time-to-botec

Benchmark sampling in different programming languages
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README.md (3772B)

      1 <!--
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      3 @license Apache-2.0
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      5 Copyright (c) 2018 The Stdlib Authors.
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      7 Licensed under the Apache License, Version 2.0 (the "License");
      8 you may not use this file except in compliance with the License.
      9 You may obtain a copy of the License at
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     11    http://www.apache.org/licenses/LICENSE-2.0
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     13 Unless required by applicable law or agreed to in writing, software
     14 distributed under the License is distributed on an "AS IS" BASIS,
     15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
     16 See the License for the specific language governing permissions and
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     19 -->
     20 
     21 # Gamma Scaled Lanczos Sum
     22 
     23 > Calculate a scaled Lanczos sum for the approximation of the [gamma function][gamma-function].
     24 
     25 <section class="intro">
     26 
     27 The [Lanczos approximation][lanczos-approximation] for the [gamma function][gamma-function] can be written in partial fraction form as follows:
     28 
     29 <!-- <equation class="equation" label="eq:lanczos_approximation" align="center" raw="\Gamma ( n ) = \frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)" alt="Lanczos approximation for gamma function."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="\Gamma ( n ) = \frac{(n+g-0.5)^{n-0.5}}{e^{n+g-0.5}} L_g(n)" data-equation="eq:lanczos_approximation">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_lanczos_approximation.svg" alt="Lanczos approximation for gamma function.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where `g` is an [arbitrary constant][@stdlib/constants/float64/gamma-lanczos-g] and `L_g(n)` is the Lanczos sum. The scaled Lanczos sum is given by 
     39 
     40 <!-- <equation class="equation" label="eq:scaled_lanczos_sum" align="center" raw="L_g(n) \cdot \exp(-g)" alt="Scaled Lanczos sum."> -->
     41 
     42 <div class="equation" align="center" data-raw-text="L_g(n) \cdot \exp(-g)" data-equation="eq:scaled_lanczos_sum">
     43     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled/docs/img/equation_scaled_lanczos_sum.svg" alt="Scaled Lanczos sum.">
     44     <br>
     45 </div>
     46 
     47 <!-- </equation> -->
     48 
     49 </section>
     50 
     51 <!-- /.intro -->
     52 
     53 <section class="usage">
     54 
     55 ## Usage
     56 
     57 ```javascript
     58 var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );
     59 ```
     60 
     61 #### gammaLanczosSumExpGScaled( x )
     62 
     63 Calculates the Lanczos sum for the approximation of the [gamma function][gamma-function] (scaled by `exp(-g)`, where `g = 10.900511`).
     64 
     65 ```javascript
     66 var v = gammaLanczosSumExpGScaled( 4.0 );
     67 // returns ~0.018
     68 
     69 v = gammaLanczosSumExpGScaled( -1.5 );
     70 // returns ~25.337
     71 
     72 v = gammaLanczosSumExpGScaled( -0.5 );
     73 // returns ~-12.911
     74 
     75 v = gammaLanczosSumExpGScaled( 0.5 );
     76 // returns ~1.772
     77 
     78 v = gammaLanczosSumExpGScaled( 0.0 );
     79 // returns Infinity
     80 
     81 v = gammaLanczosSumExpGScaled( NaN );
     82 // returns NaN
     83 ```
     84 
     85 </section>
     86 
     87 <!-- /.usage -->
     88 
     89 <section class="examples">
     90 
     91 ## Examples
     92 
     93 <!-- eslint no-undef: "error" -->
     94 
     95 ```javascript
     96 var linspace = require( '@stdlib/array/linspace' );
     97 var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );
     98 
     99 var x = linspace( -10.0, 10.0, 100 );
    100 var v;
    101 var i;
    102 
    103 for ( i = 0; i < x.length; i++ ) {
    104     v = gammaLanczosSumExpGScaled( x[ i ] );
    105     console.log( 'x: %d, f(x): %d', x[ i ], v );
    106 }
    107 ```
    108 
    109 </section>
    110 
    111 <!-- /.examples -->
    112 
    113 <section class="links">
    114 
    115 [@stdlib/constants/float64/gamma-lanczos-g]: https://www.npmjs.com/package/@stdlib/constants-float64-gamma-lanczos-g
    116 
    117 [gamma-function]: https://en.wikipedia.org/wiki/Gamma_function
    118 
    119 [lanczos-approximation]: https://en.wikipedia.org/wiki/Lanczos_approximation
    120 
    121 </section>
    122 
    123 <!-- /.links -->