pm_mathSqrt Module Reference

This module contains procedures and generic interfaces and generic interfaces for computing the square root of integers. More...

Data Types
interface	getSqrt
	Generate and return the integer square root of an input non-negative integer. More...

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

Detailed Description

This module contains procedures and generic interfaces and generic interfaces for computing the square root of integers.

In number theory, the integer square root (getSqrt) of a non-negative integer \(n\) is the non-negative integer \(m\) which is the greatest integer less than or equal to the square root of \(n\),

\begin{equation} \mbox{getSqrt}(n) = \lfloor{\sqrt{n}}\rfloor ~. \end{equation}

For example,

\begin{equation} \mbox{getSqrt}(27) = \lfloor{\sqrt{27}}\rfloor = \lfloor 5.19615242270663\ldots \rfloor = 5 ~. \end{equation}

Two methods of finding the integer square root include linear and binary search, both of which are implemented by the procedures of this module.

Note

While the procedures of this module return getSqrt(n), the greatest integer less than or equal to the square root of a given integer, one can readily compute the smallest integer a greater than or equal to the square root of n as a = 1 + getSqrt(n - 1).
This number is equivalent to the ceiling of the exact square root of n.

Benchmarks:

Benchmark :: The runtime performance of getSqrt for integer vs. real input ndim. ⛓

! Test the performance of `getSqrt()` with and without the selection `control` argument.
program benchmark
 
    use pm_bench, only: bench_type
    use pm_arrayUnique, only: getUnique
    use pm_arraySpace, only: getLogSpace
    use pm_kind, only: IK, LK, IKG => IK, RK, SK
    use iso_fortran_env, only: error_unit
 
    implicit none
 
    integer(IK)                         :: ibench
    integer(IK)                         :: iposint
    integer(IK)                         :: fileUnit
    integer(IKG)                        :: intSqrt, dumm
    integer(IKG)        , allocatable   :: posint(:)
    type(bench_type)    , allocatable   :: bench(:)
 
    bench = [ bench_type(name = SK_"getSqrtBinary", exec = getSqrtBinary, overhead = setOverhead) &
            , bench_type(name = SK_"getSqrtLinear", exec = getSqrtLinear, overhead = setOverhead) &
            , bench_type(name = SK_"floor_sqrt", exec = floor_sqrt, overhead = setOverhead) &
            ]
 
    write(*,"(*(g0,:,' '))")
    write(*,"(*(g0,:,' vs. '))") (bench(ibench)%name, ibench = 1, size(bench))
    write(*,"(*(g0,:,' '))")
 
    open(newunit = fileUnit, file = "main.out", status = "replace")
 
        write(fileUnit, "(*(g0,:,','))") "Integer", (bench(ibench)%name, ibench = 1, size(bench))
        posint = getUnique(int(getLogSpace(0._RK, log(real(huge(0_IK), RK)), count = 50_IK), IKG))
        loopOverArraySize: do iposint = 1, size(posint)
 
            write(*,"(*(g0,:,' '))") "Benchmarking with posint = ", posint(iposint)
            do ibench = 1, size(bench)
                bench(ibench)%timing = bench(ibench)%getTiming(minsec = 0.04_RK)
            end do
            write(fileUnit,"(*(g0,:,','))") posint(iposint), (bench(ibench)%timing%mean, ibench = 1, size(bench))
 
        end do loopOverArraySize
        write(*,"(*(g0,:,' '))") dumm
        write(*,"(*(g0,:,' '))")
 
    close(fileUnit)
 
contains
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! procedure wrappers.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    subroutine setOverhead()
        call getDummy()
    end subroutine
 
    subroutine getDummy()
        dumm = intSqrt - dumm
    end subroutine
 
    subroutine getSqrtBinary()
        use pm_mathSqrt, only: getSqrt, binary
        intSqrt = getSqrt(posint(iposint), binary)
        call getDummy()
    end subroutine
 
    subroutine getSqrtLinear()
        use pm_mathSqrt, only: getSqrt, linear
        intSqrt = getSqrt(posint(iposint), linear)
        call getDummy()
    end subroutine
 
    subroutine floor_sqrt()
        intSqrt = floor(sqrt(real(posint(iposint), RK)), IKG)
        call getDummy()
    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
 
import os
dirname = os.path.basename(os.getcwd()) 
 
fontsize = 14
 
df = pd.read_csv("main.out", delimiter = ",")
colnames = list(df.columns.values)
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
ax.set_title(" vs. ".join(colnames[1:])+"\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   ( colnames[1:]
           #, loc='center left'
           #, bbox_to_anchor=(1, 0.5)
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.png")
 
 
 
ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
ax = plt.subplot()
 
for colname in colnames[1:]:
    plt.plot( df[colnames[0]].values
            , df[colname].values / df[colnames[1]].values
            , linewidth = 2
            )
 
plt.xticks(fontsize = fontsize)
plt.yticks(fontsize = fontsize)
ax.set_xlabel(colnames[0], fontsize = fontsize)
ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), 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   ( colnames[1:]
           #, bbox_to_anchor = (1, 0.5)
           #, loc = "center left"
            , fontsize = fontsize
            )
 
plt.tight_layout()
plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output ⛓

Benchmark moral ⛓

1. The benchmark procedures named getSqrtBinary and getSqrtLinear call the generic interface getSqrt with a method argument of type binary_type and linear_type respectively.
   
2. From the benchmark results it appears that the binary search algorithm for computing the integer square root is significantly faster than the linear search method.
   The binary search algorithm is also faster than the naive approach based on the formula floor(sqrt(real(posint, RK))).
   

Test:

test_pm_mathSqrt

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_mathSqrt::MODULE_NAME = "@pm_mathSqrt"

Definition at line 75 of file pm_mathSqrt.F90.