Mathpack

This gem includes a collection of mathematical methods

Installation

Add this line to your application's Gemfile:

gem 'mathpack'

And then execute:

$ bundle

Or install it yourself as:

$ gem install mathpack

Information

Mathpack includes following modules:

SLE. Solves system of linear equations
Statistics. Provides methods to analyze data samples
Functions. Collects mathematical functions
Approximation. Allows to approximate table and analytical functions by polynom
NonlinearEquations. Solves nonlinear mathematical equations
IntegralEquations. Solves integral second order Fredholm and Volter equations
DifferentialEquations. Solves system of differential equations with left border initial conditions
Integration. Integrates functions
IO. Prints data
Functional. Lambda functions

Statistics

Statistics class have following methods

number

returns a number of elements in series

mean

returns a mean of series

variance

returns a variance of series

skewness

returns a skewness

kurtosis

returns a kurtosis

min

returns the minimal element of series

max

returns the maxinal element of series

raw_moment(n)

returns the nth raw moment of series

central_moment(n)

returns the nth central moment of series

empirical_cdf(x)

empirical distribution function value in x

empirical_pdf(x)

returns empirical probability density function value in x

print_empirical_cdf(filename)

allows to print empirical_cdf table function to filename

print_empirical_pdf(filename)

allows to print empirical_pdf table function to filename

trend

returns trend polynom coefficients

Usage

stat = Mathpack::Statistics.new([1, 2, 5, 6])
stat.number() #=> 4
stat.mean() #=> 3.5
stat.variance() #=> 4.25
stat.kurtosis() #=> -1.778546712802768
stat.skewness() #=> 0.0
stat.min() #=> 1
stat.max() #=> 6
stat.raw_moment(3) #=> 87.5
stat.central_moment(4) #=> 22.0625
stat.empirical_cdf(5.5) #=> 0.75
stat.empirical_pdf(3) #=> 0.07639393483317147
stat.print_empirical_cdf('cdf.csv') #=> nil
stat.print_empirical_pdf('pdf.csv') #=> nil
stat.trend(polynom_power: 1) #=> 1.7999999999999996*x - 0.9999999999999987

Functions

Functions module includes a collection of popular functions.

gamma(x)

$\Gamma(x) = \int_{0}^{\infty} x^{t-1}e^{-x}dx$

beta(x, y)

$B(x, y) = \int_{0}^{1} t^{x-1}(1-t)^{y-1}dx$

erf(x) (Laplace function)

$erf(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x}e^{-t^{2}}dt$

dawson_plus(x)

$D_{+}(x) = e^{-x^{2}}\int_{0}^{x}e^{t^{2}}dt$

dawson_minus(x)

$D_{-}(x) = e^{x^{2}}\int_{0}^{x}e^{-t^{2}}dt$

NonlinearEquations

solve(params = {})

returns solution of nonlinear equation.

Parameters

start - point to start iteration process
eps - calculations accuraccy

Usage

Now you have no problems solving nonlinear equations. If you want, for example, to solve equation $x^{2} = \sin({x+1})$

You need to complete the following steps:

Equation should look like $f(x)=0$
For our equation $f(x) = x^{2} - \sin(x+1)$
Choose the calculations accurracy. For example $0.00001$
Choose some point near the expected root of equation. For example $0$

Then to solve equation you should call

Mathpack::NonlinearEquations.solve(start: 0, eps: 0.00001){|x| x**2 - Math.sin(x+1)})

Here is some other examples of solve usage

Mathpack::NonlinearEquations.solve(start: 0, eps: 0.00001){|x| x**2 - Math.sin(x+1)})
Mathpack::NonlinearEquations.solve(start: 0.01, eps: 0.00001){|x| 1/x - Math.log(x)})
Mathpack::NonlinearEquations.solve(start: 0.01, eps: 0.00001){|x| x**2 - 2*x + 1})
Mathpack::NonlinearEquations.solve(start: 0.01, eps: 0.00001){|x| Math.exp(x-2) - Math.sin(x)})

solve_system(params = {})

returns solution of system of nonlinear equations by Newton method

Parameters

start - vector to start iteration process
eps - calculations accuraccy
f - vector of right part lambdas
w_matrix - matrix W in Newton method

Usage

If you have system of equations $f_{i}(x) = 0, i = 0, 1, ..., N-1$

$f(x) = \begin{pmatrix} f_{0}(x)\\ \cdots\\ f_{N-1}(x) \end{pmatrix}$

$W(x) = \begin{pmatrix} \frac{\partial f_{0}(x)}{\partial x_{0}}&\cdots&\frac{\partial f_{0}(x)}{\partial x_{N-1}}\\ \cdots&\ddots &\cdots \\ \frac{\partial f_{N-1}(x)}{\partial x_{0}}&\cdots& \frac{\partial f_{N-1}(x)}{\partial x_{N-1}}\ \end{pmatrix}$

For example, if you have system

$\left\{\begin{matrix} x_{1} + x_{2} - 3 = 0\\ {x_{1}}^{2} + {x_{2}}^2 - 9 = 0 \end{matrix}\right.$

W matrix and f vector is equal

$W(x) = \bigl(\begin{smallmatrix} 1 & 1\\ 2x_{1}&2x_{2} \end{smallmatrix}\bigr)$

$f(x) = \bigl(\begin{smallmatrix} x_{1} + x_{2} - 3\\ {x_{1}}^{2} + {x_{2}}^{2} - 9 \end{smallmatrix}\bigr)$

f = -> x, y { [x + y - 3.0, x**2 + y**2 - 9.0] }
w = -> x, y { [[1, 1], [2 * x, 2 * y]] }
Mathpack::NonlinearEquations.solve_system(start: [1, 5], eps: 1e-4, f: f, w_matrix: w) #=> [-1.829420851037002e-12, 3.0000000000018296]

IntegralEquations

solve_fredholm_2(params={})

returns solution of integral equation as a hash of nodes and values arrays.

Parameters

from - left border
to - right border
lambda - lambda parameter
k - kernel function (2 arguements)
f - inhomogeneity function (1 arguement)
eps - accuracy

Usage

Let we have the following integral equation

$u(x) - \frac{1}{2}\int_{1}^{2}\frac{u(t))}{\sqrt{x+t^{2}}}dt = 2x + \sqrt{x+1} - \sqrt{x+4}$

If you want to solve equation with accuracy 1e-5, you should call

k = -> x, t { 1.0 / (x + t**2)**0.5 }
f = -> x { 2*x + (x + 1)**0.5 - (x + 4)**0.5 }
Mathpack::IntegralEquations.solve_fredholm_2(from: 1.0, to: 2.0, lambda: 0.5, eps: 1e-5, k: k, f: f)
#=> {:x=>[1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, 2.0], :f=>[1.9999989550044168, 2.2499991010033416, 2.499999229316462, 2.7499993410739187, 2.9999994379022, 3.2499995215457123, 3.4999995936848047, 3.7499996558559117, 3.9999997094242845]}

solve_volter_2(params{})

returns solution of integral equation as a hash of nodes and values arrays.

Parameters

from - left border
to - right border
lambda - lambda parameter
k - kernel function (2 arguements)
f - inhomogeneity function (1 arguement)
eps - accuracy

Usage

Let we have the following integral equation

$u(x) = \frac{1}{2}\int_{0}^{x}cos(x-t)u(t)dt + \frac{3}{4}e^{x} + \frac{1}{4}cosx-\frac{1}{4}sinx, 0\leq x \leq 1$

If you want to solve equation with accuracy 1e-3, you should call

k = -> x, t { Math.cos(x - t) }
f = -> x { 0.75 * Math.exp(x) + 0.25 * Math.cos(x) - 0.25 * Math.sin(x) }
Mathpack::IntegralEquations.solve_volter_2(from: 0.0, to: 1.0, lambda: 0.5, eps: 1e-3, k: k, f: f)
#=> {:x=>[0.0, 0.0625, 0.125, 0.1875, 0.25, 0.3125, 0.375, 0.4375, 0.5, 0.5625, 0.625, 0.6875, 0.75, 0.8125, 0.875, 0.9375, 1.0], :f=>[1.0, 1.0644951210235056, 1.1331511709098798, 1.2062365414157485, 1.2840369296892897, 1.3668564541117094, 1.455018842114489, 1.5488686946157586, 1.6487728320186184, 1.755121727033003, 1.868331029922017, 1.9888431921348702, 2.117129194673052, 2.2536903879456665, 2.3990604503055315, 2.5538074729214233, 2.718536179135541]}

DifferentialEquations

solve_cauchie_system(params={})

returns solution of differential equations system as a hash of nodes and values arrays for each function. For example, { x: [], u1: [], u2: [], ... }

Parameters

from - left border
to - right border
system - array of lambdas representing each row of system
y0 - array of values of derivatives on left border. Starts with first derivative
eps - accuracy

Usage

Let we have following system of differential equations

$\left\{\begin{matrix} {u_{1}}'=u_{2}&&u_{1}(1)=1 \\ {u_{2}}'=\frac{1}{x^{2}}u_{1}-\frac{1}{x}u_{2}+3&&u_{2}(1)=1 \end{matrix}\right.$

where

$1\leq x\leq 2$

If you want to solve this system with accuracy 1e-6, you should call

cauchie_problem = [ ->(x, u1, u2) { u2 }, -> (x, u1, u2) { - 1.0 / x  * u2 + 1.0 / x**2 * u1 + 3.0 }]
Mathpack::DifferentialEquations.solve_cauchie_system(from: 1.0, to: 2.0, eps: 1e-6, system: cauchie_problem, y0: [1.0, 1.0])
#=> {:x=>[1.0,..., 2.0], :u1=>[1,...], :u2=>[1,...] }

SLE

solve(params = {})

returns solution of system of linear equations.

Parameters

matrix - system matrix
f - right part vector

Usage

Let's solve some system of linear equations. It can be written as

$AX = B$

where A is n-n matrix, X - vector of unknown, B - vector

If you want to solve this system you should call

Mathpack::SLE.solve(matrix: A, f: B)

Parameter A can be Array or Matrix class. If Array is given result will be Array class. If Matrix is given result will be Matrix class.

a = [[1, 2, 3], [4, 5, 6],[3, 5, 2]]
b = [15, 30, 15]
Mathpack::SLE.solve(matrix: a, f: b) #=> [-1.0, 2.0, 4.0]
a = Matrix[[1, 2, 3], [4, 5, 6],[3, 5, 2]]
b = Matrix.row_vector [15, 30, 15]
Mathpack::SLE.solve(matrix: a, f: b) #=> Matrix[[-1.0, 2.0, 4.0]]

Approximation

approximate_by_polynom(params = {})

returns array of coefficients of polynom, approximating given function on [from, to] segment.

Parameters

x - array of approximation nodes
polynom_power - power of approximation polynom
f - functions values in x if you have table function

generate_nodes(params = {})

returns nodes for approximation with some step.

Parameters

from - first node
to - last node
step - step between nodes

print_polynom(coefficients)

returns a string representing polynom with given coefficients.

Usage

# Function to print polynom having it's coefficients
Mathpack::Approximation.print_polynom([1, -2, 1]) #=> x^2 - 2*x + 1

# Function to generate nodes for approximation. Only choose start value, end value and step
x = Mathpack::Approximation.generate_nodes(from: 0, to: 10, step: 0.25)
#=> [0, 0.25, ..., 10]

# Function that approximate given function by polynom with power polynom_power in approximation nodes x and returns coefficients of approximating polynom
result = Mathpack::Approximation.approximate_by_polynom(x: x, polynom_power: 5){ |x| Math.sin(x) }
#=> [0.0008009744982571882, -0.030619986086758588, 0.3805927651948083, -1.8481035413353888, 2.985465287759307, -0.3873066069630569]

# May be you want to print this polynom, let's call print_polynom function
Mathpack::Approximation.print_polynom(result)
#=> 0.0008009744982571882*x^5 - 0.030619986086758588*x^4 + 0.3805927651948083*x^3 - 1.8481035413353888*x^2 + 2.985465287759307*x - 0.3873066069630569

# If you have a table of values, where x - array of arguement values and f - array of function values, call approximate by polynom function with parameter f instead of block
result = Mathpack::Approximation.approximate_by_polynom(x: [1, 2, 3], f: [1, 4, 9], polynom_power: 2) #=> [1, 0, 0]
Mathpack::Approximation.print_polynom(result) #=> x^2

Integration

integrate(params = {})

returns integral value.

Parameters

from - start of integration
to - end of integration

Usage

Let you have the following integral:

$\int_{a}^{b}f(x)dx$

Where a can be finite or equal to $-\infty$ , and b can be finite or equal to $\infty$ . To find value of integral you should call integrate method of Integration module.

Mathpack::Integration.integrate(from: a, to: b){ |x| f(x) }

Let's demostrate some examples of Integration module practical usage:

Mathpack::Integration.integrate(from: 0, to: 3.6){ |x| Math.sin(x) / x } #=> 1.8219481156495034
Mathpack::Integration.integrate(from: 0, to: Float::INFINITY){ |x| Math.exp(-x) / (x + 1) } #=> 0.5963473623136091
Mathpack::Integration.integrate(from: -Float::INFINITY, to: Float::INFINITY){ |x| Math.exp(-x**2) * Math.cos(x) } #=> 1.3803884100161075

IO

print_table_function(params = {})

writes table function values to file

Parameters

filename - name of output file
x - arguements array
y - function values array
labels - hash of labels names for x and y column

read_table_function(filename)

returns table function values hash, written to filename

Usage

If you have table function, represented by argument array and function values array, you should use print_table_function, that prints your data to filename file. If you have table function, written to filename file, you should use read_table_function, that

Mathpack::IO.print_table_function(filename: 'table.csv', x: [1, 2, 3], y: [2, 4, 6], labels: { x: 'x', y: 'f(x)'})
Mathpack::IO.read_table_function('table.csv') #=> { x: [1, 2, 3], y: [2, 4, 6] }

Contributing

Fork it ( https://github.com/[my-github-username]/mathpack/fork )
Create your feature branch (git checkout -b my-new-feature)
Commit your changes (git commit -am 'Add some feature')
Push to the branch (git push origin my-new-feature)
Create a new Pull Request