Silicium

Ruby Math Library written as exercise by MMCS students.

Installation

Add this line to your application's Gemfile:

gem 'silicium'

And then execute:

$ bundle

Or install it yourself as:

$ gem install silicium

Usage

Graphs

Graph initialization

To create an empty graph just initialize an object:

    g = OrientedGraph.new
    g = UnorientedGraph.new

Of course, you can determine vertices (name them whatever you want!). To do that, write something like:

    g = OrientedGraph.new([{v: 0,     i: [:one]},
                          {v: :one,  i: [0, 'two']},
                          {v: 'two', i: [0, 'two']}])

You have to pass an Array of Hashes, each hash consists of pair of keys:

• v: vertex name;
• i: Array of adjacent vertices

Same goes for the case with unoriented graph (note that missing edges will be added automatically):

    g = UnorientedGraph.new([{v: 0,     i: [:one]},
                           {v: :one,  i: [0, 'two']},
                           {v: 'two', i: [0, 'two']}])``

Graph Methods:

• Add vertex to your graph:

    g.add_vertex!(Vertex)

• Add edge to your graph:

    g.add_edge!(vertex_from, vertex_to)

• Get vertices adjacted with vertex:

    g.adjacted_with(vertex)

• Set label for the edge:

    g.label_edge!(vertex_from, vertex_to, label)

• Get label for the edge:

    g.get_edge_label(vertex_from, vertex_to)

• Set label for the vertex:

    g.label_vertex!(vertex, label)

• Get label for the vertex:

    g.get_vertex_label(vertex)

• Get number of vertices:

    g.vertex_number

• Get number of edges:

    g.edge_number

• Get number of vertex labels:

    g.vertex_label_number

• Get number of vertex edges:

    g.edge_label_number

• Check whether graph contains vertex:

    g.has_vertex?(vertex)

• Check whether graph contains edge:

    g.has_edge?(vertex_from, vertex_to)

• Delete vertex:

    g.delete_vertex!(vertex)

• Delete edge:

    g.delete_edge!(vertex_from, vertex_to)

• Get array of vertices:

    g.vertices

Graph algorithms:

• Check whether graph is connected:

    g.connected?(graph)

• Breadth-First Search:

    g.breadth_first_search?(graph, starting_vertex, searching_vertex)

• Algorithm of Dijkstra:

     g.dijkstra_algorythm!(graph, starting_vertex)

• Find Strongly Connected Components:

     g.find_strongly_connected_components

• Topological sort

Description

Topological sorting for Directed Acyclic Graph (DAG) is a linear ordering of vertices such that for every directed edge u v, vertex u comes before v in the ordering.

How to use

For you to have a topologically sorted graph, you need to create an object of the class Graph:

    graph = Graph.new

Then you need to add vertices to this graph using the class Node:

    graph.nodes << (node1 = Node.new(1))
    graph.nodes << (node2 = Node.new(2))

Due to the fact that only a directed graph can be sorted topologically, it is necessary to add an edge:

    graph.add_edge(node1, node2)

And finally you can type:

    TopologicalSortClass.new(graph)

Result

The result for TopologicalSortClass.new(graph).post_order.map(&:to_s) is [2, 1]

Plotter

Determine your function

def fn(x)
  x**2
end

Set scale

# 1 unit is equal 40 pixels
set_scale(40)

Draw you function

draw_fn(-20, 20) {|args| fn(args)}

Show your plot

show_window

Result

Numerical integration

Library Numerical integration includes methods for numerical integration of functions, such as 3/8 method, Simpson method, left, right and middle rectangle methods and trapezoid method.

Each function accepts 4 parameters, such as left and right integration boundaries, default accuracy of 0.0001 and the function itself. Example: three_eights_integration(4, 5, 0.01) { |x| 1 / x } or three_eights_integration(4, 5) { |x| 1 / x }

For example, to integrate 1 / x in between [4, 5] using the 3/8 method, you need to use: NumericalIntegration.three_eights_integration(4, 5) { |x| 1 / x }

using the Simpson's method: NumericalIntegration.simpson_integration(4, 5) { |x| 1 / x }

using the left rectangle method: NumericalIntegration.left_rect_integration(4, 5) { |x| 1 / x }

using the right rectangle method: NumericalIntegration.right_rect_integration(4, 5) { |x| 1 / x }

using the middle rectangle method: NumericalIntegration.middle_rectangles(4, 5) { |x| 1 / x }

using the trapezoid method: NumericalIntegration.trapezoid(4, 5) { |x| 1 / x }

Theory of probability

Combinatorics

Module with usual combinatorics formulas

    factorial(5) # 5! = 120
    combination(n, k) # C(n, k) = n! / (k! * (n-k)!)
    arrangement(n, k) # A(n, k) = n! / (n - k)!

Module Dice

Module describing both ordinary and unique dices

You can initialize a Polyhedron by two ways

first: by number - Polyhedron.new(6) - creates polyhedron with 6 sides [1,2,3,4,5,6]

second: by array - Polyhedron.new([1,3,5]) - creates polyhedron with 3 sides [1,3,5]

class Polyhedron
    csides # sides number
    sides  # array of sides
    throw # method of random getting on of the Polyhedron's sides

Example

d = Polyhedron.new(8)
d.csides # 8
d.sides # [1,2,3,4,5,6,7,8]
d.throw # getting random side (from 1 to 8)

d1 = Polyhedron.new([1,3,5,6])
d1.csides # 4
d1.sides # [1,3,5,6]
d1.throw # getting random side (from 1 or 3 or 5 or 8)

Class PolyhedronSet

You can initialize PolyhedronSet by array of:

Polyhedrons

Number of Polyhedron's sides

Array of sides

class PolyhedronSet
    percentage # hash with chances of getting definite score
    throw   # method of getting points from throwing polyhedrons
    make_graph_by_plotter # creating graph introducing chances of getting score

Example

s = PolyhedronSet.new([6, [1,2,3,4,5,6], Polyhedron.new(6)]) 

s.percentage # {3=>0.004629629629629629, 4=>0.013888888888888888, 5=>0.027777777777777776, 6=>0.046296296296296294, 
              # 7=>0.06944444444444445, 8=>0.09722222222222222, 9=>0.11574074074074074, 
              # 10=>0.125, 11=>0.125, 12=>0.11574074074074074, 13=>0.09722222222222222, 14=>0.06944444444444445, 
              # 15=>0.046296296296296294, 16=>0.027777777777777776, 17=>0.013888888888888888, 18=>0.004629629629629629}    

s.throw   # getting random score (from 3 to 18)

s.make_graph_by_plotter(xsize, ysize) # creates a graph in 'tmp/percentage.png'

Optimization

Karatsuba multiplication

The Karatsuba algorithm is a fast multiplication algorithm. It reduces the multiplication of two n-digit numbers to at most single-digit multiplications in general. It is therefore faster than the traditional algorithm, which requires single-digit products.

Example:

   karatsuba(15, 15) #returns 225

Dixon's factorialisation

Dixon's factorialisation is factorisation function, offering a way to relatively fast generate array of non-trivial divisors of a number.

Example:

   dix_factor(23449) #returns [131,179]

Euler's totient function

Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ℤ/nℤ).

Example:

   eul_f(42) #returns 12

Development

After checking out the repo, run bin/setup to install dependencies. Then, run rake test to run the tests. You can also run bin/console for an interactive prompt that will allow you to experiment.

To install this gem onto your local machine, run bundle exec rake install. To release a new version, update the version number in version.rb, and then run bundle exec rake release, which will create a git tag for the version, push git commits and tags, and push the .gem file to rubygems.org.

Contributing

Bug reports and pull requests are welcome on GitHub at https://github.com/mmcs-ruby/silicium. This project is intended to be a safe, welcoming space for collaboration, and contributors are expected to adhere to the Contributor Covenant code of conduct.

License

The gem is available as open source under the terms of the MIT License.

Code of Conduct

Everyone interacting in the Silicium project’s codebases, issue trackers, chat rooms and mailing lists is expected to follow the code of conduct.