polynomial

Polynomial.js

Polynomials are defined as the sum of variables with increasing integer power and their coefficients in a certain field. For example the following might be still known from school:

P(x) = x^2 + 4x + 3

Examples

Adding two polynomials

const p = new Polynomial("3x^2").add("-x^2"); // 2x^2

Second derivative of polynomial

const p = new Polynomial("5+3x^3+6x^5").derive(2); // 120x^3+18x

Parser

Any function (see below) as well as the constructor of the Polynomial class parses its input like this.

You can pass either Objects, Doubles or Strings. Make sure strings don't contain any white-spaces or brackets. The parser doesn't analyse the string recursively.

Objects

new Polynomial({'3': 4, '5': '9'}); // 9x^5+4x^3
new Polynomial([1,2,3]); //3x^2+2x+1

Doubles

new Polynomial(55); // 55x^0

Strings

new Polynomial("98x^2+4+23x^4");

The string parser passes every coefficient directly to the field parser, which allows to pass complex and rational coefficients as well:

// Example with complex numbers
Polynomial.setField("C");
new Polynomial("98x^2+i+23ix^4");

// Example with rational numbers
Polynomial.setField("Q");
new Polynomial("5/3x^3+4/3x");

Fields

Polynomial.js is held general in order to operate on various fields. Fraction.js and Complex.js build the perfect base to extend polynomials to rational and complex numbers.

• ℚ: Rational numbers supported by Fraction.js
• ℂ: Complex numbers supported by Complex.js
• ℍ: Quaternions supported by Quaternion.js
• ℤ_p: Field of integers mod p, with p prime
• ℝ: Field of real numbers

Accessing Coefficients

const p = new Polynomial("98x^2+4+23x^4");

console.log(p.coeff);

Examples

Polynomial.setField("Q");
Polynomial("3/2x^2-4x").mod("5x"); // 0

Polynomial.setField("Z11");
Polynomial("3x^2+x+7").gcd("3x^2+x+7"); // x^2+4x+6

Polynomial.setField("Z7");
Polynomial("9x^2+4").pow(3); // x^6+6x^4+5x^2+1

Polynomial.setField("R");
Polynomial("3x^3-1").mul(4); // 12x^3-4

// Derivative of polynomial
Polynomial.setField("Q");
Polynomial("5+3x^3+6x^5").derive(); // 30x^4+9x^2

// Integrated polynomial
Polynomial.setField("Q");
Polynomial("3x^2").integrate(); // x^3

Functions

Polynomial add(n)

Returns the sum of the actual polynomial and the parameter n

Polynomial sub(n)

Returns the difference of the actual polynomial and the parameter n

Polynomial mul(n)

Returns the product of the actual polynomial and the parameter n

Polynomial addmul(x, y)

Adds the product of x and y to the actual number

Polynomial div(n)

Returns the quotient of the actual polynomial and the parameter n

There is a global variable to enable division tracing like this, if you want to output details:

Polynomial.trace = true;
new Polynomial("x^4+3x^3+2x^2+6x")
        .div("x+3");
console.log(Polynomial.trace.map(x => x.toString())); // ["x^4+3x^3", "2x^2+6x", "0"]

Polynomial neg(n)

Returns the negated polynomial

Polynomial reciprocal(n)

Returns the reciprocal polynomial

Polynomial lc()

Gets the leading coefficient

Polynomial lm()

Gets the leading monomial

Polynomial monic()

Divide all coefficients of the polynomial by lc()

Polynomial derive(n)

Returns the n-th derivative of the polynomial

Polynomial integrate(n)

Returns the n-th integration of the polynomial

mixed eval(x)

Evaluate the polynomial at point x, using Horner's method. Type for x must be a valid value for the given field.

mixed result(x)

(Deprecated) Alias for eval.

Polynomial pow(exp)

Returns the power of the actual polynomial, raised to an integer exponent.

Polynomial mod(n)

Returns the modulus (rest of the division) of the actual polynomial and n (this % n).

Polynomial gcd(n)

Returns the greatest common divisor of two polynomials

Number degree()

Returns the degree of the polynomial

String toString()

Generates a string representation of the actual polynomial. This makes use of the toString() function of the field.

String toLatex()

Generates a LaTeX representation of the actual polynomial.

String toHorner()

Formats the actual polynomial to a Horner Scheme

Polynomial clone()

Creates a copy of the actual Polynomial object

Polynomial Polynomial::fromRoots(roots)

Creates a new (monic) Polynomial whose roots lie at the values provided in the array roots

Polynomial::setField(x)

Sets the field globally. Choose one of the following strings for x:

• "R": real numbers
• "Q": rational numbers
• "H": quaternions
• "C": complex numbers
• "Zp": with p being a prime number, like "Z7"
• or an object with the field operators. See examples folders for BigInt

Exceptions

If a really hard error occurs (parsing error, division by zero), polynomial.js throws exceptions! Please make sure you handle them correctly.

Installation

Installing Polynomial.js is as easy as cloning this repo or use one of the following command:

npm install polynomial

Using Polynomial.js with the browser

<script src="fraction.min.js"></script> <!-- Needed for field/ring Q -->
<script src="complex.min.js"></script> <!-- Needed for field C -->
<script src="polynomial.min.js"></script>
<script>
Polynomial.setField("C")
console.log(Polynomial("4x+3i"));
</script>

Coding Style

As every library I publish, Polynomial.js is also built to be as small as possible after compressing it with Google Closure Compiler in advanced mode. Thus the coding style orientates a little on maxing-out the compression rate. Please make sure you keep this style if you plan to extend the library.

Building the library

After cloning the Git repository run:

npm install
npm run build

Run a test

Testing the source against the shipped test suite is as easy as

npm run test

Copyright and Licensing

Copyright (c) 2024, Robert Eisele Licensed under the MIT license.