Overview

The focus is to find real roots of real coefficient polynomials from degree 1 up to about degree 20 as accurately and as fast as possible, e.g.

// some polynomial with double precision coefficients, i.e. x^6 - 21x^5 + 175x^4 - 735x^3 + 1624x^2 - 1764 + 720
const p = [1, -21, 175, -735, 1624, -1764, 720];  
allRoots(p); //=> [0.9999999999999997, 2.0000000000000013, 2.9999999999999316, 4.00000000000096, 5.000000000000012, 6.00000000000028]

However, the above function, allRoots, does not take error bounds into account and can thus be inaccurate if the roots have high condition numbers.

For extremely accurate (no matter how high the condition number) certified results use e.g.:

const p = [1, -21, 175, -735, 1624, -1764, 720];  // some polynomial with double precision coefficients
allRootsCertifiedSimplified(p);

or for a more flexible function that takes the input polynomial coefficients as double-double precision and the ability to specify error bounds on the coefficients in addition to a fallback function to specify exact coefficients (in the form of Shewchuk expansions) use allRootsCertified.

Though the focus is on root finding, the library include numerous useful operators on polynomials with double, double-double, Shewchuk expansion and bigint coefficients, e.g

add([1,2,3], [3,4]); //=> [1,5,7]

Why only up to about degree 20?

This isn't a hard limit. Roughly speaking, since the roots are found using Rolle's Theorem it becomes asymptotically slower (compared to Descartes Methods), i.e. roughly O(n²) vs O(n) the higher the polynomial degree, n.

Another reason is that evaluation of the polynomial at x when |x| >> 1 can result in overflow when the result is larger than about 1.8 x 10^308 (the max value a double precision floating point value can be).

Documentation

For more in-depth documentation please read the docs!.

Installation

npm install flo-poly

This package is ESM only and can be used in Node.js (or in a browser when bundled using e.g. Webpack).

Additionally, self-contained ECMAScript Module (ESM) files index.js and index.min.js in the ./browser folder is provided.

Usage

Node.js

import { allRoots } from 'flo-poly';

// some polynomial with double precision coefficients, i.e. x^6 - 21x^5 + 175x^4 - 735x^3 + 1624x^2 - 1764 + 720
const p = [1, -21, 175, -735, 1624, -1764, 720];  
const roots = allRoots(p);

if (roots.length === 6) {
    console.log('success! 😁');  // we should get to here!
} else {
    console.log('failure! 😥');  // ...and not here
}

Browsers - directly, without a bundler, using the pre-bundled minified .js file

Please note that no tree shaking will take place in this case.

<!doctype html>

<html lang="en">
<head>
    <script type="module">
        import { allRoots } from "./node_modules/flo-poly/browser/index.min.js";

        // some polynomial with double precision coefficients, i.e. x^6 - 21x^5 + 175x^4 - 735x^3 + 1624x^2 - 1764 + 720
        const p = [1, -21, 175, -735, 1624, -1764, 720];  
        const roots = allRoots(p);

        if (roots.length === 6) {
            console.log('success! 😁');  // we should get to here!
        } else {
            console.log('failure! 😥');  // ...and not here
        }
    </script>
</head>

<body>Check the console.</body>

</html>

Bundlers (Webpack, Rollup, ...)

Webpack will be taken as an example here.

Since your webpack config file might still use CommonJS you must rename webpack.config.js to webpack.config.cjs.

If you are using TypeScript:

Since this is an ESM only library you must use resolve-typescript-plugin (at least until webpack catches up with ESM?) in your webpack.config.cjs file.

npm install --save-dev resolve-typescript-plugin

and follow the instructions given at resolve-typescript-plugin.

Additionally, follow this guide.

FlorisSteenkamp/FloPoly