la lib Algorithm
The la lib Algorithm, also known as Lattice Algebra (LA) algorithm, is an innovative approach to computing that has gained significant attention in recent years. It is a mathematical framework based on lattice theory, which is a branch of abstract algebra that deals with partially ordered sets and their properties. The LA algorithm is designed to efficiently process large volumes of data, making it particularly well-suited for applications in computer science, data analytics, and engineering.
The core idea behind the la lib Algorithm is to represent data as elements of a lattice, a structure where every pair of elements has a unique least upper bound (LUB) and greatest lower bound (GLB). This representation allows for efficient computation of set operations, such as union and intersection, as well as other higher-level operations on data. By leveraging the inherent properties of lattice structures, the LA algorithm can perform complex computations in a highly parallel and scalable manner. This makes the algorithm particularly attractive for large-scale data processing tasks, such as big data analytics, machine learning, and scientific simulations, where traditional computational methods struggle to cope with the sheer volume of data.
/*
author: Christian Bender
license: MIT-license
The namespace LinearAlgebra contains useful classes and functions for dealing with
linear algebra under JavaScript.
*/
var LinearAlgebra;
(function (LinearAlgebra) {
/*
class: Vector
This class represents a vector of arbitrary size and operations on it.
*/
var Vector = /** @class */ (function () {
// constructor
function Vector (N, comps) {
if (comps === undefined) {
comps = []
}
this.components = new Array(N)
if (comps.length === 0) {
for (var i = 0; i < N; i++) {
this.components[i] = 0.0
}
} else {
if (N === comps.length) {
this.components = comps
} else {
throw new Error('Vector: invalide size!')
}
}
} // end of constructor
// returns the size of this vector.
// not the eulidean length!
Vector.prototype.size = function () {
return this.components.length
}
// computes the eulidean length.
Vector.prototype.eulideanLength = function () {
var sum = 0
for (var i = 0; i < this.components.length; i++) {
sum += this.components[i] * this.components[i]
}
return Math.sqrt(sum)
}
// getter for the components of the vector.
// returns a specified component (index)
Vector.prototype.component = function (index) {
return this.components[index]
}
// setter for a specified component of this vector.
Vector.prototype.changeComponent = function (index, value) {
if (index >= 0 && index < this.components.length) {
this.components[index] = value
} else {
throw new Error('changeComponent: index out of bounds!')
}
}
// vector addition
Vector.prototype.add = function (other) {
if (this.size() === other.size()) {
var SIZE = this.size()
var ans = new Vector(SIZE)
for (var i = 0; i < SIZE; i++) {
ans.changeComponent(i, (this.components[i] + other.component(i)))
}
return ans
} else {
throw new Error('add: vector must have same size!')
}
}
// vector subtraction
Vector.prototype.sub = function (other) {
if (this.size() === other.size()) {
var SIZE = this.size()
var ans = new Vector(SIZE)
for (var i = 0; i < SIZE; i++) {
ans.changeComponent(i, (this.components[i] - other.component(i)))
}
return ans
} else {
throw new Error('add: vector must have same size!')
}
}
// dot-product
Vector.prototype.dot = function (other) {
var sum = 0
if (other.size() === this.size()) {
var SIZE = other.size()
for (var i = 0; i < SIZE; i++) {
sum += this.components[i] * other.component(i)
}
return sum
} else {
throw new Error('dot: vectors must have same size!')
}
}
// scalar multiplication
Vector.prototype.scalar = function (s) {
var SIZE = this.size()
var ans = new Vector(SIZE)
for (var i = 0; i < SIZE; i++) {
ans.changeComponent(i, (this.components[i] * s))
}
return ans
}
// returns a string representation of this vector.
Vector.prototype.toString = function () {
var ans = '('
var SIZE = this.components.length
for (var i = 0; i < SIZE; i++) {
if (i < SIZE - 1) {
ans += this.components[i] + ','
} else {
ans += this.components[i] + ')'
}
}
return ans
}
// converts this vector in a unit basis vector and returns it.
// the One is on position 'pos'
Vector.prototype.createUnitBasis = function (pos) {
if (pos >= 0 && pos < this.components.length) {
for (var i = 0; i < this.components.length; i++) {
if (i === pos) {
this.components[i] = 1.0
} else {
this.components[i] = 0.0
}
}
} else {
throw new Error('createUnitBasis: index out of bounds')
}
return this
}
// normalizes this vector and returns it.
Vector.prototype.norm = function () {
var SIZE = this.size()
var quotient = 1.0 / this.eulideanLength()
for (var i = 0; i < SIZE; i++) {
this.components[i] = this.components[i] * quotient
}
return this
}
// returns true if the vectors are equal otherwise false.
Vector.prototype.equal = function (other) {
var ans = true
var SIZE = this.size()
var EPSILON = 0.001
if (SIZE === other.size()) {
for (var i = 0; i < SIZE; i++) {
if (Math.abs(this.components[i] - other.component(i)) > EPSILON) {
ans = false
}
}
} else {
ans = false
}
return ans
}
return Vector
}()) // end of class Vector
LinearAlgebra.Vector = Vector
// -------------- global functions ---------------------------------
// returns a unit basis vector of size N with a One on position 'pos'
function unitBasisVector (N, pos) {
var ans = new Vector(N)
for (var i = 0; i < N; i++) {
if (i === pos) {
ans.changeComponent(i, 1.0)
} else {
ans.changeComponent(i, 0)
}
}
return ans
}
LinearAlgebra.unitBasisVector = unitBasisVector
// returns a random vector with integer components (between 'a' and 'b') of size N.
function randomVectorInt (N, a, b) {
var ans = new Vector(N)
for (var i = 0; i < N; i++) {
ans.changeComponent(i, (Math.floor((Math.random() * b) + a)))
}
return ans
}
LinearAlgebra.randomVectorInt = randomVectorInt
// returns a random vector with floating point components (between 'a' and 'b') of size N.
function randomVectorFloat (N, a, b) {
var ans = new Vector(N)
for (var i = 0; i < N; i++) {
ans.changeComponent(i, ((Math.random() * b) + a))
}
return ans
}
LinearAlgebra.randomVectorFloat = randomVectorFloat
// ------------------ end of global functions -----------------------------
/*
class: Matrix
This class represents a matrix of arbitrary size and operations on it.
*/
var Matrix = /** @class */ (function () {
// constructor for zero-matrix or fix number matrix.
function Matrix (row, col, comps) {
if (comps === undefined) {
comps = []
}
if (comps.length === 0) {
this.matrix = []
var rowVector = []
for (var i = 0; i < row; i++) {
for (var j = 0; j < col; j++) {
rowVector[j] = 0
}
this.matrix[i] = rowVector
rowVector = []
}
} else {
this.matrix = comps
}
this.rows = row
this.cols = col
}
// returns the specified component.
Matrix.prototype.component = function (x, y) {
if (x >= 0 && x < this.rows && y >= 0 && y < this.cols) {
return this.matrix[x][y]
} else {
throw new Error('component: index out of bounds')
}
}
// changes the specified component with value 'value'.
Matrix.prototype.changeComponent = function (x, y, value) {
if (x >= 0 && x < this.rows && y >= 0 && y < this.cols) {
this.matrix[x][y] = value
} else {
throw new Error('changeComponent: index out of bounds')
}
}
// returns a string representation of this matrix.
Matrix.prototype.toString = function () {
var ans = ''
for (var i = 0; i < this.rows; i++) {
ans += '|'
for (var j = 0; j < this.cols; j++) {
if (j < this.cols - 1) {
ans += this.matrix[i][j] + ','
} else {
if (i < this.rows - 1) {
ans += this.matrix[i][j] + '|\n'
} else {
ans += this.matrix[i][j] + '|'
}
}
}
}
return ans
}
// returns the dimension rows x cols as number array
Matrix.prototype.dimension = function () {
var ans = []
ans[0] = this.rows
ans[1] = this.cols
return ans
}
// matrix addition. returns the result.
Matrix.prototype.add = function (other) {
if (this.rows === other.dimension()[0] &&
this.cols === other.dimension()[1]) {
var ans = new Matrix(this.rows, this.cols)
for (var i = 0; i < this.rows; i++) {
for (var j = 0; j < this.cols; j++) {
ans.changeComponent(i, j, (this.matrix[i][j] + other.component(i, j)))
}
}
return ans
} else {
throw new Error('add: matrices must have same dimension!')
}
}
// returns true if the matrices are equal, otherwise false.
Matrix.prototype.equal = function (other) {
var ans = true
var EPSILON = 0.001
if (this.rows === other.dimension()[0] &&
this.cols === other.dimension()[1]) {
for (var i = 0; i < this.rows; i++) {
for (var j = 0; j < this.cols; j++) {
if (Math.abs(this.matrix[i][j] - other.component(i, j)) > EPSILON) {
ans = false
}
}
}
} else {
ans = false
}
return ans
}
// matrix-scalar multiplication
Matrix.prototype.scalar = function (c) {
var ans = new Matrix(this.rows, this.cols)
for (var i = 0; i < this.rows; i++) {
for (var j = 0; j < this.cols; j++) {
ans.changeComponent(i, j, (this.matrix[i][j] * c))
}
}
return ans
}
return Matrix
}()) // end of class Matrix
LinearAlgebra.Matrix = Matrix
})(LinearAlgebra || (LinearAlgebra = {})) // end of namespace LinearAlgebra
