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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_specfun_gamma_log_real (s14ab)

## Purpose

nag_specfun_gamma_log_real (s14ab) returns the value of the logarithm of the gamma function, $\mathrm{ln}\Gamma \left(x\right)$, via the function name.

## Syntax

[result, ifail] = s14ab(x)
[result, ifail] = nag_specfun_gamma_log_real(x)

## Description

nag_specfun_gamma_log_real (s14ab) calculates an approximate value for $\mathrm{ln}\Gamma \left(x\right)$. 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,
 $ln⁡Γx ≈ -lnx + x R n,m 1 x+1 ;$
• for $1/2,
 $ln⁡Γx ≈ x-1 R n,m 1 x ;$
• for $3/2,
 $ln⁡Γx ≈ x-2 R n,m 2 x;$
• for $4,
 $ln⁡Γx ≈ R n,m 3 x ;$
• and for $12,
 $ln⁡Γx ≈ x-12 lnx - x + ln2π + 1x R n,m 4 1/x2 .$ (1)
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 nag_machine_decimal_digits (x02be)).
Let $\epsilon$ denote machine precision and let ${x}_{\mathrm{huge}}$ denote the largest positive model number (see nag_machine_real_largest (x02al)). For $x<0.0$ the value $\mathrm{ln}\Gamma \left(x\right)$ is not defined; nag_specfun_gamma_log_real (s14ab) returns zero and exits with ${\mathbf{ifail}}={\mathbf{1}}$. It also exits with ${\mathbf{ifail}}={\mathbf{1}}$ when $x=0.0$, and in this case the value ${x}_{\mathrm{huge}}$ is returned. For $x$ in the interval $\left(0.0,\epsilon \right]$, the function $\mathrm{ln}\Gamma \left(x\right)=-\mathrm{ln}\left(x\right)$ to machine accuracy.
Now denote by ${x}_{\mathrm{big}}$ the largest allowable argument for $\mathrm{ln}\Gamma \left(x\right)$ on the machine. For ${\left({x}_{\mathrm{big}}\right)}^{1/4} the ${R}_{n,m}^{4}\left(1/{x}^{2}\right)$ term in Equation (1) is negligible. For $x>{x}_{\mathrm{big}}$ there is a danger of setting overflow, and so nag_specfun_gamma_log_real (s14ab) exits with ${\mathbf{ifail}}={\mathbf{2}}$ and returns ${x}_{\mathrm{huge}}$.

## References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
Cody W J and Hillstrom K E (1967) Chebyshev approximations for the natural logarithm of the gamma function Math.Comp. 21 198–203

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}$ – double scalar
The argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}\le 0.0$. If ${\mathbf{x}}<0.0$ the function is undefined; on soft failure, the function value returned is zero. If ${\mathbf{x}}=0.0$ and soft failure is selected, the function value returned is the largest machine number (see nag_machine_real_largest (x02al)).
W  ${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}>{x}_{\mathrm{big}}$ (see Description). On soft failure, the function value returned is the largest machine number (see nag_machine_real_largest (x02al)).
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and result respectively, and $E$ be the absolute error in the result.
If $\delta$ is somewhat larger than machine precision, then
 $E≃ x×Ψx δ and ε≃ x×Ψx ln⁡Γ x δ$
where $\Psi \left(x\right)$ is the digamma function $\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$. Figure 1 and Figure 2 show the behaviour of these error amplification factors.
Figure 1
Figure 2
These show that relative error can be controlled, since except near $x=1\text{​ or ​}2$ relative error is attenuated by the function or at least is not greatly amplified.
For large $x$, $\epsilon \simeq \left(1+\frac{1}{\mathrm{ln}x}\right)\delta$ and for small $x$, $\epsilon \simeq \frac{1}{\mathrm{ln}x}\delta$.
The function $\mathrm{ln}\Gamma \left(x\right)$ has zeros at $x=1$ and $2$ and hence relative accuracy is not maintainable near those points. However absolute accuracy can still be provided near those zeros as is shown above.
If however, $\delta$ is of the order of machine precision, then rounding errors in the function's internal arithmetic may result in errors which are slightly larger than those predicted by the equalities. It should be noted that even in areas where strong attenuation of errors is predicted the relative precision is bounded by the effective machine precision.

None.

## Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.
```function s14ab_example

fprintf('s14ab example results\n\n');

x = [ 1   1.25    1.5    1.75    2     5    10    20   1000];
n = size(x,2);
result = x;

for j=1:n
[result(j), ifail] = s14ab(x(j));
end

disp('      x       log_gamma(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s14ab_plot;

function s14ab_plot
x = [0.1:0.2:8];

for j=1:numel(x)
[lg(j), ifail] = s14ab(x(j));
end

fig1 = figure;
plot(x,lg,'-r');
xlabel('x');
ylabel('ln \Gamma(x)');
title('Logarithm of Gamma Function ln \Gamma(x)');
axis([0 8 -1 9]);

```
```s14ab example results

x       log_gamma(x)
1.000e+00   0.000e+00
1.250e+00  -9.827e-02
1.500e+00  -1.208e-01
1.750e+00  -8.440e-02
2.000e+00   0.000e+00
5.000e+00   3.178e+00
1.000e+01   1.280e+01
2.000e+01   3.934e+01
1.000e+03   5.905e+03
```