  ## 5 Special Functions

### 5.1 Overview

The special package of Commons-Math gathers several useful special functions not provided by java.lang.Math.

### 5.2 Erf functions

Erf contains several useful functions involving the Error Function, Erf.

Function Method Reference
Error Function erf See Erf from MathWorld

### 5.3 Gamma functions

Class Gamma contains several useful functions involving the Gamma Function.

#### Gamma

Gamma.gamma(x) computes the Gamma function, Γ(x) (see MathWorld, DLMF). The accuracy of the Commons-Math implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
-5 < x < -4 x[i] = i / 1024, i = -5119, ..., -4097 0.49 ulps 0.57 ulps 3.0 ulps
-4 < x < -3 x[i] = i / 1024, i = -4095, ..., -3073 0.36 ulps 0.51 ulps 2.0 ulps
-3 < x < -2 x[i] = i / 1024, i = -3071, ..., -2049 0.41 ulps 0.53 ulps 2.0 ulps
-2 < x < -1 x[i] = i / 1024, i = -2047, ..., -1025 0.37 ulps 0.50 ulps 2.0 ulps
-1 < x < 0 x[i] = i / 1024, i = -1023, ..., -1 0.46 ulps 0.54 ulps 2.0 ulps
0 < x ≤ 8 x[i] = i / 1024, i = 1, ..., 8192 0.33 ulps 0.48 ulps 2.0 ulps
8 < x ≤ 141 x[i] = i / 64, i = 513, ..., 9024 1.32 ulps 1.19 ulps 7.0 ulps

#### Log Gamma

Gamma.logGamma(x) computes the natural logarithm of the Gamma function, log Γ(x), for x > 0 (see MathWorld, DLMF). The accuracy of the Commons-Math implementation is assessed by comparaison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
0 < x ≤ 8 x[i] = i / 1024, i = 1, ..., 8192 0.32 ulps 0.50 ulps 4.0 ulps
8 < x ≤ 1024 x[i] = i / 8, i = 65, ..., 8192 0.43 ulps 0.53 ulps 3.0 ulps
1024 < x ≤ 8192 x[i], i = 1025, ..., 8192 0.53 ulps 0.56 ulps 3.0 ulps
8933.439345993791 ≤ x ≤ 1.75555970201398e+305 x[i] = 2**(i / 8), i = 105, ..., 8112 0.35 ulps 0.49 ulps 2.0 ulps

#### Regularized Gamma

Gamma.regularizedGammaP(a, x) computes the value of the regularized Gamma function, P(a, x) (see MathWorld).

### 5.4 Beta functions

Beta contains several useful functions involving the Beta Function.

#### Log Beta

Beta.logBeta(a, b) computes the value of the natural logarithm of the Beta function, log B(a, b). (see MathWorld, DLMF). The accuracy of the Commons-Math implementation is assessed by comparison with high precision values computed with the Maxima Computer Algebra System.

Interval Values tested Average error Standard deviation Maximum error
0 < x ≤ 8
0 < y ≤ 8
x[i] = i / 32, i = 1, ..., 256
y[j] = j / 32, j = 1, ..., 256
1.80 ulps 81.08 ulps 14031.0 ulps
0 < x ≤ 8
8 < y ≤ 16
x[i] = i / 32, i = 1, ..., 256
y[j] = j / 32, j = 257, ..., 512
0.50 ulps 3.64 ulps 694.0 ulps
0 < x ≤ 8
16 < y ≤ 256
x[i] = i / 32, i = 1, ..., 256
y[j] = j, j = 17, ..., 256
1.04 ulps 139.32 ulps 34509.0 ulps
8 < x ≤ 16
8 < y ≤ 16
x[i] = i / 32, i = 257, ..., 512
y[j] = j / 32, j = 257, ..., 512
0.35 ulps 0.48 ulps 2.0 ulps
8 < x ≤ 16
16 < y ≤ 256
x[i] = i / 32, i = 257, ..., 512
y[j] = j, j = 17, ..., 256
0.31 ulps 0.47 ulps 2.0 ulps
16 < x ≤ 256
16 < y ≤ 256
x[i] = i, i = 17, ..., 256
y[j] = j, j = 17, ..., 256
0.35 ulps 0.49 ulps 2.0 ulps

(see MathWorld)