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Stdlib Authors","url":"https://github.com/stdlib-js/stdlib/graphs/contributors"},"license":"Apache-2.0 AND BSL-1.0","homepage":"https://stdlib.io","keywords":["stdlib","stdmath","mathematics","math","special function","special","function","gamma","incomplete gamma","approximation","scalar","number"],"repository":{"type":"git","url":"git://github.com/stdlib-js/math-base-special-gammainc.git"},"description":"Incomplete gamma function.","contributors":[{"name":"The Stdlib Authors","url":"https://github.com/stdlib-js/stdlib/graphs/contributors"}],"maintainers":[{"name":"stdlib-bot","email":"kgryte+stdlibbot@gmail.com"},{"name":"kgryte","email":"kgryte@gmail.com"},{"name":"planeshifter","email":"pgb@andrew.cmu.edu"},{"name":"rreusser","email":"rsreusser@gmail.com"}],"readme":"<!--\n\n@license Apache-2.0\n\nCopyright (c) 2018 The Stdlib Authors.\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n   http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n\n-->\n\n\n<details>\n  <summary>\n    About stdlib...\n  </summary>\n  <p>We believe in a future in which the web is a preferred environment for numerical computation. To help realize this future, we've built stdlib. stdlib is a standard library, with an emphasis on numerical and scientific computation, written in JavaScript (and C) for execution in browsers and in Node.js.</p>\n  <p>The library is fully decomposable, being architected in such a way that you can swap out and mix and match APIs and functionality to cater to your exact preferences and use cases.</p>\n  <p>When you use stdlib, you can be absolutely certain that you are using the most thorough, rigorous, well-written, studied, documented, tested, measured, and high-quality code out there.</p>\n  <p>To join us in bringing numerical computing to the web, get started by checking us out on <a href=\"https://github.com/stdlib-js/stdlib\">GitHub</a>, and please consider <a href=\"https://opencollective.com/stdlib\">financially supporting stdlib</a>. We greatly appreciate your continued support!</p>\n</details>\n\n# gammainc\n\n[![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] <!-- [![dependencies][dependencies-image]][dependencies-url] -->\n\n> [Incomplete gamma function][incomplete-gamma-function].\n\n<section class=\"intro\">\n\nEvaluates the regularized lower [incomplete gamma function][incomplete-gamma-function]:\n\n<!-- <equation class=\"equation\" label=\"eq:lower_incomplete_gamma\" align=\"center\" raw=\"P( x, a ) = \\frac{\\gamma(a,x)}{\\Gamma(a)} = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a-1} e^{-t} \\; dt\" alt=\"Regularized lower incomplete gamma function.\"> -->\n\n<div class=\"equation\" align=\"center\" data-raw-text=\"P( x, a ) = \\frac{\\gamma(a,x)}{\\Gamma(a)} = \\frac{1}{\\Gamma(a)} \\int_0^x t^{a-1} e^{-t} \\; dt\" data-equation=\"eq:lower_incomplete_gamma\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_lower_incomplete_gamma.svg\" alt=\"Regularized lower incomplete gamma function.\">\n    <br>\n</div> -->\n\n<!-- </equation> -->\n\nThe function can also be used to evaluate the regularized upper incomplete gamma function, which is defined as follows:\n\n<!-- <equation class=\"equation\" label=\"eq:upper_incomplete_gamma\" align=\"center\" raw=\"Q( x, a ) = \\frac{\\Gamma(a,x)}{\\Gamma(a)} = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a-1} e^{-t} \\; dt\" alt=\"Regularized upper incomplete gamma function.\"> -->\n\n<!-- <div class=\"equation\" align=\"center\" data-raw-text=\"Q( x, a ) = \\frac{\\Gamma(a,x)}{\\Gamma(a)} = \\frac{1}{\\Gamma(a)} \\int_x^\\infty t^{a-1} e^{-t} \\; dt\" data-equation=\"eq:upper_incomplete_gamma\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_upper_incomplete_gamma.svg\" alt=\"Regularized upper incomplete gamma function.\">\n    <br>\n</div> -->\n\n<!-- </equation> -->\n\nThe two functions have the relationship `Q(x,a) = 1 - P(x,a)`.\n\nIn addition, this routine can be used to evaluate the _unregularized_ gamma functions. The range of above functions is `[0,1]`, which is not the case for the _unregularized_ versions. The unregularized lower incomplete gamma function is defined as\n\n<!-- <equation class=\"equation\" label=\"eq:unreg_lower_incomplete_gamma\" align=\"center\" raw=\"\\gamma(a,x) = \\int_0^x t^{a-1} e^{-t} \\; dt\" alt=\"Unregularized lower incomplete gamma function.\"> -->\n\n<!-- <div class=\"equation\" align=\"center\" data-raw-text=\"\\gamma(a,x) = \\int_0^x t^{a-1} e^{-t} \\; dt\" data-equation=\"eq:unreg_lower_incomplete_gamma\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_unreg_lower_incomplete_gamma.svg\" alt=\"Unregularized lower incomplete gamma function.\">\n    <br>\n</div> -->\n\n<!-- </equation> -->\n\nand the upper unregularized incomplete gamma function is\n\n<!-- <equation class=\"equation\" label=\"eq:unreg_upper_incomplete_gamma\" align=\"center\" raw=\"\\Gamma(a,x)= \\int_x^\\infty t^{a-1} e^{-t} \\; dt\" alt=\"Unregularized upper incomplete gamma function.\"> -->\n\n<!-- <div class=\"equation\" align=\"center\" data-raw-text=\"\\Gamma(a,x)= \\int_x^\\infty t^{a-1} e^{-t} \\; dt\" data-equation=\"eq:unreg_upper_incomplete_gamma\">\n    <img src=\"https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gammainc/docs/img/equation_unreg_upper_incomplete_gamma.svg\" alt=\"Unregularized upper incomplete gamma function.\">\n    <br>\n</div>\n\n<!-- </equation> -->\n\nThe relationship between the two functions is `Γ(a,x) = γ(a,x) + Γ(a)`.\n\n</section>\n\n<!-- /.intro -->\n\n<section class=\"installation\">\n\n## Installation\n\n```bash\nnpm install @stdlib/math-base-special-gammainc\n```\n\n</section>\n\n<section class=\"usage\">\n\n## Usage\n\n```javascript\nvar gammainc = require( '@stdlib/math-base-special-gammainc' );\n```\n\n#### gammainc( x, s\\[, regularized\\[, upper ]] )\n\nBy default, evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `s`. The third and fourth parameters of the function can be used to specify whether instead to evaluate the non-regularized and/or upper incomplete gamma functions, respectively.\n\n```javascript\nvar y = gammainc( 6.0, 2.0 );\n// returns ~0.9826\n\ny = gammainc( 1.0, 2.0, true, true );\n// returns ~0.7358\n\ny = gammainc( 7.0, 5.0 );\n// returns ~0.8270\n\ny = gammainc( 7.0, 5.0, false );\n// returns ~19.8482\n```\n\nIf provided `NaN` as any argument, the function returns `NaN`.\n\n```javascript\nvar y = gammainc( NaN, 1.0 );\n// returns NaN\n\ny = gammainc( 1.0, NaN );\n// returns NaN\n```\n\nIf provided a negative `x`, the function returns `NaN`.\n\n```javascript\nvar y = gammainc( -2.0, 2.0 );\n// returns NaN\n```\n\nIf provided a nonpositive `s`, the function returns `NaN`.\n\n```javascript\nvar y = gammainc( 2.0, -1.0 );\n// returns NaN\n\ny = gammainc( 2.0, 0.0 );\n// returns NaN\n```\n\n</section>\n\n<!-- /.usage -->\n\n<section class=\"examples\">\n\n## Examples\n\n<!-- eslint no-undef: \"error\" -->\n\n```javascript\nvar uniform = require( '@stdlib/random-array-uniform' );\nvar logEachMap = require( '@stdlib/console-log-each-map' );\nvar gammainc = require( '@stdlib/math-base-special-gammainc' );\n\nvar opts = {\n    'dtype': 'float64'\n};\nvar x = uniform( 100, 0.0, 10.0, opts );\nvar s = uniform( 100, 0.0, 10.0, opts );\n\nlogEachMap( 'x: %0.4f, \\t s: %0.4f, \\t f(x,s): %0.4f', x, s, gammainc );\n```\n\n</section>\n\n<!-- /.examples -->\n\n<!-- C interface documentation. -->\n\n* * *\n\n<section class=\"c\">\n\n## C APIs\n\n<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->\n\n<section class=\"intro\">\n\n</section>\n\n<!-- /.intro -->\n\n<!-- C usage documentation. -->\n\n<section class=\"usage\">\n\n### Usage\n\n```c\n#include \"stdlib/math/base/special/gammainc.h\"\n```\n\n#### stdlib_base_gammainc( x, a, regularized, upper )\n\nEvaluates the [incomplete gamma function][incomplete-gamma-function] for inputs `x` and `a`.\n\n```c\ndouble out = stdlib_base_gammainc( 0.0, 1.0, true, false );\n// returns 0.0\n\nout = stdlib_base_gammainc( 6.0, 2.0, true, false );\n// returns ~0.9826\n```\n\nThe function accepts the following arguments:\n\n-   **x**: `[in] double` input value.\n-   **a**: `[in] double` input value.\n-   **regularized**: `[in] bool` boolean indicating whether to evaluate a regularized incomplete gamma function.\n-   **upper**: `[in] bool` boolean indicating whether to evaluate the upper incomplete gamma function.\n\n```c\ndouble stdlib_base_gammainc( const double x, const double a, const bool regularized, const bool upper );\n```\n\n</section>\n\n<!-- /.usage -->\n\n<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n<section class=\"notes\">\n\n</section>\n\n<!-- /.notes -->\n\n<!-- C API usage examples. -->\n\n<section class=\"examples\">\n\n### Examples\n\n```c\n#include \"stdlib/math/base/special/gammainc.h\"\n#include <stdlib.h>\n#include <stdio.h>\n\nstatic double random_uniform( const double min, const double max ) {\n    double v = (double)rand() / ( (double)RAND_MAX + 1.0 );\n    return min + ( v*(max-min) );\n}\n\nint main( void ) {\n    double x;\n    double a;\n    double y;\n    int i;\n\n    for ( i = 0; i < 10; i++ ) {\n        x = random_uniform( 0.0, 1000.0 );\n        a = random_uniform( 1.0, 1000.0 );\n        y = stdlib_base_gammainc( x, a, true, false );\n        printf( \"x: %lf, s: %lf, f(x,s): %lf\\n\", x, a, y );\n    }\n}\n```\n\n</section>\n\n<!-- /.examples -->\n\n</section>\n\n<!-- /.c -->\n\n<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->\n\n<section class=\"related\">\n\n* * *\n\n## See Also\n\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gamma`][@stdlib/math/base/special/gamma]</span><span class=\"delimiter\">: </span><span class=\"description\">gamma function.</span>\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gamma1pm1`][@stdlib/math/base/special/gamma1pm1]</span><span class=\"delimiter\">: </span><span class=\"description\">compute gamma(x+1) - 1.</span>\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gammaincinv`][@stdlib/math/base/special/gammaincinv]</span><span class=\"delimiter\">: </span><span class=\"description\">inverse incomplete gamma function.</span>\n-   <span class=\"package-name\">[`@stdlib/math-base/special/gammaln`][@stdlib/math/base/special/gammaln]</span><span class=\"delimiter\">: </span><span class=\"description\">natural logarithm of the gamma function.</span>\n\n</section>\n\n<!-- /.related -->\n\n<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n\n<section class=\"main-repo\" >\n\n* * *\n\n## Notice\n\nThis package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.\n\nFor more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].\n\n#### Community\n\n[![Chat][chat-image]][chat-url]\n\n---\n\n## Copyright\n\nCopyright &copy; 2016-2026. The Stdlib [Authors][stdlib-authors].\n\n</section>\n\n<!-- /.stdlib -->\n\n<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->\n\n<section class=\"links\">\n\n[npm-image]: http://img.shields.io/npm/v/@stdlib/math-base-special-gammainc.svg\n[npm-url]: https://npmjs.org/package/@stdlib/math-base-special-gammainc\n\n[test-image]: https://github.com/stdlib-js/math-base-special-gammainc/actions/workflows/test.yml/badge.svg?branch=v0.3.1\n[test-url]: https://github.com/stdlib-js/math-base-special-gammainc/actions/workflows/test.yml?query=branch:v0.3.1\n\n[coverage-image]: https://img.shields.io/codecov/c/github/stdlib-js/math-base-special-gammainc/main.svg\n[coverage-url]: https://codecov.io/github/stdlib-js/math-base-special-gammainc?branch=main\n\n<!--\n\n[dependencies-image]: https://img.shields.io/david/stdlib-js/math-base-special-gammainc.svg\n[dependencies-url]: https://david-dm.org/stdlib-js/math-base-special-gammainc/main\n\n-->\n\n[chat-image]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg\n[chat-url]: https://stdlib.zulipchat.com\n\n[stdlib]: https://github.com/stdlib-js/stdlib\n\n[stdlib-authors]: https://github.com/stdlib-js/stdlib/graphs/contributors\n\n[umd]: https://github.com/umdjs/umd\n[es-module]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Modules\n\n[deno-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/deno\n[deno-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/deno/README.md\n[umd-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/umd\n[umd-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/umd/README.md\n[esm-url]: https://github.com/stdlib-js/math-base-special-gammainc/tree/esm\n[esm-readme]: https://github.com/stdlib-js/math-base-special-gammainc/blob/esm/README.md\n[branches-url]: https://github.com/stdlib-js/math-base-special-gammainc/blob/main/branches.md\n\n[incomplete-gamma-function]: https://en.wikipedia.org/wiki/Incomplete_gamma_function\n\n<!-- <related-links> -->\n\n[@stdlib/math/base/special/gamma]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma\n\n[@stdlib/math/base/special/gamma1pm1]: https://www.npmjs.com/package/@stdlib/math-base-special-gamma1pm1\n\n[@stdlib/math/base/special/gammaincinv]: https://www.npmjs.com/package/@stdlib/math-base-special-gammaincinv\n\n[@stdlib/math/base/special/gammaln]: https://www.npmjs.com/package/@stdlib/math-base-special-gammaln\n\n<!-- </related-links> -->\n\n</section>\n\n<!-- /.links -->\n","readmeFilename":"README.md"}