141 lines
4.0 KiB
Markdown
141 lines
4.0 KiB
Markdown
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<!--
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# erfinv
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> [Inverse error function][inverse-error-function].
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<section class="intro">
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The [inverse error function][inverse-error-function] is defined in terms of the [Maclaurin series][maclaurin-series]
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<!-- <equation class="equation" label="eq:inverse_error_function" align="center" raw="\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1}" alt="Inverse error function."> -->
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<div class="equation" align="center" data-raw-text="\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1}" data-equation="eq:inverse_error_function">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/erfinv/docs/img/equation_inverse_error_function.svg" alt="Inverse error function.">
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<br>
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</div>
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<!-- </equation> -->
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where `c_0 = 1` and
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<!-- <equation class="equation" label="eq:inverse_error_function_series_coefficients" align="center" raw="c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}" alt="Series coefficients."> -->
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<div class="equation" align="center" data-raw-text="c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}" data-equation="eq:inverse_error_function_series_coefficients">
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<img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@591cf9d5c3a0cd3c1ceec961e5c49d73a68374cb/lib/node_modules/@stdlib/math/base/special/erfinv/docs/img/equation_inverse_error_function_series_coefficients.svg" alt="Series coefficients.">
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<br>
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</div>
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<!-- </equation> -->
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var erfinv = require( '@stdlib/math/base/special/erfinv' );
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```
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#### erfinv( x )
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Evaluates the [inverse error function][inverse-error-function].
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```javascript
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var y = erfinv( 0.5 );
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// returns ~0.4769
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y = erfinv( 0.8 );
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// returns ~0.9062
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y = erfinv( -1.0 );
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// returns -Infinity
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y = erfinv( 1.0 );
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// returns Infinity
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```
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The domain of `x` is restricted to `[-1,1]`. If `|x| > 1`, the function returns `NaN`.
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```javascript
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var y = erfinv( -3.14 );
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// returns NaN
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```
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If provided `NaN`, the function returns `NaN`.
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```javascript
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var y = erfinv( NaN );
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// returns NaN
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```
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The [inverse error function][inverse-error-function] is an [odd function][odd-function]; i.e., `erfinv(-x) = -erfinv(x)`. Thus, in accordance with the [IEEE 754][ieee754] standard, if provided `-0`, the function returns `-0`.
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```javascript
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var y = erfinv( -0.0 );
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// returns -0.0
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```
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</section>
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<!-- /.usage -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var linspace = require( '@stdlib/array/linspace' );
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var erfinv = require( '@stdlib/math/base/special/erfinv' );
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var x = linspace( -1.0, 1.0, 100 );
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var y;
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var i;
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for ( i = 0; i < x.length; i++ ) {
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y = erfinv( x[ i ] );
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console.log( 'x: %d, erfinv(x): %d', x[ i ], y );
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}
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[inverse-error-function]: https://en.wikipedia.org/wiki/Error_function#Inverse_functions
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[maclaurin-series]: http://mathworld.wolfram.com/MaclaurinSeries.html
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[odd-function]: https://en.wikipedia.org/wiki/Even_and_odd_functions
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[ieee754]: https://en.wikipedia.org/wiki/IEEE_754-1985
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</section>
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<!-- /.links -->
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