time-to-botec

Benchmark sampling in different programming languages
Log | Files | Refs | README

README.md (4687B)

      1 <!--
      2 
      3 @license Apache-2.0
      4 
      5 Copyright (c) 2018 The Stdlib Authors.
      6 
      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
     10 
     11    http://www.apache.org/licenses/LICENSE-2.0
     12 
     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 # Logarithm of Binomial Coefficient
     22 
     23 > Compute the natural logarithm of the [binomial coefficient][binomial-coefficient].
     24 
     25 <section class="intro">
     26 
     27 The natural logarithm of the [binomial coefficient][binomial-coefficient] is
     28 
     29 <!-- <equation class="equation" label="eq:binomcoefln_function" align="center" raw="f(n,k) = \ln {n \choose k}" alt="Natural logarithm of the binomial coefficient."> -->
     30 
     31 <div class="equation" align="center" data-raw-text="f(n,k) = \ln {n \choose k}" data-equation="eq:binomcoefln_function">
     32     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binomcoefln/docs/img/equation_binomcoefln_function.svg" alt="Natural logarithm of the binomial coefficient.">
     33     <br>
     34 </div>
     35 
     36 <!-- </equation> -->
     37 
     38 The [binomial coefficient][binomial-coefficient] of two nonnegative integers `n` and `k` is defined as
     39 
     40 <!-- <equation class="equation" label="eq:binomial_coefficient" align="center" raw="\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n" alt="Factorial formula for the Binomial coefficient."> -->
     41 
     42 <div class="equation" align="center" data-raw-text="\binom {n}{k} = \frac{n!}{k!\,(n-k)!} \quad \text{for }\ 0\leq k\leq n" data-equation="eq:binomial_coefficient">
     43     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/binomcoefln/docs/img/equation_binomial_coefficient.svg" alt="Factorial formula for the Binomial coefficient.">
     44     <br>
     45 </div>
     46 
     47 <!-- </equation> -->
     48 
     49 The [binomial coefficient][binomial-coefficient] can be generalized to negative integers `n` as follows:
     50 
     51 <!-- <equation class="equation" label="eq:binomial_coefficient_negative_integers" align="center" raw="\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)" alt="Generalization of the binomial coefficient to negative n."> -->
     52 
     53 <div class="equation" align="center" data-raw-text="\binom {-n}{k} = (-1)^{k} \binom{n + k - 1}{k} = (-1)^{k} \left(\!\!{\binom {n}{k}}\!\!\right)" data-equation="eq:binomial_coefficient_negative_integers">
     54     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/binomcoefln/docs/img/equation_binomial_coefficient_negative_integers.svg" alt="Generalization of the binomial coefficient to negative n.">
     55     <br>
     56 </div>
     57 
     58 <!-- </equation> -->
     59 
     60 </section>
     61 
     62 <!-- /.intro -->
     63 
     64 <section class="usage">
     65 
     66 ## Usage
     67 
     68 ```javascript
     69 var binomcoefln = require( '@stdlib/math/base/special/binomcoefln' );
     70 ```
     71 
     72 #### binomcoefln( n, k )
     73 
     74 Evaluates the [binomial coefficient][binomial-coefficient] of two integers `n` and `k`.
     75 
     76 ```javascript
     77 var v = binomcoefln( 8, 2 );
     78 // returns ~3.332
     79 
     80 v = binomcoefln( 0, 0 );
     81 // returns 0.0
     82 
     83 v = binomcoefln( -4, 2 );
     84 // returns ~2.303
     85 
     86 v = binomcoefln( 88, 3 );
     87 // returns ~11.606
     88 
     89 v = binomcoefln( NaN, 3 );
     90 // returns NaN
     91 
     92 v = binomcoefln( 5, NaN );
     93 // returns NaN
     94 
     95 v = binomcoefln( NaN, NaN );
     96 // returns NaN
     97 ```
     98 
     99 For negative `k`, the function returns `-Infinity`.
    100 
    101 ```javascript
    102 var v = binomcoefln( 2, -1 );
    103 // returns -Infinity
    104 
    105 v = binomcoefln( -3, -1 );
    106 // returns -Infinity
    107 ```
    108 
    109 The function returns `NaN` for non-integer `n` or `k`.
    110 
    111 ```javascript
    112 var v = binomcoefln( 2, 1.5 );
    113 // returns NaN
    114 
    115 v = binomcoefln( 5.5, 2 );
    116 // returns NaN
    117 ```
    118 
    119 </section>
    120 
    121 <!-- /.usage -->
    122 
    123 <section class="examples">
    124 
    125 ## Examples
    126 
    127 <!-- eslint no-undef: "error" -->
    128 
    129 ```javascript
    130 var randu = require( '@stdlib/random/base/randu' );
    131 var round = require( '@stdlib/math/base/special/round' );
    132 var binomcoefln = require( '@stdlib/math/base/special/binomcoefln' );
    133 
    134 var n;
    135 var k;
    136 var i;
    137 
    138 for ( i = 0; i < 100; i++ ) {
    139     n = round( (randu()*40.0) - 10.0 );
    140     k = round( randu()*20.0 );
    141     console.log( 'ln( %d choose %d ) = %d', n, k, binomcoefln( n, k ) );
    142 }
    143 ```
    144 
    145 </section>
    146 
    147 <!-- /.examples -->
    148 
    149 <section class="links">
    150 
    151 [binomial-coefficient]: https://en.wikipedia.org/wiki/Binomial_coefficient
    152 
    153 </section>
    154 
    155 <!-- /.links -->