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#!/usr/bin/env python3
# encoding: utf-8
"""
_BesselTransform.py
Created by Graham Dennis on 2013-11-26.
Copyright (c) 2013, Graham Dennis
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 2 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
"""
from xpdeint.Features.Transforms.MMT import MMT
from xpdeint.Features.Transforms.BesselBasis import BesselBasis
from xpdeint.Geometry.BesselDimensionRepresentation import BesselDimensionRepresentation
from xpdeint.Geometry.BesselNeumannDimensionRepresentation import BesselNeumannDimensionRepresentation
from xpdeint.Geometry.SphericalBesselDimensionRepresentation import SphericalBesselDimensionRepresentation
from xpdeint.ParserException import ParserException, error_missing_python_library
# We don't directly import mpmath so that mpmath isn't a requirement for xpdeint
# unless you use MMT's.
mpmath = None
# Again, don't directly import numpy
numpy = None
scipy = None
def require_mpmath():
global mpmath
if not mpmath:
try:
import mpmath
except ImportError:
error_missing_python_library("mpmath")
if not hasattr(mpmath, 'besselj'):
mpmath.besselj = mpmath.jn
mpmath.mp.prec = 64
def require_numpy():
global numpy
if not numpy:
try:
import numpy
except ImportError:
error_missing_python_library("numpy")
def require_scipy():
require_numpy()
global scipy
if not scipy:
try:
import scipy
import scipy.special
import scipy.optimize
except ImportError:
error_missing_python_library("scipy")
def besselJZeros(m, a, b):
require_mpmath()
if not hasattr(mpmath, 'besseljzero'):
besseljn = lambda x: mpmath.besselj(m, x)
results = [mpmath.findroot(besseljn, mpmath.pi*(kp - 1./4 + 0.5*m)) for kp in range(a, b+1)]
else:
results = [mpmath.besseljzero(m, i) for i in range(a, b+1)]
# Check that we haven't found double roots or missed a root. All roots should be separated by approximately pi
assert all([0.6*mpmath.pi < (b - a) < 1.4*mpmath.pi for a, b in zip(results[:-1], results[1:])]), "Separation of Bessel zeros was incorrect."
return results
def besselJPrimeZeros(m, a, b):
require_mpmath()
results = [mpmath.besseljzero(m, i, derivative=1) for i in range(a, b+1)]
return results
def besselNeumannMatrix(m, besseljzeros, besselValues, S):
require_scipy()
N = len(besseljzeros)
matrix = numpy.zeros([N, N])
for i in range(N):
for j in range(N):
matrix[i,j] = (2.0 / S) * scipy.special.jn(m, besseljzeros[i] * besseljzeros[j] / S) \
/ (besselValues[i] * besselValues[j])
return matrix
def besselNeumannSFactor(m, besseljzeros):
require_scipy()
besseljzeros_for_matrix = besseljzeros[:-1]
N = len(besseljzeros_for_matrix)
print("Computing the Bessel-Neumann transform S factor for lattice size %i (this will be performed once and saved for each lattice size)..." % (N))
besselValues = []
for i in range(N):
x = numpy.abs(scipy.special.jn(m, besseljzeros[i]))
if m > 0:
x *= numpy.sqrt(1.0 - m*m / (besseljzeros[i] * besseljzeros[i]))
besselValues.append(x)
def f(S):
matrix = besselNeumannMatrix(m, besseljzeros_for_matrix, besselValues, S)
determinant = numpy.linalg.det(matrix)
result = numpy.abs(determinant) - 1.0
return result
S, results = scipy.optimize.brentq(f, besseljzeros[-2], besseljzeros[-1], full_output=True)
return S
class _BesselTransform(MMT):
transformName = 'BesselTransform'
def __init__(self, *args, **KWs):
MMT.__init__(self, *args, **KWs)
dataCache = self.getVar('dataCache')
self.besselJZeroCache = dataCache.setdefault('besselJZeros', {})
self.besselJPrimeZeroCache = dataCache.setdefault('besselJPrimeZeros', {})
self.besselNeumannSCache = dataCache.setdefault('besselNeumannSFactor', {})
def newDimension(self, name, lattice, minimum, maximum,
parent, transformName, aliases = set(),
spectralLattice = None,
type = 'real', volumePrefactor = None,
xmlElement = None):
assert type == 'real'
assert transformName in ['bessel', 'spherical-bessel', 'bessel-neumann']
if not spectralLattice:
spectralLattice = lattice
dim = super(_BesselTransform, self).newDimension(name, max(lattice, spectralLattice), minimum, maximum,
parent, transformName, aliases,
type, volumePrefactor, xmlElement)
self.basisMap[dim.name] = transformName # Needs to be constructed basis here
# Bessel functions
order = 0
if xmlElement.hasAttribute('order'):
features = self.getVar('features')
orderString = xmlElement.getAttribute('order')
try:
order = int(orderString)
except ValueError:
raise ParserException(xmlElement, "Cannot understand '%s' as a meaningful order. "
"Order values must be non-negative integers." \
% xmlElement.getAttribute('order'))
else:
if order < 0:
raise ParserException(xmlElement, "The 'order' attribute for Bessel transforms must be non-negative integers.")
orderOffset = 0
dimRepClass = BesselDimensionRepresentation
if transformName == 'bessel-neumann':
weightOrder = order
dimRepClass = BesselNeumannDimensionRepresentation
else:
weightOrder = order + 1
if transformName == 'spherical-bessel':
dimRepClass = SphericalBesselDimensionRepresentation
if not self.hasattr('uselib'):
self.uselib = []
self.uselib.append('gsl')
orderOffset = 0.5
basis = BesselBasis(parent = self, **self.argumentsToTemplateConstructors)
self.basisMap[dim.name] = dict(
globalsFunction = self.globalsForDim,
order = order,
orderOffset = orderOffset,
lattice = lattice,
transformations = dict([
((name, 'k' + name), BesselBasis(parent = self, **self.argumentsToTemplateConstructors))
])
)
if not float(minimum) == 0.0:
raise ParserException(xmlElement, "The domain for Bessel transform dimensions must begin at 0.")
# Real space representation
xspace = dimRepClass(name = name, type = type, runtimeLattice = lattice,
stepSizeArray = True, parent = dim,
tag = self.coordinateSpaceTag,
_maximum = maximum, _order = order, _weightOrder = weightOrder,
**self.argumentsToTemplateConstructors)
dim.addRepresentation(xspace)
# Spectral space representation
kspace = dimRepClass(name = 'k' + name, type = type, runtimeLattice = spectralLattice,
stepSizeArray = True, parent = dim,
_maximum = '(_besseljS_%(name)s/((real)%(maximum)s))' % locals(),
_order = order, _weightOrder = weightOrder,
reductionMethod = dimRepClass.ReductionMethod.fixedStep,
tag = self.spectralSpaceTag,
**self.argumentsToTemplateConstructors)
dim.addRepresentation(kspace)
return dim
def besselJZeros(self, m, k):
if not m in self.besselJZeroCache:
self.besselJZeroCache[m] = besselJZeros(m, 1, k)
else:
if len(self.besselJZeroCache[m]) < k:
self.besselJZeroCache[m].extend(besselJZeros(m, len(self.besselJZeroCache[m])+1, k))
return self.besselJZeroCache[m][:k]
def besselJPrimeZeros(self, m, k):
if not m in self.besselJPrimeZeroCache:
self.besselJPrimeZeroCache[m] = besselJPrimeZeros(m, 1, k)
else:
if len(self.besselJPrimeZeroCache[m]) < k:
self.besselJPrimeZeroCache[m].extend(besselJPrimeZeros(m, len(self.besselJPrimeZeroCache[m])+1, k))
return self.besselJPrimeZeroCache[m][:k]
def besselNeumannSFactor(self, m, k):
require_numpy()
if not (m, k) in self.besselNeumannSCache:
zeros = list(map(numpy.double, self.besselJPrimeZeros(m, k+1)))
S = besselNeumannSFactor(m, zeros)
self.besselNeumannSCache[(m, k)] = float(S)
return self.besselNeumannSCache[(m, k)]
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