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Rudimentary symbolic math library


~> 2.0
>= 10.0
~> 3.0
 Project Readme


Rudimentary symbolic math library for Ruby. This gem is mainly intended as a coding excercise. The operations have not been optimized for speed. The current state of the project is 'under construction'.


Add this line to your application's Gemfile:

gem 'symath'

Then execute:

$ bundle

Or install it yourself as:

$ gem install symath


Using the library:

  > require 'SyMath'

Simple introduction

A convenient way to explore the SyMath library is using the interactive Ruby interpreter, irb:

  > # Load the symath library
  > require 'symath'
  => false
  > # Add the symbols module to your environment
  > extend SyMath::Definitions
  => main

You can now say, for example:

  > # Simplify an expression
  > sin(:x) + 2*sin(:x)
  => 3*sin(:x)
  > # Derivative of tan(2*y + 3)
  > (d(tan(2*:y + 3))/d(:y)).evaluate
  => 2*(tan(2*y + 3)**2 + 1)

Ruby symbols, :x and :y in the above example, are converted into symbolic math variables and Ruby numbers are converted into symbolic math numbers. Functions, operators and constants (e, pi, i, etc.) are available as methods through the SyMath::Definitions module. In some cases it is necessary to tell Ruby that your number or symbol is to be understood as a symbolic object, and not just a Ruby number or symbol. Use the to_m method to explicitly convert them to symbolic bjects:

  > # Ruby integer math
  > 3**4
  => 81
  > # SyMath symbolic math
  > 3.to_m**4
  => 3**4
  > (3.to_m**4).normalize
  => 81

An complete expression can also be converted from a string, using the same to_m method:

  > 'ln(e) + sin(pi/2)'.to_m
  => ln(e) + sin(pi/2)
  > 'ln(e) + sin(pi/2)'.to_m.normalize
  => 2

The SyMath::Definitions module

The module SyMath::Definitions is available to be included or extended to your class or code block. It gives a Ruby method for each operator, function and constant that exists, so they can be referred to by their name, as in the code examples above. If you don't want to use the module, functions, operators and constants must be referred to by the fn, op and definition methods:

  > # Using the SyMath::Definitions methods
  > sin(:x)
  => sin(x)
  > int(:x)
  => int(x)
  > e
  => e
  > sin
  => sin(...)
  > # Using the generic creator functions
  > fn(:sin, :x)
  => sin(x)
  > op(:int, :x)
  => int(x)
  > definition(:e)
  => e
  > definition(:sin)
  => sin(...)

The SyMath::Definitions module is updated dynamically after the user has defined new functions, operators and constants.

String representaton of symbolic objects

Symbolic math objects, inheriting from SyMath::Value, all have a to_s method which returns a string representation of the object. The string representation is compatible with the String.to_m method which converts a string representation into a symbolic object:

  > (ln(e) + sin(pi/2)).to_s
  => "ln(e) + sin(pi/2)"
  > 'ln(e) + sin(pi/2)'.to_m
  => ln(e) + sin(pi/2)

SyMath::Value overrides the Object.inspect method, returning the to_s representation rather than the more verbose and less readable Object.inspect output. This behaviour can be disabled with the setting 'inspect_to_s':

  > SyMath.setting(:inspect_to_s, false)
  => false
  > ln(e) + sin(pi/2)
  => "#<SyMath::Sum:0x000055e8a1d93b38 @definition=..."

Simplification and normalizing

Simple reduction rules are automatically applied when composing an expression. These can be disabled with the setting 'compose_with_simplify'. More thorough reductions are done by the use of the normalize method.

  > e*e*e*e
  => e**4
  > SyMath.setting(:compose_with_simplify, false)
  => false
  > e*e*e*e
  => e*e*e*e
  > sin(pi/2).normalize
  => 1


The library comes with a number of built-in function, which the system knows how to derivate and integrate over. The built-in functions also have a number of reduction rules which are applied by the reduce method and also as part of the 'normalize' method. A list of the defined functions is returned by the functions method. The description method gives a small description of the function:

  > SyMath::Definition::Function.functions
  => [sqrt, sin, cos, tan, sec, csc, cot, arcsin, arccos, arctan,
      arcsec, arccsc, arccot, ln, exp, abs, fact, sinh, cosh, tanh,
      coth, sech, csch, arsinh, arcosh, artanh, arcoth, arsech,
  > sin.description
  => "sin(x) - trigonometric sine"

Defining functions

User-defined functions can be added by the method define_fn:

  > define_fn('poly', [:x, :y], :x**3 + :y**2 + 1)
  => poly

The user-defined function will now be available as a method in the SyMath::Definitions module and can be used in expressions, just as the built in functions. Functions defined by an expression can be evaluated by the evaluate method, which returns the expression with each free variable replaced with the input arguments to the function:

  > poly(2, 3).evaluate
  => 2**3 + 3**2 + 1
  > poly(3).evaluate.normalize
  => 18

Lambda functions

A nameless user-defined function can be created using the lmd method. The method returns a function object which does not have a name, but otherwise works as a function. The lambda function has important usages in operators. Since they eturn a function as the result, it will typically be a lambda function. Also, the lambda function can be used for wrapping an expression into a function before doing an integral or derivative, in this way telling which variables the operator should work on. The lambda function can be called using the call method or the Ruby 'call' operator '()':

  > l = lmd(:x**3 + :y**2 + 1, :x, :y)
  > l.(2, 3)
  => (x**3 + y**2 + 1).(2,3)
  > l.(2, 3).evaluate
  => 2**3 + 3**2 + 1
  > l.(2, 3).evaluate.normalize
  => 18


The library has some built-in operators, i.e. functions which take functions as arguments and return functions. A list of the defined operators is returned by the operators method. The description method gives a small description of the operator:

  > SyMath::Definition::Operator.operators
=> [d(...), xd(...), int(...), [f](b,), #(), b(), hodge(...), grad(f),
    curl(f), div(f), laplacian(f), codiff(f), laplace(f), fourier(f),
    invfourier(f), dpart(f,t)]
  > codiff.description
  => "codiff(f) - codifferential of function f"

Defining operators

User-defined operators can be added by the method define_op:

  > define_op('d2', [:f, :x], d(d(:f)/d(:x))/d(:x))
  => d2
  > d2(:x**3 + 2, :x).evaluate
  => 6*x

The user-defined function will now be available as a method in the SyMath::Definitions module and can be used in expressions.

Evaluating functions and operators

Evaluating a functions or operators which is defined by an expression returns the expression with each free variable replaced with input arguments. Functions which do not have an expression will typically evaluate to itself (no reduction). Most operators which do not have an expression has a built in evaluation, and returns a function or expression according to the operator.


The d-operator returns the differential of a function or expresson. If a function is given, the differential is made over all the free variables of the function. If an expression is given, the operator differentiates over the first free variable found in the expression. Wrapping the expression into a lambda function makes it possible to say which variables to differentiate over. Note that the differential is an operator, so it returns the result in a a lambda function, and not just the expression.

  > d(sin(:x)).evaluate
  => cos(x)*dx.(x)
  > d(:x**2 + :y**3 + 1).evaluate.normalize
  => (2*x*dx).(x)
  > d(lmd(:x**2 + :y**3 + 1, :y)).evaluate.normalize
  => (3*y**2*dy).(y)
  > d(lmd(:x**2 + :y**3 + 1, :x, :y)).evaluate.normalize
  => (3*y**2*dy + 2*x*dx).(x,y)

As a special case, the notatonal form d(f)/d(x) is recognized as the derivative of f with regards to x. This is calculated as d(lmd(f, x))/d(x). This evaluates to the derivative expression:

  > (d(:y**2 + :x**3 + 1)/d(:x)).evaluate
  => 3*x**2

The partial derivative is available as well as 'syntactic sugar':

  > dpart(:x**2 + :y**3 + 1, :x).evaluate.normalize
  => 2*x
  > dpart(:x**2 + :y**3 + 1, :y).evaluate.normalize
  => 3*y**2


Integration is available as the int-operator. The algorithm is only a very simple one, imitating the most basic techniques of finding the anti-derivative, combined with a few well known equation patterns. The operation also has a limitation when it comes to non-commutable terms (matrices, quaternions, etc.). In that case, the result is not reliable.

With one argument, the operator evaluates to the antiderivative of the expression:

  > int(2**:x).evaluate
  => 2**x/ln(2) + C

The variable C is used by convention to represent the free constant factor of the antiderivative.

With three arguments, the int-operator evaluates to the definite integral from a to b:

  > int(2**:x, 3, 4).evaluate.normalize
  => 8/ln(2)

Complex numbers and quaternions

The imaginary unit, i, is available as a constant, and can be used for composing expressions with complex numbers. Simple reduction rules are built in which reduces i*i to -1, and so on.

The basic quaternions, i, j, k are also available as constants. The quaternion i is identical to the complex imaginary unit. Some simple reduction rules are available for the quaternions as well.

Exterior algebra

Caveat: Exterior algebra and differential forms are not well understood by the author of this code. The following has not been reviewed by any others who understand the subject better than me, and it may very well contain a lot of errors and misunderstandings.

D-forms can be defined in several ways. The following are equal:

  > # Using the to_d method on a scalar variable
  > :x.to_m.to_d
  => dx
  > # Differentiating a scalar variable
  > d(:x)
  => dx
  > # Creating a variable, and specifying the dform type
  > :dx.to_m('dform')
  => dx

D-forms can be wedged together, forming n-forms (note that the ^ operator has lower preceedence in Ruby than in math, so parantheses must be used, e.g. when adding):

  > d(:x)^d(:y)^d(:z)
  => dx^dy^dz
  > (d(:x)^d(:x)^d(:z)).normalize
  => 0

A vector variable can be defined using the to_m method, and specifying the vector type.

  > :v.to_m('vector')
  => v'

The exterior derivative and related operators all work in a local coordinate system defined by set of basic vectors of names :x1, :x2, x3. The names can be changed by setting the built-in variable 'basis':

  > SyMath.get_variable('basis')
  => [x1, x2, x3]
  > SyMath.assign_variable('basis', [:x, :y, :z])
  => {dx=>x', dy=>y', dz=>z'}
  > SyMath.get_variable('basis')
  => [x, y, z]

The rest of this section assumes that the following scalars, vectors and d-forms are defined:

  > SyMath.assign_variable('basis', [:x1, :x2, :x3])
  => {dx1=>x1', dx2=>x2', dx3=>x3'}
  > x1  = :x1.to_m
  > x2  = :x2.to_m
  > x3  = :x3.to_m
  > x1v = :x1.to_m('vector')
  > x2v = :x2.to_m('vector')
  > x3v = :x3.to_m('vector')
  > dx1 = :dx1.to_m('dform')
  > dx2 = :dx2.to_m('dform')
  > dx3 = :dx3.to_m('dform')

The exterior derivative is available as the xd-operator:

  > xd(:x1 - :x1*:x2 + :x3**2).evaluate
  => dx1 - (dx1*x2 + x1*dx2) + 2*x3*dx3

The musical isomorphisms are available as the flat and sharp operators:

  > flat(x1v^x2v).evaluate
  => dx1^dx2
  > sharp(dx1^dx2).evaluate
  => x1'^x2'

The flat and sharp operators use the metric tensor in their calculations. This is available a the built-in 'g' variable:

  > SyMath.get_variable(:g)
  => [1, 0, 0; 0, 1, 0; 0, 0, 1]

It can be changed using the SyMath.assign_variable(:g, [value]) method.

The hodge star operator is available as well:

  > hodge(dx1^dx2).evaluate
  => dx3
  > hodge(3).evaluate
  => 3*dx1^dx2^dx3

Gradient, curl, divergence, laplacian and co-differential are defined from the above operators in the usual way:

  > grad(x1 - x1*x2 + x3**2).evaluate
  => 2*x3*x3' - x2*x1' - x1*x2' + x1'
  > curl(-x2*x1v + x1*x2*x2v + x3*x3v).evaluate
  => x2*x3' + x3'
  > div(-x2*x1v + x1*x2*x2v + x3*x3v).evaluate
  => x1 + 1
  > laplacian(x1**2 + x2**2 + x3**2).evaluate
  => 6
  > codiff(x1**2*(dx1^dx3) + x2**2*(dx3^dx1) + x3**2*(dx1^dx2)).evaluate
  => 2*x1*dx3

Vectors and matrices

Vectors can be defined in the coordinate array form by converting an array to a math object, using the to_m method. Matrices can be created the same way from two dimensional arrays:

  > m = [[1, 2, 3], [4, 5, 6]].to_m
  > v = [-3, 4, 1].to_m
  > m*v.transpose
  => [1, 2, 3; 4, 5, 6]*[- 3; 4; 1]
  > (m*v.transpose).evaluate
  => [8; 14]

The vector and matrix cells can of course contain symbolic expressions instead of just numbers.

Methods for manipulating expressions

The library contains a few more complex expression manipulation methods which are available to all math expression objects inheriting from the SyMath::Value class (the root class of the expression components).


The normalization method tries to put an expression on a normal form, based on some heuristics.

  • Expressions formed by natural numbers are calculated.
  • Fractions of natural numbers are simplified as far as possible.
  • Products of equal factors are collapsed to power expressions.
  • Products of powers with equal base are collapsed.
  • Sums of equal terms are simplified to integer products.
  • Product factors are ordered if permitted by commutativity.
  • Sum terms are ordered.
  > # FIXME: Find some better examples
  > (:x*4*:x*3*:y*:y**10).normalize
  => 12*x**2*y**11

Variable replacement

The replace method replaces takes a map of 'variable => expression' as argument. It looks up all instances of the variables in the original expression, and replaces them with the expressions given by the map:

  > (:x**:y).replace({:x.to_m => :a + 2, :y.to_m => :b + 3})
  => (a + 2)**(b + 3)

Matching and pattern replacement

The match method can be seen as a 'reverse' operation to replace-method covered in the last section. It compares an expression to a template expression containing some free variables. It returns an array of all possible maps for the free variables in the template so that it matches the original expression:

  > (:x**2 + :y**2 + 3).match(:a + :b, [:a.to_m, :b.to_m])
  => [{a=>x**2, b=>y**2 + 3},
      {a=>y**2, b=>x**2 + 3},
      {a=>3, b=>x**2 + y**2},
      {a=>x**2 + y**2, b=>3},
      {a=>x**2 + 3, b=>y**2},
      {a=>y**2 + 3, b=>x**2}]

Match/replace operation

The match_replace method tries to find an occurence of a pattern in the expression, and replaces it if it is found. The method can be used together with the iterate method. The latter repeats a method until there are no more changes:

  > a = sin(sin(sin(:a + :b) - sin(:f)))
  > a.match_replace(sin(:x), :e*:x, [:x])
  => e*sin(sin(a + b) - sin(f))
  > a.iterate('match_replace', sin(:x), :e*:x, [:x])
  => e*e*(e*(a + b) - e*f)

Factorization and product expansion

The factorization method has been ripped from the python Py-library. It factorizes a polynomial of one variable:

  > (6*:x**2 + 24*:x**3 - 27*:x**4 + 18*:x**5 + 72*:x**6 - 9*:x).factorize
  => 3*x*(2*x - 1)*(4*x + 3)*(3*x**3 + 1)

The expand method expands the polynomial:

  > (3*:x*(2*:x - 1)*(4*:x + 3)*(3*:x**3 + 1)).expand.normalize
  => 72*x**6 + 18*x**5 - 27*x**4 + 24*x**3 + 6*x**2 - 9*x


The library has some global settings which change the behaviour of the system:

  > # List all settings
  > SyMath.settings
  => {
      # Symbol used when a d-form is created
      :d_symbol                 => "d",
      # Symbol used when a vector is stringified
      :vector_symbol            => "'",
      # Co-vector symbol used in a tensor type signature
      :covector_symbol          => ".",
      # Show all parentheses when stringifying an expression
      :expl_parentheses         => false,
      # Put square roots on exponent form
      :sq_exponent_form         => false,
      # Put fractions on exponent form
      :fraction_exponent_form   => false,
      # In latex strings, insert a product sign between the factors
      :ltx_product_sign         => false,
      # Use simplification rules when expressions are composed
      :compose_with_simplify    => true,
      # Use oo as complex infinity
      :complex_arithmetic       => true,
      # Return to_s representation by the inspect method
      :inspect_to_s             => true,
      # Maximum value of factorial which is normalized to a number
      :max_calculated_factorial => 100
  > # Show one setting
  > SyMath.setting(:vector_symbol)
  => "'"
  > # Change a setting
  > SyMath.setting(:vector_symbol, '¤')
  => "¤"


After checking out the repo, run bin/setup to install dependencies. Then, run rake spec 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


Bug reports and pull requests are welcome on GitHub at 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 SyMath project’s codebases, issue trackers, chat rooms and mailing lists is expected to follow the code of conduct.