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algebra

means completeness and balancing, from the Arabic word الجبر

algebra

Vectors, Matrices; Real, Complex, Quaternion; custom groups and rings for Node.js

New: checkout matrices and vectors made of strings, with cyclic algebra.

NOTA BENE Immagine all code examples below as written in some REPL where expected output is documented as a comment.

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Algebra OnQuaternionsAndOctonions

Table Of Contents

Status

algebra is under development, but API should not change until version 1.0.

I am currently adding more tests and examples to achieve a stable version.

Many functionalities of previous versions are now in separated atomic packages:

Features

  • Real, Complex, Quaternion, Octonion numbers.
  • Vector and Matrix spaces over any field (included Real numbers, of course :).
  • Expressive syntax.
  • Everything is a Tensor.
  • Immutable objects.
  • math blog with articles explaining algebra concepts and practical examples. I started blogging about math hoping it can help other people learning about the Queen of Science.

Installation

With npm do

npm install algebra

With bower do

bower install algebra

or use a CDN adding this to your HTML page

<script src="https://cdn.rawgit.com/fibo/algebra/master/dist/algebra.js"></script>

Quick start

This is a 60 seconds tutorial to get your hands dirty with algebra.

First of all, import algebra package.

const algebra = require('algebra')

Try it out

All code in the examples below should be contained into a single file, like test/quickStart.js.

view on requirebin

Scalars

Use the Real numbers as scalars.

const R = algebra.Real

Every operator is implemented both as a static function and as an object method.

Static operators return raw data, while class methods return object instances.

Use static addition operator to add three numbers.

R.add(1, 2, 3) // 1 + 2 + 3 = 6 

Create two real number objects: x = 2, y = -2

const x = new R(2)
const y = new R(-2)

The value r is the result of x multiplied by y.

// 2 * (-2) = -4 
var r = x.mul(y)
 
// Scalar { data: -4 } 
 
// x and y are not changed 
x.data // 2 
y.data // -2 

Raw numbers are coerced, operators can be chained when it makes sense. Of course you can reassign x, for example, x value will be 0.1: x -> x + 3 -> x * 2 -> x ^-1

// ((2 + 3) * 2)^(-1) = 0.1 
= x.add(3).mul(2).inv()
 
// Scalar { data: 0.1 } 

Comparison operators equal and notEqual are available, but they cannot be chained.

x.equal(0.1) // true 
x.notEqual(Math.PI) // true 

You can also play with Complexes.

const C = algebra.Complex
 
let z1 = new C([1, 2])
const z2 = new C([3, 4])
 
z1 = z1.mul(z2)
 
z1 // Scalar { data: [-5, 10] } 
 
z1 = z1.conj().mul([2, 0])
 
z1.data // [-10, -20] 

Vectors

Create vector space of dimension 2 over Reals.

const R2 = algebra.VectorSpace(R)(2)

Create two vectors and add them.

let v1 = new R2([0, 1])
const v2 = new R2([1, -2])
 
// v1 -> v1 + v2 -> [0, 1] + [1, -2] = [1, -1] 
v1 = v1.add(v2)
 
v1 // Vector { data: [1, -1] } 

Matrices

Create space of matrices 3 x 2 over Reals.

const R3x2 = algebra.MatrixSpace(R)(3, 2)

Create a matrix.

//       | 1 1 | 
//  m1 = | 0 1 | 
//       | 1 0 | 
// 
const m1 = new R3x2([1, 1,
                   0, 1,
                   1, 0])

Multiply m1 by v1, the result is a vector v3 with dimension 3. In fact we are multiplying a 3 x 2 matrix by a 2 dimensional vector, but v1 is traited as a column vector so it is like a 2 x 1 matrix.

Then, following the row by column multiplication law we have

//  3 x 2  by  2 x 1  which gives a   3 x 1 
//      ↑      ↑ 
//      +------+----→  by removing the middle indices. 
// 
//                   | 1 1 | 
//    v3 = m1 * v1 = | 0 1 | * [1 , -1] = [0, -1, 1] 
//                   | 1 0 | 
 
const v3 = m1.mul(v1)
 
v3.data // [0, -1, 1] 

Let's try with two square matrices 2 x 2.

const R2x2 = algebra.MatrixSpace(R)(2, 2)
 
let m2 = new R2x2([1, 0,
                   0, 2])
 
const m3 = new R2x2([0, -1,
                     1, 0])
 
m2 = m2.mul(m3)
 
m2 // Matrix { data: [0, -1, 2, 0] } 

Since m2 is a square matrix we can calculate its determinant.

m2.determinant // Scalar { data: 2 } 

API

About operators

All operators are implemented as static methods and as object methods. In both cases, operands are coerced to raw data. As an example, consider addition of vectors in a plane.

const R2 = algebra.R2
 
const vector1 = new R2([1, 2])
const vector2 = new R2([3, 4])

The following static methods, give the same result: [4, 6].

R2.addition(vector1, [3, 4])
R2.addition([1, 2], vector2)
R2.addition(vector1, vector2)

The following object methods, give the same result: a vector instance with data [4, 6].

const vector3 = vector1.addition([3, 4])
const vector4 = vector1.addition(vector2)
 
R2.equal(vector3, vector4) // true 

Operators can be chained and accept multiple arguments when it makes sense.

vector1.addition(vector1, vector1).equality([4, 6]) // true 

Objects are immutable

vector1.data // still [1, 2] 

Cyclic

Cyclic(elements)

Create an algebra cyclic ring, by passing its elements. The elements are provided as a string or an array, which lenght must be a prime number. This is necessary, otherwise the result would be a wild land where you can find zero divisor beasts.

Let's create a cyclic ring containing lower case letters, numbers and the blank char. How many are they? They are 26 + 10 + 1 = 37, that is prime! We like it.

const Cyclic = algebra.Cyclic
 
// The elements String or Array length must be prime. 
const elements = ' abcdefghijklmnopqrstuvwyxz0123456789'
 
const Alphanum = Cyclic(elements)

Operators derive from modular arithmetic

const a = new Alphanum('a')
 
Alphanum.addition('a', 'b') // 'c' 

You can also create element instances, and do any kind of operations.

const x = new Alphanum('a')
 
const y = x.add('c', 'a', 't')
           .mul('i', 's')
           .add('o', 'n')
           .sub('t', 'h', 'e')
           .div('t', 'a', 'b', 'l', 'e')
 
y.data // 's' 

Yes, they are scalars so you can build vector or matrix spaces on top of them.

const VectorStrings2 = algebra.VectorSpace(Alphanum)(2)
const MatrixStrings2x2 = algebra.MatrixSpace(Alphanum)(2)
const 
const vectorOfStrings = new VectorStrings2(['o', 'k'])
const matrixOfStrings = new MatrixStrings2x2(['c', 'o',
                                              'o', 'l'])
 
matrixOfStrings.mul(vectorOfStrings).data // ['x', 'y'] 

Note that, in the particular example above, since the matrix is simmetric it commutes with the vector, hence changing the order of the operands the result is still the same.

vectorOfStrings.mul(matrixOfStrings).data // ['x', 'y'] 

CompositionAlgebra

A composition algebra is one of ℝ, ℂ, ℍ, O: Real, Complex, Quaternion, Octonion. A generic function is provided to iterate the Cayley-Dickson construction over any field.

CompositionAlgebra(field[, num])

  • num can be 1, 2, 4 or 8

Let's use for example the [src/binaryField][binaryField] which exports an object with all the stuff needed by algebra-ring npm package.

const CompositionAlgebra = algebra.CompositionAlgebra
 
const binaryField = require('algebra/src/binaryField')
 
const Bit = CompositionAlgebra(binaryField)
 
Bit.contains(1) // true 
Bit.contains(4) // false 
 
const bit = new Bit(1)
Bit.addition(0).data // 1 

Not so exciting, let's build something more interesting. Let's pass a second parameter, that is used to build a Composition algebra over the given field. It is something experimental also for me, right now I am writing this but I still do not know how it will behave. My idea is that

A byte is an octonion of bits

Maybe we can discover some new byte operator, taken from octonion rich algebra structure. Create an octonion algebra over the binary field, a.k.a Z2 and create the eight units.

// n must be a power of two 
const Byte = CompositionAlgebra(binaryField, 8)
 
const byte1 = new Byte([1, 0, 0, 0, 0, 0, 0, 0])
const byte2 = new Byte([0, 1, 0, 0, 0, 0, 0, 0])
const byte3 = new Byte([0, 0, 1, 0, 0, 0, 0, 0])
const byte4 = new Byte([0, 0, 0, 1, 0, 0, 0, 0])
const byte5 = new Byte([0, 0, 0, 0, 1, 0, 0, 0])
const byte6 = new Byte([0, 0, 0, 0, 0, 1, 0, 0])
const byte7 = new Byte([0, 0, 0, 0, 0, 0, 1, 0])
const byte8 = new Byte([0, 0, 0, 0, 0, 0, 0, 1])

The first one corresponds to one, while the rest are immaginary units, but since the underlying field is Z2, -1 corresponds to 1.

byte1.mul(byte1).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte2.mul(byte2).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte3.mul(byte3).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte4.mul(byte4).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte5.mul(byte5).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte6.mul(byte6).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte7.mul(byte7).data // [1, 0, 0, 0, 0, 0, 0, 0] 
byte8.mul(byte8).data // [1, 0, 0, 0, 0, 0, 0, 0] 

Keeping in mind that Byte space defined above is an algebra, i.e. it has composition laws well defined, you maybe already noticed that, for example byte2 could be seen as corresponding to 4, but in this strange structure we created, 4 * 4 = 2.

You can play around with this structure.

const max = byte1.add(byte2).add(byte3).add(byte4)
                 .add(byte5).add(byte6).add(byte7).add(byte8)
 
max.data // [1, 1, 1, 1, 1, 1, 1, 1] 

Scalar

Scalar attributes

Scalar.one
Scalar.zero

Scalar order

It is always 0 for scalars, see also tensor order.

Scalar.order
scalar.order
scalar.data

Scalar operators

Scalar set operators

Scalar.contains(scalar1, scalar2[, scalar3, … ])
scalar1.belongsTo(Scalar)

Scalar equality

Scalar.equality(scalar1, scalar2)
scalar1.equality(scalar2)

Scalar disequality

Scalar.disequality(scalar1, scalar2)
scalar1.disequality(scalar2)

Scalar addition

Scalar.addition(scalar1, scalar2[, scalar3, … ])
scalar1.addition(scalar2[, scalar3, … ])

Scalar subtraction

Scalar.subtraction(scalar1, scalar2[, … ])
scalar1.subtraction(scalar2[, scalar3, … ])

Scalar multiplication

Scalar.multiplication(scalar1, scalar2[, scalar3, … ])
scalar1.multiplication(scalar2[, scalar3, … ])

Scalar division

Scalar.division(scalar1, scalar2[, scalar3, … ])
scalar1.division(scalar2[, scalar3, … ])

Scalar negation

Scalar.negation(scalar)
scalar.negation()

Scalar inversion

Scalar.inversion(scalar)
scalar.inversion()

Scalar conjugation

Scalar.conjugation(scalar)
scalar.conjugation()

Real

Inherits everything from Scalar.

const Real = algebra.Real
 
Real.addition(1, 2) // 3 
 
const pi = new Real(Math.PI)
const twoPi = pi.mul(2)
 
Real.subtraction(twoPi, 2 * Math.PI) // 0 

Complex

Inherits everything from Scalar.

const Complex = algebra.Complex
 
const complex1 = new Complex([1, 2])
 
complex1.conjugation() // Complex { data: [1, -2] } 

Quaternion

Inherits everything from Scalar.

Octonion

Inherits everything from Scalar.

Common spaces

R

The real line.

It is in alias of Real.

const R = algebra.R

R2

The real plane.

const R2 = algebra.R2

It is in alias of VectorSpace(Real)(2).

R3

The real space.

const R3 = algebra.R3

It is in alias of VectorSpace(Real)(3).

R2x2

Real square matrices of rank 2.

const R2x2 = algebra.R2x2

It is in alias of MatrixSpace(Real)(2).

C

The complex numbers.

It is in alias of Complex.

const C = algebra.C

H

Usually it is used the H in honour of Sir Hamilton.

It is in alias of Quaternion.

const H = algebra.H

Vector

Inherits everything from Tensor.

VectorSpace(Scalar)(dimension)

Vector attributes

Vector dimension

Vector.dimension
R2.dimension // 2 
vector.dimension
const vector = new R2([1, 1])
 
vector.dimension // 2 

Vector operators

Addition

Vector.addition(vector1, vector2)
R2.addition([2, 1], [1, 2]) // [3, 3] 
vector1.addition(vector2)
const vector1 = new R2([2, 1])
const vector2 = new R2([2, 2])
 
const vector3 = vector1.addition(vector2)
 
vector3 // Vector { data: [4, 3] } 

Cross product

It is defined only in dimension three. See Cross product on wikipedia.

Vector.crossProduct(vector1, vector2)
R3.crossProduct([3, -3, 1], [4, 9, 2]) // [-15, 2, 39] 
vector1.crossProduct(vector2)
const vector1 = new R3([3, -3, 1])
const vector2 = new R3([4, 9, 2])
 
const vector3 = vector1.crossProduct(vector2)
 
vector3 // Vector { data: [-15, 2, 39] } 

Matrix

Inherits everything from Tensor.

MatrixSpace(Scalar)(numRows[, numCols])

Matrix attributes

Matrix.isSquare
Matrix.numCols
Matrix.numRows

Matrix operators

Matrix multiplication

Matrix.multiplication(matrix1, matrix2)
matrix1.multiplication(matrix2)

Matrix inversion

It is defined only for square matrices which determinant is not zero.

Matrix.inversion(matrix)
matrix.inversion

Matrix determinant

It is defined only for square matrices.

Matrix.determinant(matrix)
matrix.determinant

Matrix adjoint

Matrix.adjoint(matrix1)
matrix.adjoint

Tensor

TensorSpace(Scalar)(indices)

Tensor attributes

Tensor.one
Tensor.zero
tensor.data

Tensor indices

Tensor.indices
tensor.indices

Tensor order

It represents the number of varying indices.

  • A scalar has order 0.
  • A vector has order 1.
  • A matrix has order 2.
Tensor.order
tensor.order

Tensor operators

Tensor.contains(tensor1, tensor2[, tensor3, … ])

Tensor equality

const T2x2x2 = TensorSpace(Real)([2, 2, 2])
 
const tensor1 = new T2x2x2([1, 2, 3, 4, 5, 6, 7, 8])
const tensor2 = new T2x2x2([2, 3, 4, 5, 6, 7, 8, 9])
Tensor.equality(tensor1, tensor2)
T2x2x2.equality(tensor1, tensor1) // true 
T2x2x2.equality(tensor1, tensor2) // false 
tensor1.equality(tensor2)
tensor1.equality(tensor1) // true 
tensor2.equality(tensor2) // false 
Tensor.disequality(tensor1, tensor2)
tensor1.disequality(tensor2)
Tensor.addition(tensor1, tensor2[, tensor3, … ])
tensor1.addition(tensor2[, tensor3, … ])
Tensor.subtraction(tensor1, tensor2[, tensor3, … ])
tensor1.subtraction(tensor2[, tensor3, … ])
Tensor.product(tensor1, tensor2)
tensor1.product(tensor2)
Tensor.contraction()
tensor.contraction()
Tensor.negation(tensor1)
tensor.negation()

Tensor multiplication

Tensor.scalarMultiplication(tensor, scalar)
tensor.scalarMultiplication(scalar)

License

MIT