pm_mathFactorial Module Reference

This module contains procedures and generic interfaces for the Factorial function. More...

Data Types
interface	getFactorial
	Generate and return the factorial of the input positive integer. More...
interface	getLogFactorial
	Generate and return the natural logarithm of the factorial of the input positive whole real number. More...

Variables
character(*, SK), parameter	MODULE_NAME = "@pm_mathFactorial"

Detailed Description

This module contains procedures and generic interfaces for the Factorial function.

This module provides a method getFactorial to calculate the factorial of a number. However, note that factorial can easily overflow the integer representations of computers and even real numbers. As such, a better safer method of computing the factorial is to compute its natural logarithm via getLogFactorial.

Benchmarks:

Benchmark :: The runtime performance of getFactorial vs. getLogFactorial ⛓

! Test the performance of `getFactorial()` vs. `getLogFactorial()`.
program benchmark
 
    use iso_fortran_env, only: error_unit
    use pm_kind, only: IK, LK, SK, RKD, IKD
    use pm_arrayRange, only: getRange
    use pm_bench, only: bench_type
 
    implicit none
 
    integer(IK)                         :: i
    integer(IK)                         :: ipnt
    integer(IK)                         :: fileUnit
    integer(IK)     , parameter         :: NPNT = 10_IK
    integer(IK)     , parameter         :: NBENCH = 2_IK
    real(RKD)                           :: Point_RKD(NPNT)
    real(RKD)                           :: dummy = 0._RKD
    real(RKD)                           :: factorial_RKD = 0._RKD
    integer(IKD)                        :: factorial_IKD = 0_IKD
    integer(IKD)                        :: point_IKD(NPNT)
    type(bench_type)                    :: bench(NBENCH)
 
    bench(1) = bench_type(name = SK_"getFactorial", exec = getFactorial , overhead = setOverhead)
    bench(2) = bench_type(name = SK_"getLogFactorial", exec = getLogFactorial , overhead = setOverhead)
 
    point_IKD = getRange(start = 2_IKD, stop = 20_IKD, step = 2_IKD)
    Point_RKD = real(point_IKD, RKD)
    
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' vs. '))") (bench(i)%name, i = 1, NBENCH)
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "Point", (bench(i)%name, i = 1, NBENCH)
 
        loopOverPoint: do ipnt = 1, NPNT
 
            write(*,"(*(g0,:,' '))") "Benchmarking with point", point_IKD(ipnt)
 
            do i = 1, NBENCH
                bench(i)%timing = bench(i)%getTiming()
            end do
 
            write(fileUnit,"(*(g0,:,','))") point_IKD(ipnt), (bench(i)%timing%mean, i = 1, NBENCH)
 
        end do loopOverPoint
 
        write(*,"(*(g0,:,' '))") dummy
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call finalize()
    end subroutine
 
    subroutine finalize()
        dummy = dummy + factorial_IKD + factorial_RKD
    end subroutine
 
    subroutine getFactorial()
        block
            use pm_mathFactorial, only: getFactorial
            factorial_IKD = getFactorial(point_IKD(ipnt))
            call finalize()
        end block
    end subroutine
 
    subroutine getLogFactorial()
        block
            use pm_mathFactorial, only: getLogFactorial
            factorial_RKD = log(getLogFactorial(Point_RKD(ipnt)))
            call finalize()
        end block
    end subroutine
 
end program benchmark

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Postprocessing of the benchmark output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
 
fontsize = 14
 
methods = ["getLogFactorial", "getFactorial"]
labels = [label+"(Whole Number)" for label in methods]
 
df = pd.read_csv("main.out")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for method in methods:
    plt.plot( df["Point"].values
            , df[method].values
            , linewidth = 2
            , linestyle = "-"
            , marker = "*"
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Whole Number", fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title("getLogFactorial() vs. getFactorial()\nLower is better.", fontsize = fontsize)
ax.set_xscale("log")
ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( labels
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.getFactorial_vs_getLogFactorial.runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
plt.plot( df["Point"].values
        , np.ones(len(df["Point"].values))
        , linestyle = "-"
        , marker = "*"
       #, color = "black"
        , linewidth = 2
        )
plt.plot( df["Point"].values
        , df["getFactorial"].values / df["getLogFactorial"].values
        , linestyle = "-"
        , marker = "*"
        , linewidth = 2
        )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel("Whole Number", fontsize = fontsize)
ax.set_ylabel("Runtime Ratio", fontsize = fontsize)
ax.set_title("getFactorial() / getLogFactorial(), Lower means faster.\nLower than 1 means faster than getLogFactorial().", fontsize = fontsize)
ax.set_xscale("log")
#ax.set_yscale("log")
plt.minorticks_on()
plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
ax.tick_params(axis = "y", which = "minor")
ax.tick_params(axis = "x", which = "minor")
ax.legend   ( labels
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark.getFactorial_vs_getLogFactorial.runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

1. The procedures under the generic interface getFactorial compute the factorial using its default definition while the procedures under the generic interface getLogFactorial use the Fortran intrinsic log_gamma() to compute the log(factorial) which is then converted to factorial in the benchmark code.
   Based on the benchmark results, the safe method of computing the factorial as a real number (thus avoiding the potential numerical overflow) is about 3-5 times slower than the direct definition of the factorial, although the performance gap closes at large input whole numbers.
   

Test:

test_pm_mathFactorial

Final Remarks ⛓

---

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
   
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
   

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, April 23, 2017, 1:36 AM, Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin

Variable Documentation

MODULE_NAME

character(*, SK), parameter pm_mathFactorial::MODULE_NAME = "@pm_mathFactorial"

Definition at line 57 of file pm_mathFactorial.F90.