# NAG FL Interfaces14apf (gamma_​log_​real_​vector)

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## 1Purpose

s14apf returns an array of values of the logarithm of the gamma function, $\mathrm{ln}\Gamma \left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s14apf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s14apf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s14apf or nagf_specfun_gamma_log_real_vector.

## 3Description

s14apf calculates an approximate value for $\mathrm{ln}\Gamma \left(x\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 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,
 $ln⁡Γ(x) ≈ -ln(x) + x R n,m 1 (x+1) ;$
• 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; s14apf 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 s14apf 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\mathrm{ln}\Gamma \left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}\le 0$.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}$ is too large and positive. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{2}}$ in s14abf , as defined in the the Users' Note for your implementation.
5: $\mathbf{ifail}$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, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
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

s14apf is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s14apfe.f90)

### 10.2Program Data

Program Data (s14apfe.d)

### 10.3Program Results

Program Results (s14apfe.r)