gamma_f77

ckormanyos/gamma_f77 implements the real-valued Gamma function in quadruple-precision using the classic Fortran77 language.

Mathematical Background

The gamma function $\Gamma\left(z\right)$ is the complex-valued extension of the well-known integer factorial function.

For $\mathbb{Re}\left(z\right) > 0$, $\Gamma\left(z\right)$ is defined by

$$\Gamma\left(z\right)=\int_{0}^{\infty}t^{z-1} e^{t} dt\text{,}$$

$\Gamma\left(z\right)$ has a value of complex-infinity at the origin and also has poles at integer values along the negative real axis.

Reflection is given by

$$ \Gamma(-z)= \frac{\pi}{z\Gamma\left(z\right)\sin\left(\pi z\right)}\text{.}$$

Reccurence is given by

$$ \Gamma\left(z+1\right)= z\Gamma\left(z\right)\text{.}$$

Calculation Method

Let's look at some background information regarding computations of the real-valued gamma function.

The real-valued gamma function, $\Gamma\left(x\right)$ can be readily calculated using a series expansion for its reciprocal near the origin. Large arguments valued greater than one use recurrence. For negative argument, the function value for the corresponding positive-valued argument is first calculated and the value for negative argument is obtaind via reflection.

Consider the series expansion of the reciprocal of the gamma function near the origin

$$ \frac{1}{\Gamma\left(z\right)}\approx \sum_{k=1}^{n} a^{k} z^{k}\text{.}$$

In the subroutine GAMMA in Sect. 3.1.5 on pages 49-50 of [1], the coefficients $a_{k}$ are given to $26$ terms. These are used in a series calculation of $\Gamma\left(x\right)$ for real-valued $x$ using Fortran77's double-precision data type REAL*8. Further information on this coefficient expansion can be found in Sect. 6.1.34 of [2], in Sect. 5.7.1 of [3] and in additional references therein.

See also Wolfram Alpha(R) for brief mathematical insight into the fascinating series expansion of the reciprocal of the gamma function near the origin.

Expanded Quadruple-Precision Implementation

In this repository, the series calculation mentioned above has been extended to quadruple-precision.

The coefficients $a_{k}$ have been expanded (via computer algebra) to $48$ terms having $51$ decimal digits of precision. With this coefficient list, it is possible to reach the quadruple-precision of Fortran77's data type REAL*16. These higher-precision coefficients can be found in the table G in the source code.

The implementation uses the gfortran dialect that is available in g++.

Test-Run and CI

Continuous Integration

Continuous integration (CI) runs with gfortran using GHA ubuntu-latest and macos-latest runners. CI exercises both building gamma as well as running several straightforward gamma test cases.

A (growing) test suite is present in both the build workflows as well as in cover.sh. These tests are used in CI to verify the expected functionality and also to obtain code coverage results.

Testing

The test-run computes $9$ gamma values $\Gamma\left(x\right)$ for positive, real-valued arguments at

$$x = 1.11, 2.21, 3.31, {\ldots} 9.91\text{.}$$

Negative reflection is tested at

$$x=-4.56\text{.}$$

Integral-valued argument is checked for

$$x=31\text{,}$$

which is used to compute $\Gamma\left(31\right)$, the result of which is expected to be equal to the integral factorial

$$30 ! = 265,252,859,812,191,058,636,308,480,000,000 \text{.}$$

CI runs on Ubuntu and MacOS using g++. Correct numerical results are verified on the OS-level up to the $33$ decimal digit precision using the built-in program grep.

The program can also be compiled and executed at this short link to godbolt.

Long Standards-Conforming Time-Span

The program compiles with language standards legacy (i.e., Fortran77) as well as all the way up to modern f2023.

This is a remarkably long standards-conforming time-span. It exceeds 40 years - with hopefully more to come!

Proving this longevity in this repository was achieved in part through contributions from @Beliavsky. These were initially proposed in gamma_f77/issues/13. Thank you for these contributions.

References

[1] Shanjie Zhang and Jianming Jin, Computation of Special Functions, Wiley, 1996, ISBN: 0-471-11963-6, LC: QA351.C45

[2] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, 9th Printing, Dover Publications, 1970.

[3] F.W.J. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.