#*********************************************
#*********************************************
#' Calculates the probability density function of the pressure amplitude from a finite number of sine waves of arbitrary individual amplitudes, given in Equation 56 of Barakat 1974, using the terminology of that paper.
#'
#' @param r is the argument (superimposed pressure amplitude).
#' @param beta_n is the vector of individual pressure amplitudes, or a single number giving the number of significant scatterers.
#' @param N is the number of positive roots of the Bessel function of the first kind.
#' @param max.memory is the maximum memory occupied by the function before splitting into a loop for each value of 'r'.
#'
#' @return
#'
#' @examples
#' \dontrun{}
#'
#' @importFrom gsl bessel_zero_J0
#'
#' @export
#' @rdname dBarakatA
#'
dBarakatA=function(r,beta_n,magn=1,N=100,betadistr=c("seq","flat"),max.memory=1e9){
############ AUTHOR(S): ############
# Arne Johannes Holmin
############ LANGUAGE: #############
# English
############### LOG: ###############
# Start: 2012-07-27 - Clean version.
########### DESCRIPTION: ###########
# Calculates the probability density function of the pressure amplitude from a finite number of sine waves of arbitrary individual amplitudes, given in Equation 56 of Barakat 1974, using the terminology of that paper.
########## DEPENDENCIES: ###########
#
############ VARIABLES: ############
# ---r--- is the argument (superimposed pressure amplitude).
# ---beta_n--- is the vector of individual pressure amplitudes, or a single number giving the number of significant scatterers.
# ---N--- is the number of positive roots of the Bessel function of the first kind.
# ---max.memory--- is the maximum memory occupied by the function before splitting into a loop for each value of 'r'.
##################################################
##################################################
##### Preparation #####
# Function calculating the characteristic function of the sum of scatterers (Barakat 1974, equation 31):
phi_y=function(omega,beta_n){
apply(besselJ(outer(beta_n,omega,"*"),0),2,prod)
}
# Generate the 'beta_n' vector:
beta_n=magn*getSeqBarakat(beta_n,betadistr=betadistr)
# Max of r:
R=sum(beta_n)
# Set the lower limit for r as the maximum of 0 and max(beta_n)-sum(beta_n[-which.max(beta_n)])
R0=max(0,max(beta_n)-sum(beta_n[-which.max(beta_n)]))
# Orders of the Bessel function of the first kind:
N=seq_len(N)
##### Execution #####
# Roots of the Bessel function of the first kind and zeroth order:
gamma_n=bessel_zero_J0(N)
# NUMERATOR of the expression in Barakat, equation 56 (dependent on 55 and 31):
NUMERATOR=phi_y(gamma_n/R,beta_n)
# Denominator of the expression in Barakat, equation 56 (dependent on 55 and 31):
DENOMINATOR=(R*besselJ(gamma_n,1))^2
# Factor of the expression in Barakat, equation 56 (dependent on 55 and 31):
if(.Platform$r_arch=="x86_64" && length(gamma_n)*length(r)*16<max.memory || .Platform$r_arch=="i386" && length(gamma_n)*length(r)*8<max.memory){
FACTOR=besselJ(outer(gamma_n,r,"*")/R,0)
# The expression in Barakat, equation 56 (dependent on 55 and 31):
out=2*r*colSums(NUMERATOR / DENOMINATOR * FACTOR)
out[r<R0 | r>R]=0
}
else{
out=r
for(i in seq_along(r)){
# Factor of the expression in Barakat, equation 56 (dependent on 55 and 31):
FACTOR=besselJ(gamma_n*r[i]/R,0)
# The expression in Barakat, equation 56 (dependent on 55 and 31):
out[i]=2*r[i]*sum(NUMERATOR / DENOMINATOR * FACTOR)
}
out[r<R0 | r>R]=0
}
##### Output #####
out
##################################################
##################################################
}
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