Project

## silicium

0.09
The project is in a healthy, maintained state
Long list of applied functions
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Dependencies

~> 2.0
~> 5.0
>= 12.3.3
= 0.17

# Silicium

Ruby Math Library written as exercise by MMCS students.

## Installation

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:

• Get vertices 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)
• Algorithm of Dijkstra:
g.dijkstra_algorythm!(graph, starting_vertex)
• Find Strongly Connected Components:
g.find_strongly_connected_components
• Algorithm of Dijkstra: dijkstra_algorythm!(graph, starting_vertex)

• 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:

And finally you can type:

TopologicalSortClass.new(graph)

#### Result

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

Algorithm of Dijkstra: dijkstra_algorythm!(graph, starting_vertex)

Algorithm of Kruskal: kruskal_mst(graph)

### GraphVisualiser

#### Set window size

change_window_size(1000, 600)

#### Set graph

graph = OrientedGraph.new([{v: :one, i:  [:one, :two, :four]},
{v: :two, i:[ :one, :two]},
{v: :five, i:[ :one,:three, :four]},
{v: :four, i:[ :one, :four]},
{v: :three, i:[ :one, :two]}])
set_graph(graph)

show_window

### Plotter

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_window

### 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 }

### Polynomial interpolation

Library polynomial_interpolation includes methods for two types of polynomial such Lagrange polynomial and Newton polynomial

Each function accepts 3 parameters, such as array of data points, array returned by function and the node to interpolate.

using the lagrange_polynomials method: lagrange_polynomials([-1, 0, 1, 4], [-7, -1, 1, 43], 2 )

using the newton_polynomials method: newton_polynomials([-1, 0, 1, 2], [-9, -4, 11, 78], 0.1 )

###Geometry Module with geometry functions and geometry structures How to initialize the line with two points:

line = Line2dCanon.new(point1, point2)

How to initialize the line with coefficients:

line.initialize_with_coefficients(a, b, c)

How to check if two lines are parallel:

line1.parallel?(line2)

How to check if two lines are intersecting:

line1.intersecting?(line2)

How to check if two lines are perpendicular:

line1.perpendicular?(line2)

How to get the distance between two parallel lines:

line1.distance_between_parallel_lines(line2)

How to check if the point is on segment:

line.check_point_on_segment(point)

How to check if array of points is on the same line:

line.array_of_points_is_on_line(array_of_points)

How to get a distance from point to line:

distance_point_to_line(point)

How to get a distance from point to plane:

plane.distance_point_to_plane(point)

How to check if the point is on plane:

plane.point_is_on_plane?(point)

How to initialize a plane with 3 points:

plane = Plane3d.new(point1, point2, point3)

How to initialize a plane with coefficients:

plane.initialize_with_coefficients(a,b,c,d)

How to get the distance between parallel planes:

plane1.distance_between_parallel_planes(plane2)

How to check if two planes are perpendicular:

perpendicular?(other_plane)

How to check if two planes are intersecting in 3-dimensional space:

plane1.intersecting?(plane2)

How to check if two planes are parallel in 3-dimensional space:

plane1.parallel?(plane2)

How to get a normal vector:

norm = vector_a.norm_vector(point2, point3)

How to check if two vectors are collinear:

vector1.collinear?(vector2)

How to get a vector multiplication of two vectors:

vector1.vector_multiplication(vector2)

### 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'

## Module BernoulliTrials

Module allows find the probability of an event occurring a certain number of times for any number of independent trials.

n - count of independent trials
k - count of successful events
p - probability of succesful event (k / n)
q - probability of bad event (1 - p)

We have either the probability of event (p) or datas to calculate it (p = suc / all)

For small n probability is calculated by the Bernoulli formula C(n,k) * (p ^ k) * (q ^ (n-k))
For big n probability is calsulated by the Laplace theorem f((k - n*p)/sqrt(n*p*q)) / sqrt(n*p*q)
Auxiliary Gaussian function F(x) = exp(-(x^2/2)) / sqrt(2*PI), F(-x) = F(x)

Laplace theorem give satisfactory approximation for n*p*q > 9

Example

--- Number 1 ---
Probability of making a detail of excellent quality is 0.75.
Probability that out of 400 parts, 280 will be of high quality.

n = 400, k = 280, p = 0.75, q = 0.25

n * p * q > 9, that Laplace theorem

F((280-300) / sqrt(75)) = F(-2.31) = F(2.31) = F(exp(-(2.31^2)/2) / sqrt(2*3.14)) = 0.0277
P = 0.0277 / sqrt(75) = 0.0032

--- Number 2 ---
Of 100 batteries, 7 breaks down during a year of storage.
Choose 5 batteries at random.
Probability that among them 3 are serviceable.

n = 5, k = 3, all = 100, suc = 7
p = 7 / 100 = 0.07, q = 0.93

n * p * q < 9, that Bernoulli formula
P = C(5,3) * (0.93^3) * (0.07^2) = 0.0394

### Matrix

#### Method Gauss and Kramer

We have added Two methods for solving a system of linear equations: Gauss and Kramer.

The Gauss method is implemented as a function, and the Kramer rule is implemented as a method for the Matrix class.

To use the Gauss method, you need to call it with a single argument-the matrix whose roots you want to find.

##### Example
gauss_method_sol(Matrix[[1,2,3,4,5],[0,1,-1,2,3],[0,1,-1,2,3],[0,2,-2,4,6]].row_vectors
[-1,3,0,0]

To use Kramer's rule, you need to call it as a method of the Matrix class with an array argument containing the values of each expression of a system of linear equations

##### Example
Matrix[[2, -5, 3], [4, 1, 4], [1, 2, -8]].kramer([7,21,-11]
[3,1,2]

### Backpropogation

When you need to compute a gradient value for a really huge expression, that a good practise to use a backpropogation algorithm to enhance the speed and quality of work. First, you needed a construct a Computational Graph, what makes our works more effective than it will be by using a common Gradient Decent

my_graph = Comp_Graph.new("(x*W1+b1)*W2+b2")

Than, we initialize our parametrs:

variables = Hash["x",1.0,"W1",1.0,"b1",1.0,"W2",1.0,"b2",1.0]

Finally, we can start to start training! The values will pass forward throw the graph and return the result of results of neural net(in theory)

computed_value = my_graph.ForwardPass(variables)

When it's done, we can use it to compute the curreny of result by loss function(at this example it's just a half of difference between values) and than start to move back, but now we compute the gradient value

trivial_loss = (expected_value - computed_value) * 0.5

That's it! The last thing to do is apply gradient value to inserted parametrs, depended on value of learning speed(learn_rate)

learn_rate = 0.01

After a lot of repeating we will move closer to the perfect values of hyperparametrs in the net

### 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

## 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.

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.

### Method Gauss–Seidel

Use the-Gauss Seidel Method to solve a system of linear equations

Members containing x are written to an array of arrays in a. Free members are written in b. Condition for ending the Seidel iteration process when the epsilon accuracy is reached.

Example

gauss_seidel(a,b,eps)
g = gauss_seidel(([[0.13,0.22,-0.33,-0.07],[0,0.45,-0.23,0.07],[0.11,0,-0.08,0.18],[0.08,0.09,0.33,0.21]]),[-0.11,0.33,-0.85,1.7], 0.001)