ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
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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

udf = getBandUDF(energy, alpha, beta, ebreak, zeta = zeta, invEfold = invEfold)
Generate and return the unnormalized density function (UDF) of the Band spectral model/distribution.
This module contains procedures and generic interfaces for computing the Band photon distribution wid...
Definition: pm_distBand.F90:97
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

1program example
2
3 use pm_kind, only: SK, IK
4 use pm_distBand, only: getBandUDF
5 use pm_distBand, only: getBandZeta
8 use pm_io, only: display_type
9
10 implicit none
11
12 integer(IK), parameter :: NP = 999
13 real :: udf(NP), point(NP), alpha, beta, ebreak
14
15 type(display_type) :: disp
16 disp = display_type(file = "main.out.F90")
17
18 call setLinSpace(point, x1 = 0.01, x2 = 10.)
19
20 call disp%skip()
21 call disp%show("point(1)")
22 call disp%show( point(1) )
23 call disp%show("alpha = -.5; beta = -1.5; ebreak = 100.")
24 alpha = -.5; beta = -1.5; ebreak = 100.
25 call disp%show("udf(1) = getBandUDF(point(1), alpha, beta, ebreak)")
26 udf(1) = getBandUDF(point(1), alpha, beta, ebreak)
27 call disp%show("udf(1)")
28 call disp%show( udf(1) )
29 call disp%skip()
30
31 call disp%skip()
32 call disp%show("point(1)")
33 call disp%show( point(1) )
34 call disp%show("alpha = -.5; beta = -1.5; ebreak = 100.")
35 alpha = -.5; beta = -1.5; ebreak = 100.
36 call disp%show("udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)")
37 udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)
38 call disp%show("udf(1)")
39 call disp%show( udf(1) )
40 call disp%skip()
41
42 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
43 ! Output an example udf array for visualization.
44 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
45
46 block
47 integer(IK) :: fileUnit, i
48 open(newunit = fileUnit, file = "getBandUDF.RK.txt")
49 do i = 1, NP
50 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.])
51 end do
52 close(fileUnit)
53 end block
54
55end program example
Generate count evenly spaced points over the interval [x1, x2] if x1 < x2, or [x2,...
Return the linSpace output argument with size(linSpace) elements of evenly-spaced values over the int...
Generate and return the coefficient of continuity of the Band spectral model/distribution from the Ba...
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11726
This is a generic method of the derived type display_type with pass attribute.
Definition: pm_io.F90:11508
This module contains procedures and generic interfaces for generating arrays with linear or logarithm...
This module contains classes and procedures for input/output (IO) or generic display operations on st...
Definition: pm_io.F90:252
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
Definition: pm_io.F90:11393
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
Generate and return an object of type display_type.
Definition: pm_io.F90:10282

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example output
1
2point(1)
3+0.999999978E-2
4alpha = -.5; beta = -1.5; ebreak = 100.
5udf(1) = getBandUDF(point(1), alpha, beta, ebreak)
6udf(1)
7+9.99899960
8
9
10point(1)
11+0.999999978E-2
12alpha = -.5; beta = -1.5; ebreak = 100.
13udf(1) = getBandUDF(point(1), alpha, beta, ebreak, zeta = getBandZeta(alpha, beta, ebreak), invEfold = (alpha - beta) / ebreak)
14udf(1)
15+9.99899960
16
17

Postprocessing of the example output
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6import glob
7import sys
8
9linewidth = 2
10fontsize = 17
11
12marker ={ "CK" : "-"
13 , "IK" : "."
14 , "RK" : "-"
15 }
16xlab = { "CK" : "X ( real/imaginary components )"
17 , "IK" : "X ( integer-valued )"
18 , "RK" : "X ( real-valued )"
19 }
20label = [ r"$\alpha, \beta = +0.5, -0.5, x_b = 0.5$"
21 , r"$\alpha, \beta = +1.5, -1.0, x_b = 1.0$"
22 , r"$\alpha, \beta = -0.5, -2.0, x_b = 2.0$"
23 , r"$\alpha, \beta = +1.5, -3.0, x_b = 5.0$"
24 ]
25
26for kind in ["IK", "CK", "RK"]:
27
28 pattern = "*." + kind + ".txt"
29 fileList = glob.glob(pattern)
30 if len(fileList) == 1:
31
32 df = pd.read_csv(fileList[0], delimiter = " ")
33
34 fig = plt.figure(figsize = 1.25 * np.array([6.4, 4.8]), dpi = 200)
35 ax = plt.subplot()
36
37 if kind == "CK":
38 plt.plot( df.values[:, 0]
39 , df.values[:,1:]
40 , marker[kind]
41 , linewidth = linewidth
42 #, color = "r"
43 )
44 plt.plot( df.values[:, 1]
45 , df.values[:,1:]
46 , marker[kind]
47 , linewidth = linewidth
48 #, color = "blue"
49 )
50 else:
51 plt.plot( df.values[:, 0]
52 , df.values[:,1:]
53 , marker[kind]
54 , linewidth = linewidth
55 #, color = "r"
56 )
57 ax.legend ( label
58 , fontsize = fontsize
59 )
60
61 plt.xticks(fontsize = fontsize - 2)
62 plt.yticks(fontsize = fontsize - 2)
63 ax.set_xscale("log")
64 ax.set_yscale("log")
65 ax.set_xlabel(xlab[kind], fontsize = 17)
66 ax.set_ylabel("Unnormalized Density Function (UDF)", fontsize = 17)
67
68 plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
69 ax.tick_params(axis = "y", which = "minor")
70 ax.tick_params(axis = "x", which = "minor")
71 plt.tight_layout()
72
73 plt.savefig(fileList[0].replace(".txt",".png"))
74
75 elif len(fileList) > 1:
76
77 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.
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Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 656 of file pm_distBand.F90.


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