, 15]
```

Implementation of the Convex Hull Gift Wrapping Algorithm

Parameters

pts – A list containing 2D points

Returns

A list of points consisting of the convex hull starting from leftmost point going around

mathslib.algorithms.convex_hull_DC(pts)[source]

Implementation of the Convex Hull Divide and conquer Algorithm

Parameters

pts – A list containing 2D points

Returns

A list of points consisting of the convex hull starting from leftmost point going around

mathslib.simple module

Various simple functions

Author: Igor van Loo

mathslib.simple.bin_exp(a, b, c, n, m=None)[source]

If (a + b√(c))^n (mod m) = x + y√(c), then this function finds x, y by using binary exponentiation.

Parameters
• a – An integer, coefficient of nonsqrt term

• b – An integer, coefficient of sqrt

• c – An integer, inside the sqrt

• n – An integer, exponent

• m – An integer, the modulus

Returns

x, y such that (a + b√(c))^n (mod m) = x + y√(c)

```#Using fibonacci relation to golden ratio we know
#(((1 + sqrt(5))/2)^n - ((1 + sqrt(5))/2)^n)/sqrt(5) = F(n)

x = bin_exp(1/2, 1/2, 5, 10)  #x = (61.5, 27.5) represents 61.5 + 27.5*sqrt(5)
y = bin_exp(1/2, -1/2, 5, 10) #y = (61.5, -27.5) represents 61.5 - 27.5*sqrt(5)

#Therefore, F(10) = (61.5 + 27.5*sqrt(5) - (61.5 - 27.5*sqrt(5)))/sqrt(5)) = 55

print(x[1] - y[1]) #55.0
```
mathslib.simple.number_to_base(n, b)[source]

Changes n from base 10 to base b

Parameters
• n – An integer, number to be changed

• b – An integer, base in question

Returns

n in base b

```print(number_to_base(10, 2)) #[1, 0, 1, 0]
print(number_to_base(10, 3)) #[1, 0, 1]
```
mathslib.simple.extended_euclidean_algorithm(a, b)[source]

Standard Extended Euclidean Algorithm

Parameters
• a – An integer

• b – An integer

Returns

A tuple (g, s, t) where gcd(a, b) = g = as + bt

```print(extended_euclidean_algorithm(240, 46)) #(2, -9, 47)
```
mathslib.simple.lcm(a_list)[source]

Finds the lcm of a list of numbers

Parameters

alist – A list containing integers

Returns

The lcm of all numbers in the list

```print(lcm([2, 3])) #6
print(lcm([2, 4, 5, 7])) #140
print(lcm([8345, 23579, 174])) #34237415370
```
mathslib.simple.mod_division(a, b, m)[source]

Finds a/b mod m

Parameters
• a – An integer, the numerator

• b – An integer, the denominator

• m – An integer, the modulus

Returns

a/b mod m

```print(mod_division(8, 4, 5)) #2
```
mathslib.simple.bisect(alist, goal)[source]

This function is equivalent to bisect_right from the bisect module

Parameters
• alist – A list

• goal – A number

Returns

index i of A such that A[i - 1] < g <= A[i]

```print(bisect([2, 3, 5, 7], 6)) #3 since A[2] = 5 < 6 <= A[3] = 7
```
mathslib.simple.is_clockwise(a, b, c)[source]

Finds if 3 points a going to b going to c are in clockwise order. It is used in convex hull algorithm

Parameters
• a – A tuple, representing a point in 2D

• b – A tuple, representing a point in 2D

• c – A tuple, representing a point in 2D

Returns

True if point are in clockwise direction, otherwise False

Module contents

Library of Mathematical functions and Algorithms

Author: Igor van Loo