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Schmidt","url":"https://thi.ng"},"license":"Apache-2.0","homepage":"https://thi.ng/dual-algebra","keywords":["algebra","datastructure","derivative","descend","differentiation","dual","gradient","math","multivariate","typescript","vector"],"repository":{"type":"git","url":"git+https://codeberg.org/thi.ng/umbrella.git","directory":"packages/dual-algebra"},"description":"Multivariate dual number algebra, automatic differentiation","maintainers":[{"name":"thi.ng","email":"k+npm@thi.ng"}],"readme":"<!-- This file is generated - DO NOT EDIT! -->\n<!-- Please see: https://codeberg.org/thi.ng/umbrella/src/branch/develop/CONTRIBUTING.md#changes-to-readme-files -->\n# ![@thi.ng/dual-algebra](https://codeberg.org/thi.ng/umbrella/media/branch/develop/assets/banners/thing-dual-algebra.svg?0530ab72)\n\n[![npm version](https://img.shields.io/npm/v/@thi.ng/dual-algebra.svg)](https://www.npmjs.com/package/@thi.ng/dual-algebra)\n![npm downloads](https://img.shields.io/npm/dm/@thi.ng/dual-algebra.svg)\n[![Mastodon Follow](https://img.shields.io/mastodon/follow/109331703950160316?domain=https%3A%2F%2Fmastodon.thi.ng&style=social)](https://mastodon.thi.ng/@toxi)\n\n> [!NOTE]\n\n> This is one of 217 standalone projects. LLM-free, human-made and\n> cared for software, maintained as part of the\n> [@thi.ng/umbrella](https://codeberg.org/thi.ng/umbrella/) ecosystem and\n> anti-framework.\n>\n> 🚀 Please help me to work full-time on these projects by [sponsoring\n> me](https://codeberg.org/thi.ng/umbrella/src/branch/develop/CONTRIBUTING.md#donations).\n> Thank you! ❤️\n\n- [About](#about)\n- [Status](#status)\n- [Related packages](#related-packages)\n- [Installation](#installation)\n- [Dependencies](#dependencies)\n- [Usage examples](#usage-examples)\n- [API](#api)\n- [Authors](#authors)\n- [License](#license)\n\n## About\n\nMultivariate dual number algebra, automatic differentiation.\n\n- [Wikipedia: Dual numbers](https://en.wikipedia.org/wiki/Dual_number)\n- [Wikipedia: Automatic_differentiation](https://en.wikipedia.org/wiki/Automatic_differentiation#Automatic_differentiation_using_dual_numbers)\n\n(Package name with hat tip to [@paniq](https://www.shadertoy.com/view/4dVGzw))\n\nDual numbers are an elegant solution to compute **precise**<sup>(1)</sup> derivatives of\nfunctions which otherwise require complex & brittle numerical solutions.\nFurthermore, multivariate dual numbers can be used to obtain (in parallel)\nderivatives of multiple variables within a single function execution.\n\nIn this package, dual numbers are encoded as vanilla JS arrays with the internal\nstructure: `[real, d1 .. dn]`, where `real` is the real-valued part of the\nnumber and `d1`..`dn` multivariate derivatives. At minimum, at least `d1`\nexists, but the number (of derivatives) depends on usage and the number of\nvariables in a function one wishes to compute derivatives for.\n\n<small><sup>(1)</sup> Here *\"precise\"* within the realm of IEEE-754</small>\n\nSome examples (see further below for code example):\n\n```ts\n[Math.PI, 0] // the scalar π as 1-dual number\n[Math.PI, 1] // π as the current value of a 1-dual variable\n\n[5, 1, 0] // 5 as first variable in 2-variable function\n[3, 0, 1] // 3 as second variable in a 2-var function\n\n[5, 1, 0, 0] // 1st var in 3-var fn\n[3, 0, 1, 0] // 2nd var in 3-var fn\n[2, 0, 0, 1] // 3rd var in 3-var fn\n```\n\nAlternatively, use convenience fns to create dual numbers:\n\n```ts\nimport { $, $2, $3, dual } from \"@thi.ng/dual-algebra\";\n\n$(5)     // [5, 0]\n$(5, 1)  // [5, 1]\n\n$2(5)    // [5, 0, 0]\n$2(5, 2) // [5, 0, 1]\n\n$3(5)    // [5, 0, 0, 0]\n$3(5, 2) // [5, 0, 1, 0]\n\ndual(5, 6)    // [5, 0, 0, 0, 0, 0, 0]\ndual(5, 6, 4) // [5, 0, 0, 0, 1, 0, 0]\n```\n\nThe following operations are available so far. Each operation takes one or more\nmultivariate dual number(s) and computes the actual real-valued results as well\nas the 1st derivatives. Each op has an optimized/loop-free impl for 1-dual\nnumbers.\n\n- `add(a, b)`\n- `sub(a, b)`\n- `mul(a, b)`\n- `div(a, b)`\n- `neg(a)`\n- `abs(a)`\n\nExponentials:\n\n- `pow(a, k)` (k = scalar)\n- `sqrt(a)`\n- `exp(a)`\n- `log(a)`\n\nTrigonometry:\n\n- `sin(a)`\n- `cos(a)`\n- `tan(a)`\n- `atan(a)`\n\nPolynomials:\n\n- `quadratic(x, a, b, c)` ⇒ _ax^2 + bx + c_\n- `cubic(x, a, b, c, d)` ⇒ _ax^3 + bx^2 + cx + d_\n- `quartic(x, a, b, c, d, e)` ⇒ _ax^4 + bx^3 + cx^2 + dx + e_\n\nFor each polynomial, there're scalar versions available too, taking only\nrational numbers as arguments (rather than dual numbers already). These versions\nare suffixed with `S` (for \"scalar\"): `quadraticS`, `cubicS` and `quarticS`...\n\n## Status\n\n**ALPHA** - bleeding edge / work-in-progress\n\n[Search or submit any issues for this package](https://codeberg.org/thi.ng/umbrella/issues?q=%5Bdual-algebra%5D)\n\n## Related packages\n\n- [@thi.ng/math](https://codeberg.org/thi.ng/umbrella/src/branch/develop/packages/math) - Assorted common math functions & utilities\n\n## Installation\n\n```bash\nyarn add @thi.ng/dual-algebra\n```\n\nESM import:\n\n```ts\nimport * as da from \"@thi.ng/dual-algebra\";\n```\n\nBrowser ESM import:\n\n```html\n<script type=\"module\" src=\"https://esm.run/@thi.ng/dual-algebra\"></script>\n```\n\n[JSDelivr documentation](https://www.jsdelivr.com/)\n\nFor Node.js REPL:\n\n```js\nconst da = await import(\"@thi.ng/dual-algebra\");\n```\n\nPackage sizes (brotli'd, pre-treeshake): ESM: 992 bytes\n\n## Dependencies\n\n- [@thi.ng/api](https://codeberg.org/thi.ng/umbrella/src/branch/develop/packages/api)\n\nNote: @thi.ng/api is in _most_ cases a type-only import (not used at runtime)\n\n## Usage examples\n\nOne project in this repo's\n[/examples](https://codeberg.org/thi.ng/umbrella/src/branch/develop/examples)\ndirectory is using this package:\n\n| Screenshot                                                                                                            | Description                                                | Live demo                                            | Source                                                                                    |\n|:----------------------------------------------------------------------------------------------------------------------|:-----------------------------------------------------------|:-----------------------------------------------------|:------------------------------------------------------------------------------------------|\n| <img src=\"https://codeberg.org/thi.ng/umbrella/media/branch/develop/assets/examples/spline-tangent.png\" width=\"240\"/> | 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) |\n\n## API\n\n[Generated API docs](https://docs.thi.ng/umbrella/dual-algebra/)\n\n```ts\nimport { $2, add, mul, neg, sin, evalFn2 } from \"@thi.ng/dual-algebra\";\n\n// compute the actual result and derivatives of X & Y\n// of this function with 2 variables:\n// z = -x^2 + 3 * sin(y)\n\nconst f = (x: number, y: number) => {\n    // convert to multivariate dual numbers\n    const xx = $2(x, 1);\n    const yy = $2(y, 2);\n    // compute...\n    return add(neg(mul(xx, xx)), mul($2(3), sin(yy)));\n}\n\n// `evalFn2()` is higher order fn syntax sugar to simplify\n// dealing w/ scalars, here same with that wrapper:\nconst g = evalFn2((x, y) => add(neg(mul(x, x)), mul($2(3), sin(y))));\n\nf(0, 0);\n// [0, 0, 3] => [f(x,y), dFdx(f(x,y)), dFdy(f(x,y))]\n\ng(0, 0);\n// [0, 0, 3]\n\nf(1, Math.PI);\n// [-0.9999999999999997, -2, -3]\n```\n\nPolynomial example (see [interactive\ngraph](https://www.desmos.com/calculator/5ot2dpgv0a) of this function):\n\n```ts\nimport { add, mul, pow, cubicS } from \"@thi.ng/dual-algebra\";\n\n// compute the cubic polynomial: f(x) = 2x^3 - 3x^2 - 4x + 5\n\n// using `cubicS()` polynomial helper\nconst f1 = (x: number) => cubicS(x, 2, -3, -4, 5);\n\n// ...or expanded out\nconst f2 = (x: number) =>\n    add(\n        add(\n            add(\n                mul([2, 0], pow([x, 1], 3)),\n                mul([-3, 0], pow([x, 1], 2))\n            ),\n            mul([-4, 0], [x, 1])\n        ),\n        [5, 0]\n    );\n\nf2(0) // [5, -4] [f(x), dFdx(f(x))]\nf2(1) // [0, -4]\nf2(2) // [1, 8]\n```\n\n## Authors\n\n- [Karsten Schmidt](https://thi.ng)\n\nIf this project contributes to an academic publication, please cite it as:\n\n```bibtex\n@misc{thing-dual-algebra,\n  title = \"@thi.ng/dual-algebra\",\n  author = \"Karsten Schmidt\",\n  note = \"https://thi.ng/dual-algebra\",\n  year = 2020\n}\n```\n\n## License\n\n&copy; 2020 - 2026 Karsten Schmidt // Apache License 2.0\n","readmeFilename":"README.md"}