Generate and return the mean of the Band distribution for an input set of parameters.
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Generate and return the mean of the Band distribution for an input set of parameters.
See the documentation of pm_distBand for more information on the Band distribution.
This generic interface performs one of the following quantities:

The mean of the Band distribution.
The mean of the Band distribution with an unknown normalization (amplitude) over an arbitrary range \((\ms{lb}, \ms{ub})\) is defined by the following integral ratio,
\begin{equation}
\large
\mu_E = \frac{ \int_{\ms{lb}}^{\ms{ub}} E ~ f_{\ms{BAND}}(E  \alpha, \beta, \ebreak) dE }{ \int_{\ms{lb}}^{\ms{ub}} f_{\ms{BAND}}(E  \alpha, \beta, \ebreak) dE } ~,
\end{equation}
where \(\mu_E\) is the mean of the Band distribution.
The mean of the Band distribution is particularly important for converting the photon fluence of a Band spectrum to the corresponding energy fluence.
For example, the energy fluence \(\sergs\) of the Band distribution with the same physical unit as \(E\) can be computed from the corresponding the Band photon fluence \(\sphot\) as,
\begin{equation}
\large
\sergs = \sphot \mu_E ~,
\end{equation}

The generalized mean of the Band distribution.
Optionally, this generic interface also computes the above integral with new support \((\ms{lbnew}, \ms{ubnew})\),
\begin{equation}
\large
\mu_E = \frac{ \int_{\ms{lbnew}}^{\ms{ubnew}} E ~ f_{\ms{BAND}}(E  \alpha, \beta, \ebreak) dE }{ \int_{\ms{lb}}^{\ms{ub}} f_{\ms{BAND}}(E  \alpha, \beta, \ebreak) dE } ~,
\end{equation}
where \(\mu_E\) is not anymore the common definition of the distribution mean, but a generalization of the concept.
This generalized mean facilitates the computation of the energy or photon fluence over a different range from the range of the original photon or energy fluence.
 Warning
 The input arguments
lbnew
, ubnew
, lb
, ub
, ebreak
must all be unitless (without physical dimensions) or all have the same physical units (typically, \(\kev\)).
 Note
 The physical units of the input or output units can be changed via the facilities of module pm_physUnit.
 Parameters

[out]  mean  : The output positive scalar or array of the same shape as any input arraylike argument, of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128).

If the input arguments
lbnew and ubnew are missing, then mean will contain the mean of the Band distribution on return.

If the input arguments
lbnew and ubnew are present, then mean will contain the ratio of the UCDF of the Band distribution in the range \((\ms{lbnew}, \ms{ubnew})\) to the UCDF of the Band distribution in the \((\ms{lb}, \ms{ub})\) as defined above.

[in]  lb  : The input positive scalar or array of the same shape as any input arraylike argument, of the same type and kind as the output argument mean , representing the lower bound of the Band distribution.

[in]  ub  : The input positive scalar or array of the same shape as any input arraylike argument, of the same type and kind as the output argument mean , representing the upper bound of the Band distribution.

[in]  alpha  : The input scalar or array of the same shape as other arraylike arguments of the same type and kind as mean , containing the first shape parameter of the distribution.

[in]  beta  : The input scalar or array of the same shape as other arraylike arguments of the same type and kind as mean , containing the second shape parameter of the distribution.

[in]  ebreak  : The input scalar or array of the same shape as other arraylike arguments of the same type and kind as mean , containing the normalized spectral break energy values: \(\ebreak = \frac{\ebreak}{100\kev}\).

[out]  info  : The output scalar of type integer of default kind IK.
On output, it is set to positive the number of iterations taken for the series representation of the Gamma function to converge.
If the algorithm fails to converge, then info is set to the negative of the number of iterations taken by the algorithm or, to the output error returned by brute force integrator getQuadErr.
An negative output value signifies the lack of convergence and failure to compute the UCDF.
This is likely to happen if the input value for alpha or beta are too extreme.

[in]  lbnew  : The input positive scalar or array of the same shape as any input arraylike argument, of the same type and kind as the output argument mean , representing the new lower bound of the Band distribution.
(optional, default = lb ) 
[in]  ubnew  : The input positive scalar or array of the same shape as any input arraylike argument, of the same type and kind as the output argument mean , representing the new upper bound of the Band distribution.
(optional, default = ub ) 
Possible calling interfaces ⛓
call setBandMean(mean, lb, ub, alpha, beta, ebreak, info)
call setBandMean(mean, lb, ub, alpha, beta, ebreak, info, lbnew, ubnew)
Generate and return the mean of the Band distribution for an input set of parameters.
This module contains procedures and generic interfaces for computing the Band photon distribution wid...
 Warning
 The condition
0 < lb
must hold for the corresponding input arguments.
The condition lb < ub
must hold for the corresponding input arguments.
The condition 0 < lbnew
must hold for the corresponding input arguments.
The condition lbnew < ubnew
must hold for the corresponding input arguments.
The condition 0 < fluence
must hold for the corresponding input arguments.
The condition 0 < lbnew
must hold for the corresponding input arguments.
The condition 0 < ubnew
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
.
 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
setBandMean
getBandZeta
getBandEpeak
getBandEbreak
setBandPhoton
setBandEnergy
Example usage ⛓
13 type(display_type) :: disp
19 real(RKG) :: mean, fluence, lbnew, ubnew, lb, ub, alpha, beta, ebreak
22 call disp%show(
"fluence = 2.044544e07_RKG; lbnew = 50._RKG; ubnew = 300._RKG; lb = 50._RKG; ub = 300._RKG; alpha = 9.469590e01_RKG; beta = 3.722981_RKG; ebreak = getBandEbreak(alpha, beta, 1.928073e+02_RKG);")
23 fluence
= 2.044544e07_RKG; lbnew
= 50._RKG; ubnew
= 300._RKG; lb
= 50._RKG; ub
= 300._RKG; alpha
= 9.469590e01_RKG; beta
= 3.722981_RKG; ebreak
= getBandEbreak(alpha, beta,
1.928073e+02_RKG);
24 call disp%show(
"call setBandMean(mean, lb, ub, alpha, beta, ebreak, info)")
25 call setBandMean(mean, lb, ub, alpha, beta, ebreak, info)
26 call disp%show(
"if (info < 0) error stop")
27 if (info
< 0)
error stop
28 call disp%show(
"real(ERGS2KEV * fluence / mean, RKG) ! photon fluence 1.084876")
Generate and return the spectral break energy parameter of the Band spectral model/distribution from ...
This is a generic method of the derived type display_type with pass attribute.
This is a generic method of the derived type display_type with pass attribute.
This module contains classes and procedures for input/output (IO) or generic display operations on st...
type(display_type) disp
This is a scalar module variable an object of type display_type for general display.
This module defines the relevant Fortran kind typeparameters frequently used in the ParaMonte librar...
integer, parameter LK
The default logical kind in the ParaMonte library: kind(.true.) in Fortran, kind(....
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in CFortran Interoper...
integer, parameter RKD
The double precision real kind in Fortran mode. On most platforms, this is an 64bit real kind.
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in CFortran Intero...
integer, parameter RKH
The scalar integer constant of intrinsic default kind, representing the highestprecision real kind t...
This module contains relevant physical constants.
real(RKB), parameter ERGS2KEV
The scalar constant of type real of kind RKB representing one ergs of energy in kiloelectronvolts.
real(RKB), parameter KEV2ERGS
The scalar constant of type real of kind RKB representing one kiloelectronvolts of energy in ergs.
Generate and return an object of type display_type.
Example Unix compile command via Intel ifort
compiler ⛓
3ifort fpp standardsemantics O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example Windows Batch compile command via Intel ifort
compiler ⛓
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standardsemantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
Example Unix / MinGW compile command via GNU gfortran
compiler ⛓
3gfortran cpp ffreelinelengthnone O3 Wl,rpath,../../../lib I../../../inc main.F90 ../../../lib/libparamonte* o main.exe
Example output ⛓
2fluence
= 2.044544e07_RKG; lbnew
= 50._RKG; ubnew
= 300._RKG; lb
= 50._RKG; ub
= 300._RKG; alpha
= 9.469590e01_RKG; beta
= 3.722981_RKG; ebreak
= getBandEbreak(alpha, beta,
1.928073e+02_RKG);
3call setBandMean(mean, lb, ub, alpha, beta, ebreak, info)
4if (info
< 0)
error stop
5real(ERGS2KEV
* fluence
/ mean, RKG)
 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.

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.

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 1008 of file pm_distBand.F90.