# polynomial

A polynomial library

## Usage no npm install needed!

``````<script type="module">
import polynomial from 'https://cdn.skypack.dev/polynomial';
</script>``````

# 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

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

## Second derivative of polynomial

``````var 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
• p: Field of integers mod p, with p prime
• ℝ: Field of real numbers

## 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

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

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 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 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 commands:

``````bower install polynomial.js
``````

or

``````npm install polynomial
``````

# Using Polynomial.js with the browser

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

# Using Polynomial.js with require.js

``````<script src="require.js"></script>
<script>
requirejs(['polynomial.js'],
function(Polynomial) {
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.

# Testing

If you plan to enhance the library, make sure you add test cases and all the previous tests are passing. You can test the library with

``````npm test
``````