# 
[](https://www.npmjs.com/package/@thi.ng/dual-algebra)
[](https://mastodon.thi.ng/@toxi)
> [!NOTE]
> This is one of 215 standalone projects. LLM-free, human-made and
> cared for software, maintained as part of the
> [@thi.ng/umbrella](https://codeberg.org/thi.ng/umbrella/) ecosystem and
> anti-framework.
>
> 🚀 Please help me to work full-time on these projects by [sponsoring
> me](https://codeberg.org/thi.ng/umbrella/src/branch/develop/CONTRIBUTING.md#donations).
> Thank you! ❤️
- [About](#about)
- [Status](#status)
- [Related packages](#related-packages)
- [Installation](#installation)
- [Dependencies](#dependencies)
- [Usage examples](#usage-examples)
- [API](#api)
- [Authors](#authors)
- [License](#license)
## About
Multivariate dual number algebra, automatic differentiation.
- [Wikipedia: Dual numbers](https://en.wikipedia.org/wiki/Dual_number)
- [Wikipedia: Automatic_differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers)
(Package name with hat tip to [@paniq](https://www.shadertoy.com/view/4dVGzw))
Dual numbers are an elegant solution to compute **precise**(1) derivatives of
functions which otherwise require complex & brittle numerical solutions.
Furthermore, multivariate dual numbers can be used to obtain (in parallel)
derivatives of multiple variables within a single function execution.
In this package, dual numbers are encoded as vanilla JS arrays with the internal
structure: `[real, d1 .. dn]`, where `real` is the real-valued part of the
number and `d1`..`dn` multivariate derivatives. At minimum, at least `d1`
exists, but the number (of derivatives) depends on usage and the number of
variables in a function one wishes to compute derivatives for.
(1) Here *"precise"* within the realm of IEEE-754
Some examples (see further below for code example):
```ts
[Math.PI, 0] // the scalar π as 1-dual number
[Math.PI, 1] // π as the current value of a 1-dual variable
[5, 1, 0] // 5 as first variable in 2-variable function
[3, 0, 1] // 3 as second variable in a 2-var function
[5, 1, 0, 0] // 1st var in 3-var fn
[3, 0, 1, 0] // 2nd var in 3-var fn
[2, 0, 0, 1] // 3rd var in 3-var fn
```
Alternatively, use convenience fns to create dual numbers:
```ts
import { $, $2, $3, dual } from "@thi.ng/dual-algebra";
$(5) // [5, 0]
$(5, 1) // [5, 1]
$2(5) // [5, 0, 0]
$2(5, 2) // [5, 0, 1]
$3(5) // [5, 0, 0, 0]
$3(5, 2) // [5, 0, 1, 0]
dual(5, 6) // [5, 0, 0, 0, 0, 0, 0]
dual(5, 6, 4) // [5, 0, 0, 0, 1, 0, 0]
```
The following operations are available so far. Each operation takes one or more
multivariate dual number(s) and computes the actual real-valued results as well
as the 1st derivatives. Each op has an optimized/loop-free impl for 1-dual
numbers.
- `add(a, b)`
- `sub(a, b)`
- `mul(a, b)`
- `div(a, b)`
- `neg(a)`
- `abs(a)`
Exponentials:
- `pow(a, k)` (k = scalar)
- `sqrt(a)`
- `exp(a)`
- `log(a)`
Trigonometry:
- `sin(a)`
- `cos(a)`
- `tan(a)`
- `atan(a)`
Polynomials:
- `quadratic(x, a, b, c)` ⇒ _ax^2 + bx + c_
- `cubic(x, a, b, c, d)` ⇒ _ax^3 + bx^2 + cx + d_
- `quartic(x, a, b, c, d, e)` ⇒ _ax^4 + bx^3 + cx^2 + dx + e_
For each polynomial, there're scalar versions available too, taking only
rational numbers as arguments (rather than dual numbers already). These versions
are suffixed with `S` (for "scalar"): `quadraticS`, `cubicS` and `quarticS`...
## Status
**ALPHA** - bleeding edge / work-in-progress
[Search or submit any issues for this package](https://codeberg.org/thi.ng/umbrella/issues?q=%5Bdual-algebra%5D)
## Related packages
- [@thi.ng/math](https://codeberg.org/thi.ng/umbrella/src/branch/develop/packages/math) - Assorted common math functions & utilities
## Installation
```bash
yarn add @thi.ng/dual-algebra
```
ESM import:
```ts
import * as da from "@thi.ng/dual-algebra";
```
Browser ESM import:
```html
```
[JSDelivr documentation](https://www.jsdelivr.com/)
For Node.js REPL:
```js
const da = await import("@thi.ng/dual-algebra");
```
Package sizes (brotli'd, pre-treeshake): ESM: 992 bytes
## Dependencies
- [@thi.ng/api](https://codeberg.org/thi.ng/umbrella/src/branch/develop/packages/api)
Note: @thi.ng/api is in _most_ cases a type-only import (not used at runtime)
## Usage examples
One project in this repo's
[/examples](https://codeberg.org/thi.ng/umbrella/src/branch/develop/examples)
directory is using this package:
| Screenshot | Description | Live demo | Source |
|:----------------------------------------------------------------------------------------------------------------------|:-----------------------------------------------------------|:-----------------------------------------------------|:------------------------------------------------------------------------------------------|
| 
| Compute cubic spline position & tangent using Dual Numbers | [Demo](https://demo.thi.ng/umbrella/spline-tangent/) | [Source](https://codeberg.org/thi.ng/umbrella/src/branch/develop/examples/spline-tangent) |
## API
[Generated API docs](https://docs.thi.ng/umbrella/dual-algebra/)
```ts
import { $2, add, mul, neg, sin, evalFn2 } from "@thi.ng/dual-algebra";
// compute the actual result and derivatives of X & Y
// of this function with 2 variables:
// z = -x^2 + 3 * sin(y)
const f = (x: number, y: number) => {
// convert to multivariate dual numbers
const xx = $2(x, 1);
const yy = $2(y, 2);
// compute...
return add(neg(mul(xx, xx)), mul($2(3), sin(yy)));
}
// `evalFn2()` is higher order fn syntax sugar to simplify
// dealing w/ scalars, here same with that wrapper:
const g = evalFn2((x, y) => add(neg(mul(x, x)), mul($2(3), sin(y))));
f(0, 0);
// [0, 0, 3] => [f(x,y), dFdx(f(x,y)), dFdy(f(x,y))]
g(0, 0);
// [0, 0, 3]
f(1, Math.PI);
// [-0.9999999999999997, -2, -3]
```
Polynomial example (see [interactive
graph](https://www.desmos.com/calculator/5ot2dpgv0a) of this function):
```ts
import { add, mul, pow, cubicS } from "@thi.ng/dual-algebra";
// compute the cubic polynomial: f(x) = 2x^3 - 3x^2 - 4x + 5
// using `cubicS()` polynomial helper
const f1 = (x: number) => cubicS(x, 2, -3, -4, 5);
// ...or expanded out
const f2 = (x: number) =>
add(
add(
add(
mul([2, 0], pow([x, 1], 3)),
mul([-3, 0], pow([x, 1], 2))
),
mul([-4, 0], [x, 1])
),
[5, 0]
);
f2(0) // [5, -4] [f(x), dFdx(f(x))]
f2(1) // [0, -4]
f2(2) // [1, 8]
```
## Authors
- [Karsten Schmidt](https://thi.ng)
If this project contributes to an academic publication, please cite it as:
```bibtex
@misc{thing-dual-algebra,
title = "@thi.ng/dual-algebra",
author = "Karsten Schmidt",
note = "https://thi.ng/dual-algebra",
year = 2020
}
```
## License
© 2020 - 2026 Karsten Schmidt // Apache License 2.0