# NAG FL Interfaces14abf (gamma_​log_​real)

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## 1Purpose

s14abf returns the value of the logarithm of the gamma function, $\mathrm{ln}\Gamma \left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s14abf ( x,
 Real (Kind=nag_wp) :: s14abf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
C Header Interface
#include <nag.h>
 double s14abf_ (const double *x, Integer *ifail)
The routine may be called by the names s14abf or nagf_specfun_gamma_log_real.

## 3Description

s14abf 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,
 $ln⁡Γ(x) ≈ -ln(x) + x R n,m 1 (x+1) ;$
• for $1/2,
 $ln⁡Γ(x) ≈ (x-1) R n,m 1 (x) ;$
• for $3/2,
 $ln⁡Γ(x) ≈ (x-2) R n,m 2 (x);$
• for $4,
 $ln⁡Γ(x) ≈ R n,m 3 (x) ;$
• and for $12,
 $ln⁡Γ(x) ≈ (x-12) ln(x) - x + ln(2π) + 1x R n,m 4 (1/x2) .$ (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 x02bef).
Let $\epsilon$ denote machine precision and let ${x}_{\mathrm{huge}}$ denote the largest positive model number (see x02alf). For $x<0.0$ the value $\mathrm{ln}\Gamma \left(x\right)$ is not defined; s14abf 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 s14abf exits with ${\mathbf{ifail}}={\mathbf{2}}$ and returns ${x}_{\mathrm{huge}}$. The value of ${x}_{\mathrm{big}}$ is given in the Users' Note for your implementation.

## 4References

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

## 5Arguments

1: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$ and the constant ${x}_{\mathrm{big}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\le {x}_{\mathrm{big}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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
 $E≃ |x×Ψ(x)| δ and ε≃ | x×Ψ(x) ln⁡Γ (x) | δ$
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.
These show that relative error can be controlled, since except near $x=1$ 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 routine'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.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
s14abf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s14abfe.f90)

### 10.2Program Data

Program Data (s14abfe.d)

### 10.3Program Results

Program Results (s14abfe.r)