compute-erfc
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Complementary error function.
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erfc
===
[![NPM version][npm-image]][npm-url] [![Build Status][travis-image]][travis-url] [![Coverage Status][coveralls-image]][coveralls-url] [![Dependencies][dependencies-image]][dependencies-url]
> Computes the [complementary error function](https://en.wikipedia.org/wiki/Error_function).
The [complementary error function](https://en.wikipedia.org/wiki/Error_function) is defined as
<div class="equation" align="center" data-raw-text="
\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\, dt" data-equation="eq:complementary_error_function">
<img src="https://cdn.rawgit.com/compute-io/erfc/0e95731781097162b7be21124e02c1aa5b436a8c/docs/img/eqn.svg" alt="Equation of the complementary error function.">
<br>
</div>
Equivalently, the [complementary error function](https://en.wikipedia.org/wiki/Error_function) can be expressed using Craig's formula:
<div class="equation" align="center" data-raw-text="\operatorname{erfc}(x) & = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta" data-equation="eq:craigs_formula">
<img src="https://cdn.rawgit.com/compute-io/erfc/473f8144e996f868eccbc6c136f6727762a8a9b8/docs/img/eqn2.svg" alt="Craig's formula of the complementary error function.">
<br>
</div>
## Installation
``` bash
$ npm install compute-erfc
```
For use in the browser, use [browserify](https://github.com/substack/node-browserify).
## Usage
``` javascript
var erfc = require( 'compute-erfc' );
```
#### erfc( x[, options] )
Evaluates the [complementary error function](http://en.wikipedia.org/wiki/Error_function). `x` may be either a [`number`](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number), an [`array`](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array), a [`typed array`](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays), or a [`matrix`](https://github.com/dstructs/matrix). Values may include `NaN`, `+infinity`, and `-infinity`.
``` javascript
var matrix = require( 'dstructs-matrix' ),
data,
mat,
out,
i;
out = erfc( -1 );
// returns ~1.8427
out = erfc( [ -10, -1, 0, 1, 10 ] );
// returns [ 2, 1.8427, 1, 0.1573, 2.0885e-45 ]
data = [ 0, 1, 2 ];
out = erfc( data );
// returns [ 1, ~0.1573, ~0.0047 ]
data = new Int8Array( data );
out = erfc( data );
// returns Float64Array( [ 1, ~0.1573, ~0.0047 ] )
data = new Float64Array( 6 );
for ( i = 0; i < 6; i++ ) {
data[ i ] = i / 2;
}
mat = matrix( data, [3,2], 'float64' );
/*
[ 0 0.5
1 1.5
2 2.5 ]
*/
out = erfc( mat );
/*
[ 1 ~0.48
~0.16 ~0.03
~0 ~0 ]
*/
```
The function accepts the following `options`:
* __accessor__: accessor `function` for accessing `array` values.
* __dtype__: output [`typed array`](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays) or [`matrix`](https://github.com/dstructs/matrix) data type. Default: `float64`.
* __copy__: `boolean` indicating if the `function` should return a new data structure. Default: `true`.
* __path__: [deepget](https://github.com/kgryte/utils-deep-get)/[deepset](https://github.com/kgryte/utils-deep-set) key path.
* __sep__: [deepget](https://github.com/kgryte/utils-deep-get)/[deepset](https://github.com/kgryte/utils-deep-set) key path separator. Default: `'.'`.
For non-numeric `arrays`, provide an accessor `function` for accessing `array` values.
``` javascript
var data = [
['beep', -10],
['boop', -1],
['bip', 0],
['bap', 1],
['baz', 10]
];
function getValue( d, i ) {
return d[ 1 ];
}
var out = erfc( data, {
'accessor': getValue
});
// returns [ 2, 1.8427, 1, 0.1573, 2.0885e-45 ]
```
To [deepset](https://github.com/kgryte/utils-deep-set) an object `array`, provide a key path and, optionally, a key path separator.
``` javascript
var data = [
{'x':[0,-10]},
{'x':[1,-1]},
{'x':[2,0]},
{'x':[3,1]},
{'x':[4,10]}
];
var out = erfc( data, 'x|1', '|' );
/*
[
{'x':[0,2]},
{'x':[1,1.8427]},
{'x':[2,1]},
{'x':[3,0.1573]},
{'x':[4,2.0885e-45]}
]
*/
var bool = ( data === out );
// returns true
```
By default, when provided a [`typed array`](https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays) or [`matrix`](https://github.com/dstructs/matrix), the output data structure is `float64` in order to preserve precision. To specify a different data type, set the `dtype` option (see [`matrix`](https://github.com/dstructs/matrix) for a list of acceptable data types).
``` javascript
var data, out;
data = new Int8Array( [0, 1, 2] );
out = erfc( data, {
'dtype': 'int32'
});
// returns Int32Array( [1,0,0] )
// Works for plain arrays, as well...
out = erfc( [0, 1, 2], {
'dtype': 'uint8'
});
// returns Uint8Array( [1,0,0] )
```
By default, the function returns a new data structure. To mutate the input data structure (e.g., when input values can be discarded or when optimizing memory usage), set the `copy` option to `false`.
``` javascript
var data,
bool,
mat,
out,
i;
var data = [ -10, -1, 0, 1, 10 ];
var out = erfc( data, {
'copy': false
});
// returns [ 2, 1.8427, 1, 0.1573, 2.0885e-45 ]
bool = ( data === out );
// returns true
data = new Float64Array( 6 );
for ( i = 0; i < 6; i++ ) {
data[ i ] = i / 2;
}
mat = matrix( data, [3,2], 'float64' );
/*
[ 0 0.5
1 1.5
2 2.5 ]
*/
out = erfc( mat, {
'copy': false
});
/*
[ 1 ~0.48
~0.16 ~0.03
~0 ~0 ]
*/
bool = ( mat === out );
// returns true
```
## Notes
* If an element is __not__ a numeric value, the evaluated [complementary error function](http://en.wikipedia.org/wiki/Error_function) is `NaN`.
``` javascript
var data, out;
out = erfc( null );
// returns NaN
out = erfc( true );
// returns NaN
out = erfc( {'a':'b'} );
// returns NaN
out = erfc( [ true, null, [] ] );
// returns [ NaN, NaN, NaN ]
function getValue( d, i ) {
return d.x;
}
data = [
{'x':true},
{'x':[]},
{'x':{}},
{'x':null}
];
out = erfc( data, {
'accessor': getValue
});
// returns [ NaN, NaN, NaN, NaN ]
out = erfc( data, {
'path': 'x'
});
/*
[
{'x':NaN},
{'x':NaN},
{'x':NaN,
{'x':NaN}
]
*/
```
* Be careful when providing a data structure which contains non-numeric elements and specifying an `integer` output data type, as `NaN` values are cast to `0`.
``` javascript
var out = erfc( [ true, null, [] ], {
'dtype': 'int8'
});
// returns Int8Array( [0,0,0] );
```
## Examples
``` javascript
var matrix = require( 'dstructs-matrix' ),
erfc = require( 'compute-erfc' );
var data,
mat,
out,
tmp,
i;
// Plain arrays...
data = new Array( 10 );
for ( i = 0; i < data.length; i++ ) {
data[ i ] = Math.random()*20 - 10;
}
out = erfc( data );
// Object arrays (accessors)...
function getValue( d ) {
return d.x;
}
for ( i = 0; i < data.length; i++ ) {
data[ i ] = {
'x': data[ i ]
};
}
out = erfc( data, {
'accessor': getValue
});
// Deep set arrays...
for ( i = 0; i < data.length; i++ ) {
data[ i ] = {
'x': [ i, data[ i ].x ]
};
}
out = erfc( data, {
'path': 'x/1',
'sep': '/'
});
// Typed arrays...
data = new Int32Array( 10 );
for ( i = 0; i < data.length; i++ ) {
data[ i ] = Math.random()*20 - 10;
}
tmp = erfc( data );
out = '';
for ( i = 0; i < data.length; i++ ) {
out += tmp[ i ];
if ( i < data.length-1 ) {
out += ',';
}
}
// Matrices...
mat = matrix( data, [5,2], 'int32' );
out = erfc( mat );
// Matrices (custom output data type)...
out = erfc( mat, {
'dtype': 'uint8'
});
```
To run the example code from the top-level application directory,
``` bash
$ node ./examples/index.js
```
## Tests
### Unit
Unit tests use the [Mocha](http://mochajs.org) test framework with [Chai](http://chaijs.com) assertions. To run the tests, execute the following command in the top-level application directory:
``` bash
$ make test
```
All new feature development should have corresponding unit tests to validate correct functionality.
### Test Coverage
This repository uses [Istanbul](https://github.com/gotwarlost/istanbul) as its code coverage tool. To generate a test coverage report, execute the following command in the top-level application directory:
``` bash
$ make test-cov
```
Istanbul creates a `./reports/coverage` directory. To access an HTML version of the report,
``` bash
$ make view-cov
```
---
## License
[MIT license](http://opensource.org/licenses/MIT).
## Copyright
Copyright © 2014-2015. The [Compute.io](https://github.com/compute-io) Authors.
[npm-image]: http://img.shields.io/npm/v/compute-erfc.svg
[npm-url]: https://npmjs.org/package/compute-erfc
[travis-image]: http://img.shields.io/travis/compute-io/erfc/master.svg
[travis-url]: https://travis-ci.org/compute-io/erfc
[coveralls-image]: https://img.shields.io/coveralls/compute-io/erfc/master.svg
[coveralls-url]: https://coveralls.io/r/compute-io/erfc?branch=master
[dependencies-image]: http://img.shields.io/david/compute-io/erfc.svg
[dependencies-url]: https://david-dm.org/compute-io/erfc
[dev-dependencies-image]: http://img.shields.io/david/dev/compute-io/erfc.svg
[dev-dependencies-url]: https://david-dm.org/dev/compute-io/erfc
[github-issues-image]: http://img.shields.io/github/issues/compute-io/erfc.svg
[github-issues-url]: https://github.com/compute-io/erfc/issues