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gamma.js
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194 lines (160 loc) · 4.77 KB
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'use strict'
const deepMap = require('../../utils/collection/deepMap')
const isInteger = require('../../utils/number').isInteger
function factory (type, config, load, typed) {
const multiply = load(require('../arithmetic/multiply'))
const pow = load(require('../arithmetic/pow'))
const product = require('./product')
/**
* Compute the gamma function of a value using Lanczos approximation for
* small values, and an extended Stirling approximation for large values.
*
* For matrices, the function is evaluated element wise.
*
* Syntax:
*
* math.gamma(n)
*
* Examples:
*
* math.gamma(5) // returns 24
* math.gamma(-0.5) // returns -3.5449077018110335
* math.gamma(math.i) // returns -0.15494982830180973 - 0.49801566811835596i
*
* See also:
*
* combinations, factorial, permutations
*
* @param {number | Array | Matrix} n A real or complex number
* @return {number | Array | Matrix} The gamma of `n`
*/
const gamma = typed('gamma', {
'number': function (n) {
let t, x
if (isInteger(n)) {
if (n <= 0) {
return isFinite(n) ? Infinity : NaN
}
if (n > 171) {
return Infinity // Will overflow
}
return product(1, n - 1)
}
if (n < 0.5) {
return Math.PI / (Math.sin(Math.PI * n) * gamma(1 - n))
}
if (n >= 171.35) {
return Infinity // will overflow
}
if (n > 85.0) { // Extended Stirling Approx
const twoN = n * n
const threeN = twoN * n
const fourN = threeN * n
const fiveN = fourN * n
return Math.sqrt(2 * Math.PI / n) * Math.pow((n / Math.E), n) *
(1 + 1 / (12 * n) + 1 / (288 * twoN) - 139 / (51840 * threeN) -
571 / (2488320 * fourN) + 163879 / (209018880 * fiveN) +
5246819 / (75246796800 * fiveN * n))
}
--n
x = p[0]
for (let i = 1; i < p.length; ++i) {
x += p[i] / (n + i)
}
t = n + g + 0.5
return Math.sqrt(2 * Math.PI) * Math.pow(t, n + 0.5) * Math.exp(-t) * x
},
'Complex': function (n) {
let t, x
if (n.im === 0) {
return gamma(n.re)
}
n = new type.Complex(n.re - 1, n.im)
x = new type.Complex(p[0], 0)
for (let i = 1; i < p.length; ++i) {
const real = n.re + i // x += p[i]/(n+i)
const den = real * real + n.im * n.im
if (den !== 0) {
x.re += p[i] * real / den
x.im += -(p[i] * n.im) / den
} else {
x.re = p[i] < 0
? -Infinity
: Infinity
}
}
t = new type.Complex(n.re + g + 0.5, n.im)
const twoPiSqrt = Math.sqrt(2 * Math.PI)
n.re += 0.5
const result = pow(t, n)
if (result.im === 0) { // sqrt(2*PI)*result
result.re *= twoPiSqrt
} else if (result.re === 0) {
result.im *= twoPiSqrt
} else {
result.re *= twoPiSqrt
result.im *= twoPiSqrt
}
const r = Math.exp(-t.re) // exp(-t)
t.re = r * Math.cos(-t.im)
t.im = r * Math.sin(-t.im)
return multiply(multiply(result, t), x)
},
'BigNumber': function (n) {
if (n.isInteger()) {
return (n.isNegative() || n.isZero())
? new type.BigNumber(Infinity)
: bigFactorial(n.minus(1))
}
if (!n.isFinite()) {
return new type.BigNumber(n.isNegative() ? NaN : Infinity)
}
throw new Error('Integer BigNumber expected')
},
'Array | Matrix': function (n) {
return deepMap(n, gamma)
}
})
/**
* Calculate factorial for a BigNumber
* @param {BigNumber} n
* @returns {BigNumber} Returns the factorial of n
*/
function bigFactorial (n) {
if (n.isZero()) {
return new type.BigNumber(1) // 0! is per definition 1
}
const precision = config.precision + (Math.log(n.toNumber()) | 0)
const Big = type.BigNumber.clone({ precision: precision })
let res = new Big(n)
let value = n.toNumber() - 1 // number
while (value > 1) {
res = res.times(value)
value--
}
return new type.BigNumber(res.toPrecision(type.BigNumber.precision))
}
gamma.toTex = { 1: `\\Gamma\\left(\${args[0]}\\right)` }
return gamma
}
// TODO: comment on the variables g and p
const g = 4.7421875
const p = [
0.99999999999999709182,
57.156235665862923517,
-59.597960355475491248,
14.136097974741747174,
-0.49191381609762019978,
0.33994649984811888699e-4,
0.46523628927048575665e-4,
-0.98374475304879564677e-4,
0.15808870322491248884e-3,
-0.21026444172410488319e-3,
0.21743961811521264320e-3,
-0.16431810653676389022e-3,
0.84418223983852743293e-4,
-0.26190838401581408670e-4,
0.36899182659531622704e-5
]
exports.name = 'gamma'
exports.factory = factory