pm_distBand::getBandUDF Interface Reference

Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.
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Detailed Description

Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.

See the documentation of pm_distBand for more information on the Band distribution.
The unnormalized unit-less density function of the Band model can be written as,

\begin{equation} \large f_{\ms{BAND}}(E | \alpha, \beta, \ebreak) = \begin{cases} E^\alpha \exp\left(-\frac{E}{\efold}\right) &,~ \ms{if} & 0 < \ms{lb} \leq E < \ebreak \\ \zeta E^\beta &,~ \ms{if} & \ebreak \leq E < \ms{ub} < +\infty \\ \end{cases} \end{equation}

where,

1. \(E\) is the (presumably unitless) energy at which the distribution must be computed,
2. \(\ebreak\) is the (presumably unitless or of the same unit as \(E\)) spectral break energy,
3. \(\efold = \frac{\alpha - \beta}{\ebreak}\) is the e-folding energy, a measure of the scale of the distribution below and above which the distribution approaches power-law behavior with exponents \(\alpha\) and \(\beta\) respectively,
4. the factor \(\eta(\alpha, \beta, \ebreak)\) is a normalization constant that properly normalizes the PDF,
5. the factor \(\zeta = \ebreak^{\alpha - \beta} \exp\left(\beta - \alpha\right)\) is the coefficient of continuity of the distribution that makes the distribution continuously differentiable.
   
6. the constants \((\ms{lb}, \ms{ub})\) represent the lower and upper bounds of the PDF respectively.
   

Parameters

[in]	energy	: The input scalar or array of the same shape as other array like arguments of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the energy at which the UDF must be computed.
[in]	alpha	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy, containing the first shape parameter of the distribution.
[in]	beta	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy, containing the second shape parameter of the distribution.
[in]	ebreak	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy, containing the spectral break energy values.
[in]	zeta	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy, containing the containing the coefficient of continuity of the distribution. (optional. default = getBandZeta(alpha, beta, ebreak). Its presence can expedite the computations.) This means that the output udf is computed from the lower tail of the distribution.)
[in]	invEfold	: The input scalar or array of the same shape as other array-like arguments of the same type and kind as energy, containing the inverse of the e-folding energy of the Band model: \(\frac{1}{\efold} = \frac{\alpha - \beta}{\ebreak}\). (optional. default = (alpha - beta) / ebreak. Its presence can expedite the computations.)

Returns

udf : The output scalar or array of the same shape as any input array-like argument, of the same type and kind as the input argument energy, containing the distribution UDF.

Possible calling interfaces ⛓

use pm_distBand, only: getBandUDF
 
udf = getBandUDF(energy, alpha, beta, ebreak, zeta = zeta, invEfold = invEfold)

Warning

The condition 0 < energy must hold for the corresponding input arguments.
The condition alpha /= -2 must hold for the corresponding input arguments.
The condition 0 < ebreak must hold for the corresponding input arguments.
The condition 0 < invEfold must hold for the corresponding input arguments.
The condition beta < alpha must hold for the corresponding input arguments.
The condition ebreak = (alpha - beta) * invEfold must hold for the corresponding input arguments.
The condition zeta = getZeta(alpha, beta, ebreak) must hold for the corresponding input arguments.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

The pure procedure(s) documented herein become impure when the ParaMonte library is compiled with preprocessor macro CHECK_ENABLED=1.
By default, these procedures are pure in release build and impure in debug and testing builds.

Remarks

The procedures under discussion are elemental.

Note

The normalization (and the physical units) of the input energy is irrelevant as long as the input values ebreak and zeta are computed in the same physical dimensions and with the same normalizations.

See also

getBandUDF
setBandUCDF
getBandZeta
getBandEpeak
getBandEbreak

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK
    use pm_distBand, only: getBandUDF
    use pm_distBand, only: getBandZeta
    use pm_arraySpace, only: setLinSpace
    use pm_arraySpace, only: getLinSpace
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK), parameter  :: NP = 999
    real :: udf(NP), point(NP), alpha, beta, ebreak
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call setLinSpace(point, x1 = 0.01, x2 = 10.)
 
    call disp%skip()
    call disp%show("point(1)")
    call disp%show( point(1) )
    call disp%show("alpha = -.5; beta = -1.5; ebreak = 100.")
                    alpha = -.5; beta = -1.5; ebreak = 100.
    call disp%show("udf(1) = getBandUDF(point(1), alpha, beta, ebreak)")
                    udf(1) = getBandUDF(point(1), alpha, beta, ebreak)
    call disp%show("udf(1)")
    call disp%show( udf(1) )
    call disp%skip()
 
    call disp%skip()
    call disp%show("point(1)")
    call disp%show( point(1) )
    call disp%show("alpha = -.5; beta = -1.5; ebreak = 100.")
                    alpha = -.5; beta = -1.5; ebreak = 100.
    call disp%show("udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)")
                    udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)
    call disp%show("udf(1)")
    call disp%show( udf(1) )
    call disp%skip()
 
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    ! Output an example udf array for visualization.
    !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
    block
        integer(IK) :: fileUnit, i
        open(newunit = fileUnit, file = "getBandUDF.RK.txt")
        do i = 1, NP
            write(fileUnit,"(*(g0,:,' '))") point(i), getBandUDF(point(i), [.5, 1.5, -.5, +1.5], -[.5, 1.0, 2., 3.], [.5, 1.0, 2.0, 5.])
        end do
        close(fileUnit)
    end block
 
end program example

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

Example output ⛓

 
point(1)
+0.999999978E-2
alpha = -.5; beta = -1.5; ebreak = 100.
udf(1) = getBandUDF(point(1), alpha, beta, ebreak)
udf(1)
+9.99899960
 
 
point(1)
+0.999999978E-2
alpha = -.5; beta = -1.5; ebreak = 100.
udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)
udf(1)
+9.99899960
 
 

Postprocessing of the example output ⛓

#!/usr/bin/env python
 
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import glob
import sys
 
linewidth = 2
fontsize = 17
 
marker ={ "CK" : "-"
        , "IK" : "."
        , "RK" : "-"
        }
xlab =  { "CK" : "X ( real/imaginary components )"
        , "IK" : "X ( integer-valued )"
        , "RK" : "X ( real-valued )"
        }
label = [ r"$\alpha, \beta = +0.5, -0.5, x_b = 0.5$"
        , r"$\alpha, \beta = +1.5, -1.0, x_b = 1.0$"
        , r"$\alpha, \beta = -0.5, -2.0, x_b = 2.0$"
        , r"$\alpha, \beta = +1.5, -3.0, x_b = 5.0$"
        ]
 
for kind in ["IK", "CK", "RK"]:
 
    pattern = "*." + kind + ".txt"
    fileList = glob.glob(pattern)
    if len(fileList) == 1:
 
        df = pd.read_csv(fileList[0], delimiter = " ")
 
        fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
        ax = plt.subplot()
 
        if kind == "CK":
            plt.plot( df.values[:, 0]
                    , df.values[:,1:]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            plt.plot( df.values[:,1]
                    , df.values[:,1:]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "blue"
                    )
        else:
            plt.plot( df.values[:, 0]
                    , df.values[:,1:]
                    , marker[kind]
                    , linewidth = linewidth
                   #, color = "r"
                    )
            ax.legend   ( label
                        , fontsize = fontsize
                        )
 
        plt.xticks(fontsize = fontsize - 2)
        plt.yticks(fontsize = fontsize - 2)
        ax.set_xscale("log")
        ax.set_yscale("log")
        ax.set_xlabel(xlab[kind], fontsize = 17)
        ax.set_ylabel("Unnormalized Density Function (UDF)", fontsize = 17)
 
        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")
        plt.tight_layout()
 
        plt.savefig(fileList[0].replace(".txt",".png"))
 
    elif len(fileList) > 1:
 
        sys.exit("Ambiguous file list exists.")

Visualization of the example output ⛓

Test:

test_pm_distBand

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, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 656 of file pm_distBand.F90.

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The documentation for this interface was generated from the following file:

• src/fortran/main/pm_distBand.F90