# -*- coding: utf-8 -*-
# mathtoolspy
# -----------
# A fast, efficient Python library for mathematically operations, like
# integration, solver, distributions and other useful functions.
#
# Author: sonntagsgesicht, based on a fork of Deutsche Postbank [pbrisk]
# Version: 0.3, copyright Wednesday, 18 September 2019
# Website: https://github.com/sonntagsgesicht/mathtoolspy
# License: Apache License 2.0 (see LICENSE file)
[docs]class GaussKronrodIntegrator:
_MAX_INTEGRAL_LENGTH = 50
'''
nsteps = 100,
max_number_of_iterations = 255,
min_number_of_iterations = 7,
initial_order = 3,
abs_tolerance = 1E-10,
rel_tolerance = 1E-4,
check_abs_tolerance = True,
check_rel_tolerance = False)
'''
def __init__(self, max_number_of_iterations=255, initial_order=3, min_number_of_iterations=7,
abs_tolerance=1E-10, rel_tolerance=1E-4,
check_abs_tolerance=True, check_rel_tolerance=False):
self.max_number_of_iterations = max_number_of_iterations
self.initial_order = initial_order
self.abs_tolerance = abs_tolerance
self.rel_tolerance = rel_tolerance
self.check_abs_tolerance = check_abs_tolerance
self.check_rel_tolerance = check_rel_tolerance
self.min_number_of_iterations = self._init_min_iterations(min_number_of_iterations)
def _init_min_iterations(self, min_number_of_iterations):
max_iter = self.max_number_of_iterations
if not min_number_of_iterations <= max_iter:
raise IndentationError('Number of %d iterations exceeded.' % max_iter)
order = self.initial_order
for k in range(max_iter):
if GaussKronrodConstants.gaussKronrodPattersonRule[k] >= order:
return max(k, min_number_of_iterations)
return max_iter
[docs] def integrate(self, function, lower_bound, upper_bound):
if upper_bound - lower_bound > GaussKronrodIntegrator._MAX_INTEGRAL_LENGTH:
bounds = self._get_bounds(lower_bound, upper_bound)
return sum([self.integrate(function, a, b) for a, b in bounds])
interval_length = upper_bound - lower_bound
interval_length_half = interval_length / 2
fct_values = 17 * [0]
# apply the 1-point Gauss rule (which is the midpoint rule) and store the function value:
fct_value = function(lower_bound + interval_length_half)
fct_values[0] = fct_value
value_for_refinement = fct_value * interval_length
value = value_for_refinement
previous_accumulation_value = 0
ip = 0
jh = 0
max_iters = min(self.max_number_of_iterations, GaussKronrodConstants.min_max_iter)
min_iters = self.min_number_of_iterations
for k in range(1, max_iters + 1):
value = value_for_refinement
previous_accumulation_value = value_for_refinement
value_for_refinement = 0
for i in range(GaussKronrodConstants.KX[k - 1], GaussKronrodConstants.KX[k]):
for j in range(GaussKronrodConstants.KL[i], GaussKronrodConstants.KH[i] + 1):
value_for_refinement += GaussKronrodConstants.coefficients[ip] * fct_values[j]
ip += 1
# compute constribution from the new function values
jl = jh
jh = jl + jl
j1 = GaussKronrodConstants.FL[k - 1]
j2 = GaussKronrodConstants.FH[k - 1]
for j in range(jl, jh + 1):
x = GaussKronrodConstants.coefficients[ip] * interval_length_half
lf = function(lower_bound + x)
uf = function(upper_bound - x)
fct_value = lf + uf
value_for_refinement += GaussKronrodConstants.coefficients[ip + 1] * fct_value
if j1 <= j2:
fct_values[j1] = fct_value
j1 += 1
ip += 2
value_for_refinement = interval_length_half * value_for_refinement + 0.5 * previous_accumulation_value
jh += 1
if k >= min_iters and self._tolerance_is_reached(value_for_refinement, value):
return value_for_refinement
return value_for_refinement
def __call__(self, fct, lower_bound, upper_bound):
return self.integrate(fct, lower_bound, upper_bound)
def _tolerance_is_reached(self, newValue, oldValue):
if self.check_rel_tolerance:
return abs(newValue - oldValue) < self.rel_tolerance * abs(newValue)
if self.check_abs_tolerance:
return abs(newValue - oldValue) < self.abs_tolerance
return False
def _get_bounds(self, lower_bound, upper_bound):
ret = []
step = GaussKronrodIntegrator._MAX_INTEGRAL_LENGTH
a = lower_bound
while a + step < upper_bound:
b = a + step
ret.append((a, b))
a = b
ret.append((a, upper_bound))
return ret
[docs]class GaussKronrodConstants:
"""
Constants for Gauss Kronrod Integrator
"""
min_max_iter = 7
gaussKronrodPattersonRule = [3, 7, 15, 31, 63, 127, 255]
FL = [1, 2, 4, 8, 11, 13, 0]
FH = [1, 3, 7, 15, 16, 16, -1]
KL = [0, 0, 0, 0, 0, 2, 4, 8, 4, 8, 11]
KH = [0, 1, 3, 7, 15, 2, 5, 16, 4, 8, 16]
KX = [0, 1, 2, 3, 4, 5, 8, 11]
coefficients = [
# Corrections
-.11111111111111111111e+00,
# nodes and weights
+.22540333075851662296e+00,
+.55555555555555555556e+00,
# Corrections
+.647209421402969791e-02,
-.928968790944433705e-02,
# nodes and weights
+.39508731291979716579e-01,
+.10465622602646726519e+00,
+.56575625065319744200e+00,
+.40139741477596222291e+00,
# Corrections
+.5223046896961622e-04,
+.17121030961750000e-03,
-.724830016153892898e-03,
-.7017801099209042e-04,
# nodes and weights
+.61680367872449777899e-02,
+.17001719629940260339e-01,
+.11154076712774300110e+00,
+.92927195315124537686e-01,
+.37889705326277359705e+00,
+.17151190913639138079e+00,
+.77661331357103311837e+00,
+.21915685840158749640e+00,
# corrections
+.682166534792e-08,
+.12667409859336e-06,
+.59565976367837165e-05,
+.1392330106826e-07,
-.6629407564902392e-04,
-.704395804282302e-06,
-.34518205339241e-07,
-.814486910996e-08,
# nodes and weights
+.90187503233240234038e-03,
+.25447807915618744154e-02,
+.18468850446259893130e-01,
+.16446049854387810934e-01,
+.70345142570259943330e-01,
+.35957103307129322097e-01,
+.16327406183113126449e+00,
+.56979509494123357412e-01,
+.29750379350847292139e+00,
+.76879620499003531043e-01,
+.46868025635562437602e+00,
+.93627109981264473617e-01,
+.66886460674202316691e+00,
+.10566989358023480974e+00,
+.88751105686681337425e+00,
+.11195687302095345688e+00,
# corrections
+.371583e-15,
+.21237877e-12,
+.10522629388435e-08,
+.1748029e-14,
+.3475718983017160e-06,
+.90312761725e-11,
+.12558916e-13,
+.54591e-15,
-.72338395508691963e-05,
-.169699579757977e-07,
-.854363907155e-10,
-.12281300930e-11,
-.462334825e-13,
-.42244055e-14,
-.88501e-15,
-.40904e-15,
# nodes and weights
+.12711187964238806027e-03,
+.36322148184553065969e-03,
+.27937406277780409196e-02,
+.25790497946856882724e-02,
+.11315242452570520059e-01,
+.61155068221172463397e-02,
+.27817125251418203419e-01,
+.10498246909621321898e-01,
+.53657141626597094849e-01,
+.15406750466559497802e-01,
+.89628843042995707499e-01,
+.20594233915912711149e-01,
+.13609206180630952284e+00,
+.25869679327214746911e-01,
+.19305946804978238813e+00,
+.31073551111687964880e-01,
+.26024395564730524132e+00,
+.36064432780782572640e-01,
+.33709033997521940454e+00,
+.40715510116944318934e-01,
+.42280428994795418516e+00,
+.44914531653632197414e-01,
+.51638197305415897244e+00,
+.48564330406673198716e-01,
+.61664067580126965307e+00,
+.51583253952048458777e-01,
+.72225017797817568492e+00,
+.53905499335266063927e-01,
+.83176474844779253501e+00,
+.55481404356559363988e-01,
+.94365568695340721002e+00,
+.56277699831254301273e-01,
# corrections
+.1041098e-15,
+.249472054598e-10,
+.55e-20,
+.290412475995385e-07,
+.367282126e-13,
+.5568e-18,
-.871176477376972025e-06,
-.8147324267441e-09,
-.8830920337e-12,
-.18018239e-14,
-.70528e-17,
-.506e-19,
# nodes and weights
+.17569645108401419961e-04,
+.50536095207862517625e-04,
+.40120032808931675009e-03,
+.37774664632698466027e-03,
+.16833646815926074696e-02,
+.93836984854238150079e-03,
+.42758953015928114900e-02,
+.16811428654214699063e-02,
+.85042788218938676006e-02,
+.25687649437940203731e-02,
+.14628500401479628890e-01,
+.35728927835172996494e-02,
+.22858485360294285840e-01,
+.46710503721143217474e-02,
+.33362148441583432910e-01,
+.58434498758356395076e-02,
+.46269993574238863589e-01,
+.70724899954335554680e-02,
+.61679602220407116350e-01,
+.83428387539681577056e-02,
+.79659974529987579270e-01,
+.96411777297025366953e-02,
+.10025510022305996335e+00,
+.10955733387837901648e-01,
+.12348658551529473026e+00,
+.12275830560082770087e-01,
+.14935550523164972024e+00,
+.13591571009765546790e-01,
+.17784374563501959262e+00,
+.14893641664815182035e-01,
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+.16173218729577719942e-01,
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+.18631848256138790186e-01,
+.31701256890892077191e+00,
+.19795495048097499488e-01,
+.35772335749024048622e+00,
+.20905851445812023852e-01,
+.40059606975775710702e+00,
+.21956366305317824939e-01,
+.44550486736806745112e+00,
+.22940964229387748761e-01,
+.49231224246628339785e+00,
+.23854052106038540080e-01,
+.54086998801016766712e+00,
+.24690524744487676909e-01,
+.59102017877011132759e+00,
+.25445769965464765813e-01,
+.64259616216846784762e+00,
+.26115673376706097680e-01,
+.69542355844328595666e+00,
+.26696622927450359906e-01,
+.74932126969651682339e+00,
+.27185513229624791819e-01,
+.80410249728889984607e+00,
+.27579749566481873035e-01,
+.85957576684743982540e+00,
+.27877251476613701609e-01,
+.91554595991628911629e+00,
+.28076455793817246607e-01,
+.97181535105025430566e+00,
+.28176319033016602131e-01,
# corrections
+.3326e-18,
+.114094770478e-11,
+.2952436056970351e-08,
+.51608328e-15,
-.110177219650597323e-06,
-.58656987416475e-10,
-.23340340645e-13,
-.1248950e-16,
# nodes and weights
+.24036202515353807630e-05,
+.69379364324108267170e-05,
+.56003792945624240417e-04,
+.53275293669780613125e-04,
+.23950907556795267013e-03,
+.13575491094922871973e-03,
+.61966197497641806982e-03,
+.24921240048299729402e-03,
+.12543855319048853002e-02,
+.38974528447328229322e-03,
+.21946455040427254399e-02,
+.55429531493037471492e-03,
+.34858540851097261500e-02,
+.74028280424450333046e-03,
+.51684971993789994803e-02,
+.94536151685852538246e-03,
+.72786557172113846706e-02,
+.11674841174299594077e-02,
+.98486295992298408193e-02,
+.14049079956551446427e-02,
+.12907472045965932809e-01,
+.16561127281544526052e-02,
+.16481342421367271240e-01,
+.19197129710138724125e-02,
+.20593718329137316189e-01,
+.21944069253638388388e-02,
+.25265540247597332240e-01,
+.24789582266575679307e-02,
+.30515340497540768229e-01,
+.27721957645934509940e-02,
+.36359378430187867480e-01,
+.30730184347025783234e-02,
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+.49884702478705123440e-01,
+.36933779170256508183e-02,
+.57588434808916940190e-01,
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+.43326409680929828545e-02,
+.74921067092924347640e-01,
+.46573172997568547773e-02,
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+.53130866051870565663e-02,
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