time-to-botec

Benchmark sampling in different programming languages
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README.md (3403B)

      1 <!--
      2 
      3 @license Apache-2.0
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      5 Copyright (c) 2019 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
     17 limitations under the License.
     18 
     19 -->
     20 
     21 # ellipe
     22 
     23 > Compute the [complete elliptic integral of the second kind][elliptic-integral].
     24 
     25 <section class="intro">
     26 
     27 The [complete elliptic integral of the second kind][elliptic-integral] is defined as
     28 
     29 <!-- <equation class="equation" label="eq:complete_elliptic_integral_second_kind" align="center" raw="E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta" alt="Complete elliptic integral of the second kind."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="E(m)=\int_0^{\pi/2} \sqrt{1 - m (\sin\theta)^2} d\theta" data-equation="eq:complete_elliptic_integral_second_kind">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@129e5a32ce2af2ed694daf2e9d4214255e60c42a/lib/node_modules/@stdlib/math/base/special/ellipe/docs/img/equation_complete_elliptic_integral_second_kind.svg" alt="Complete elliptic integral of the second kind.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 where the parameter `m` is related to the modulus `k` by `m = k^2`.
     39 
     40 </section>
     41 
     42 <!-- /.intro -->
     43 
     44 <section class="usage">
     45 
     46 ## Usage
     47 
     48 ```javascript
     49 var ellipe = require( '@stdlib/math/base/special/ellipe' );
     50 ```
     51 
     52 #### ellipe( m )
     53 
     54 Computes the [complete elliptic integral of the second kind][elliptic-integral].
     55 
     56 ```javascript
     57 var v = ellipe( 0.5 );
     58 // returns ~1.351
     59 
     60 v = ellipe( -1.0 );
     61 // returns ~1.910
     62 
     63 v = ellipe( 2.0 );
     64 // returns NaN
     65 
     66 v = ellipe( Infinity );
     67 // returns NaN
     68 
     69 v = ellipe( -Infinity );
     70 // returns NaN
     71 
     72 v = ellipe( NaN );
     73 // returns NaN
     74 ```
     75 
     76 </section>
     77 
     78 <!-- /.usage -->
     79 
     80 <section class="notes">
     81 
     82 ## Notes
     83 
     84 -   This function is valid for `-∞ < m <= 1`.
     85 
     86 </section>
     87 
     88 <!-- /.notes -->
     89 
     90 <section class="examples">
     91 
     92 ## Examples
     93 
     94 <!-- eslint no-undef: "error" -->
     95 
     96 ```javascript
     97 var randu = require( '@stdlib/random/base/randu' );
     98 var ellipe = require( '@stdlib/math/base/special/ellipe' );
     99 
    100 var m;
    101 var i;
    102 
    103 for ( i = 0; i < 100; i++ ) {
    104     m = -1.0 + ( randu() * 2.0 );
    105     console.log( 'ellipe(%d) = %d', m, ellipe( m ) );
    106 }
    107 ```
    108 
    109 </section>
    110 
    111 <!-- /.examples -->
    112 
    113 * * *
    114 
    115 <section class="references">
    116 
    117 ## References
    118 
    119 -   Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." _Celestial Mechanics and Dynamical Astronomy_ 105 (4): 305. doi:[10.1007/s10569-009-9228-z][@fukushima:2009a].
    120 -   Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." _Journal of Computational and Applied Mathematics_ 282 (July): 71–76. doi:[10.1016/j.cam.2014.12.038][@fukushima:2015a].
    121 
    122 </section>
    123 
    124 <!-- /.references -->
    125 
    126 <section class="links">
    127 
    128 [elliptic-integral]: https://en.wikipedia.org/wiki/Elliptic_integral
    129 
    130 [@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z
    131 
    132 [@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038
    133 
    134 </section>
    135 
    136 <!-- /.links -->