naginterfaces.library.specfun.gamma_​log_​real_​vector

naginterfaces.library.specfun.gamma_log_real_vector(x)

gamma_log_real_vector returns an array of values of the logarithm of the gamma function, $ln\left(\Gamma \right)\left(x\right)$.

For full information please refer to the NAG Library document for s14ap

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s14apf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The argument $x_{\text{i}}$ of the function, for $\text{i}=1,2,\dots ,n$.

Returns

ffloat, ndarray, shape $\left(n\right)$

$ln\left(\Gamma \right)\left(x_i\right)$, the function values.

ivalidint, ndarray, shape $\left(n\right)$

$ivalid[\text{i}-1]$ contains the error code for $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

$x_i\le 0$.

$ivalid[i-1]=2$

$x_i$ is too large and positive. The threshold value is the same as for $errno$ = 2 in gamma_log_real().

Raises

NagValueError

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $x$ was invalid.

Check $ivalid$ for more information.

Notes

gamma_log_real_vector calculates an approximate value for $ln\left(\Gamma \right)\left(x\right)$ for an array of arguments $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. It is based on rational Chebyshev expansions.

Denote by $R_{n,m}^i\left(x\right)=P_n^i\left(x\right)/Q_m^i\left(x\right)$ a ratio of polynomials of degree $n$ in the numerator and $m$ in the denominator. Then:

for $0<x\le 1/2$,

$$ln\left(\Gamma \right)\left(x\right)\approx -ln\left(x\right)+x{R}_{n,m}^1(x+1)\text{;}$$

for $1/2<x\le 3/2$,

$$ln\left(\Gamma \right)\left(x\right)\approx (x-1)R_{n,m}^1\left(x\right)\text{;}$$

for $3/2<x\le 4$,

$$ln\left(\Gamma \right)\left(x\right)\approx (x-2)R_{n,m}^2\left(x\right)\text{;}$$

for $4<x\le 12$,

$$ln\left(\Gamma \right)\left(x\right)\approx R_{n,m}^3\left(x\right)\text{;}$$

and for $12<x$,

$$ln\left(\Gamma \right)\left(x\right)\approx (x-\frac{1}{2})ln\left(x\right)-x+ln\left(\sqrt{2\pi }\right)+\frac{1}{x}{R}_{n,m}^4\left(1/x^2\right)\text{.}$$

For each expansion, the specific values of $n$ and $m$ are selected to be minimal such that the maximum relative error in the expansion is of the order ${10}^{-d}$, where $d$ is the maximum number of decimal digits that can be accurately represented for the particular implementation (see machine.decimal_digits).

Let $ϵ$ denote machine precision and let $x_{huge}$ denote the largest positive model number (see machine.real_largest). For $x<0.0$ the value $ln\left(\Gamma \right)\left(x\right)$ is not defined; gamma_log_real_vector returns zero and exits with $errno$ = 1. It also exits with $errno$ = 1 when $x=0.0$, and in this case the value $x_{huge}$ is returned. For $x$ in the interval $(0.0,ϵ]$, the function $ln\left(\Gamma \right)\left(x\right)=-ln\left(x\right)$ to machine accuracy.

Now denote by $x_{big}$ the largest allowable argument for $ln\left(\Gamma \right)\left(x\right)$ on the machine. For ${\left(x_{big}\right)}^{1/4}<x\le x_{big}$ the $R_{n,m}^4\left(1/x^2\right)$ term in Equation (1) is negligible. For $x>x_{big}$ there is a danger of setting overflow, and so gamma_log_real_vector exits with $errno$ = 2 and returns $x_{huge}$.

References

NIST Digital Library of Mathematical Functions

Cody, W J and Hillstrom, K E, 1967, Chebyshev approximations for the natural logarithm of the gamma function, Math.Comp. (21), 198–203