Incomplete gamma function.
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> [Incomplete gamma function][incomplete-gamma-function].
Evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function]:
The function can also be used to evaluate the regularized upper incomplete gamma function, which is defined as follows:
The two functions have the relationship Q(x,a) = 1 - P(x,a).
In addition, this routine can be used to evaluate the _unregularized_ gamma functions. The range of above functions is [0,1], which is not the case for the _unregularized_ versions. The unregularized lower incomplete gamma function is defined as
and the upper unregularized incomplete gamma function is
The relationship between the two functions is Γ(a,x) = γ(a,x) + Γ(a).
``bash`
npm install @stdlib/math-base-special-gammainc
`javascript`
var gammainc = require( '@stdlib/math-base-special-gammainc' );
#### gammainc( x, s\[, regularized\[, upper ]] )
By default, evaluates the regularized lower [incomplete gamma function][incomplete-gamma-function] for inputs x and s. The third and fourth parameters of the function can be used to specify whether instead to evaluate the non-regularized and/or upper incomplete gamma functions, respectively.
`javascript
var y = gammainc( 6.0, 2.0 );
// returns ~0.9826
y = gammainc( 1.0, 2.0, true, true );
// returns ~0.7358
y = gammainc( 7.0, 5.0 );
// returns ~0.8270
y = gammainc( 7.0, 5.0, false );
// returns ~19.8482
`
If provided NaN as any argument, the function returns NaN.
`javascript
var y = gammainc( NaN, 1.0 );
// returns NaN
y = gammainc( 1.0, NaN );
// returns NaN
`
If provided a negative x, the function returns NaN.
`javascript`
var y = gammainc( -2.0, 2.0 );
// returns NaN
If provided a nonpositive s, the function returns NaN.
`javascript
var y = gammainc( 2.0, -1.0 );
// returns NaN
y = gammainc( 2.0, 0.0 );
// returns NaN
`
`javascript
var uniform = require( '@stdlib/random-array-uniform' );
var logEachMap = require( '@stdlib/console-log-each-map' );
var gammainc = require( '@stdlib/math-base-special-gammainc' );
var opts = {
'dtype': 'float64'
};
var x = uniform( 100, 0.0, 10.0, opts );
var s = uniform( 100, 0.0, 10.0, opts );
logEachMap( 'x: %0.4f, \t s: %0.4f, \t f(x,s): %0.4f', x, s, gammainc );
`
*
`c`
#include "stdlib/math/base/special/gammainc.h"
#### stdlib_base_gammainc( x, a, regularized, upper )
Evaluates the [incomplete gamma function][incomplete-gamma-function] for inputs x and a.
`c
double out = stdlib_base_gammainc( 0.0, 1.0, true, false );
// returns 0.0
out = stdlib_base_gammainc( 6.0, 2.0, true, false );
// returns ~0.9826
`
The function accepts the following arguments:
- x: [in] double input value.[in] double
- a: input value.[in] bool
- regularized: boolean indicating whether to evaluate a regularized incomplete gamma function.[in] bool
- upper: boolean indicating whether to evaluate the upper incomplete gamma function.
`c`
double stdlib_base_gammainc( const double x, const double a, const bool regularized, const bool upper );
`c
#include "stdlib/math/base/special/gammainc.h"
#include
#include
static double random_uniform( const double min, const double max ) {
double v = (double)rand() / ( (double)RAND_MAX + 1.0 );
return min + ( v*(max-min) );
}
int main( void ) {
double x;
double a;
double y;
int i;
for ( i = 0; i < 10; i++ ) {
x = random_uniform( 0.0, 1000.0 );
a = random_uniform( 1.0, 1000.0 );
y = stdlib_base_gammainc( x, a, true, false );
printf( "x: %lf, s: %lf, f(x,s): %lf\n", x, a, y );
}
}
`
*
- [@stdlib/math-base/special/gamma][@stdlib/math/base/special/gamma]: gamma function.
- [@stdlib/math-base/special/gamma1pm1][@stdlib/math/base/special/gamma1pm1]: compute gamma(x+1) - 1.
- [@stdlib/math-base/special/gammaincinv][@stdlib/math/base/special/gammaincinv]: inverse incomplete gamma function.
- [@stdlib/math-base/special/gammaln][@stdlib/math/base/special/gammaln]: natural logarithm of the gamma function.
*
This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.
For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib].
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[incomplete-gamma-function]: https://en.wikipedia.org/wiki/Incomplete_gamma_function
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