# -*- coding: utf-8 -*-
#
# This file is part of SpectralToolbox.
#
# SpectralToolbox is free software: you can redistribute it and/or modify
# it under the terms of the LGNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# SpectralToolbox 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
# LGNU Lesser General Public License for more details.
#
# You should have received a copy of the LGNU Lesser General Public License
# along with SpectralToolbox. If not, see <http://www.gnu.org/licenses/>.
#
# DTU UQ Library
# Copyright (C) 2012-2015 The Technical University of Denmark
# Scientific Computing Section
# Department of Applied Mathematics and Computer Science
#
# Copyright (C) 2015-2016 Massachusetts Institute of Technology
# Uncertainty Quantification group
# Department of Aeronautics and Astronautics
#
# Author: Daniele Bigoni
#
import numpy as np
from numpy.fft import ifft
import itertools
[docs]class SparseGrid:
QRULE = ''
DIM = 0
K = 0
SYM = 1
def __init__(self, qrule, dim, k, sym):
"""
Initialization of the Sparse Grid instance.
Syntax:
sg = SparseGrid(qrule, dim, k, sym)
Input:
* qrule = Function of 1D integration rule
* dim = dimension of the integration problem
* k = Accuracy level. The rule will be exact for polynomial up to total order 2k-1
* sym = (optional) only used for own 1D quadrature rule (type not "KPU",...). If sym is supplied and not=0, the code will run faster but will produce incorrect results if 1D quadrature rule is asymmetric.
Description:
Several 1D integration rules are available to be chosen for the ``qrule`` input parameter
* KPU = Nested rule for unweighted integral over [0,1]
* KPN = Nested rule for integral with Gaussian weight
* GQU = Gaussian quadrature for unweighted integral over [0,1] (Gauss-Legendre)
* GQN = Gaussian quadrature for integral with Gaussian weight (Gauss-Hermite)
* CC = Clenshaw-Curtis quadrature for unweighted integral over [-1,1]
* FEJ = Fejer's quadrature for unweighted integral over [-1,1]
* func = any function provided by the user that accept level l and returns nodes n and weights w for univariate quadrature rule with polynomial exactness 2l-1 as [n w] = feval(func,level)
"""
self.QRULE = qrule
self.DIM = dim
self.K = k
self.SYM = sym
def __binomial(self,n,k):
accum = 1
for m in range(1,k+1):
accum = accum*(n-k+m)/m
return accum
[docs] def sparseGrid(self):
"""
sparseGrid(): main function for generating nodes & weights for sparse grids intergration
Syntax:
(n,w) = sparseGrid()
Output:
* n = matrix of nodes with dim columns
* w = row vector of corresponding weights
"""
builtinfct = (self.QRULE == GQU) or (self.QRULE == GQN) or (self.QRULE == KPU) or (self.QRULE == KPN) or (self.QRULE == CC) or (self.QRULE == FEJ)
if self.SYM == None:
if builtinfct:
self.SYM = True
else:
self.SYM = False
# Evaluate the 1D nodes and weights
n1D = []
w1D = []
R1D = np.zeros(self.K)
for level in range(1,self.K+1):
(n,w) = self.QRULE(level)
# If user supplied symmetric 1D rule: keep only positive orthant
if (not builtinfct) and self.SYM:
perm = np.argsort(n)
n = n[perm]
w = w[perm]
n = n[np.floor(len(n)/2):]
w = w[np.floor(len(n)/2):]
R1D[level-1] = len(n)
n1D.append(n)
w1D.append(w)
# Initialization
minq = max(0,self.K - self.DIM)
maxq = self.K - 1
nodes = np.empty( (0,self.DIM) )
weights = np.empty( (0,1) )
# Outer loop over q
for q in range(minq,maxq+1):
r = len(weights)
bq = (-1)**(maxq-q) * self.__binomial(self.DIM-1, self.DIM + q - self.K)
# Matrix of all rowvectors in N**D_{q}
allrv = self.__spgrGetSeq(self.DIM, self.DIM + q)
# Preallocate new rows for nodes and weights
Rq = np.prod(R1D[allrv],1)
sRq = sum(Rq)
nodes = np.vstack( (nodes, np.zeros( (sRq,self.DIM) ) ) )
weights = np.vstack( ( weights, np.zeros((sRq,1)) ) )
# Inner loop collecting product rules
for j in range(0,allrv.shape[0]):
midx = allrv[j,:]
(newn,neww) = self.__spgrKronProd(np.asarray(n1D)[midx], np.asarray(w1D)[midx])
nodes[r:(r+Rq[j]),] = newn
weights[r:(r+Rq[j]),0] = bq * neww
r = r + Rq[j]
# Collect equal node
# 1. Sort rows
perm = np.argsort(nodes.view([('',nodes.dtype)]*nodes.shape[1]),0).ravel()
nodes = nodes[perm,]
weights = weights[perm]
keep = [0]
lastkeep = 0
# 2. make a list of rows to keep and sum weights of equal nodes
for j in range(1,nodes.shape[0]):
if np.prod(nodes[j,] == nodes[j-1,]):
weights[lastkeep] = weights[lastkeep] + weights[j]
else:
lastkeep = j
keep.append(j)
# 3. drop equal rows
nodes = nodes[keep,]
weights = weights[keep]
# If symmetric rule: so far we computed for positive orthant.
# Now copy to other orthants
if self.SYM:
nr = len(weights)
m = n1D[0][0]
for j in range(0,self.DIM):
keep = np.zeros(nr,dtype=int)
numnew = 0
for r in range(0,nr):
if nodes[r,j] != m:
numnew = numnew + 1
keep[numnew-1] = r
if numnew > 0:
nodes = np.vstack( (nodes, nodes[keep[0:numnew],]) )
nodes[nr:nr+numnew,j] = np.tile(2*m,(numnew)) - nodes[nr:nr+numnew,j]
weights = np.vstack( (weights, weights[keep[0:numnew]]) )
nr = nr + numnew
# Final sorting
perm = np.argsort(nodes.view([('',nodes.dtype)]*nodes.shape[1]),0).ravel()
nodes = nodes[perm,]
weights = weights[perm]
# Normalize weights to account for rounding errors
weights = weights / sum(weights)
return (nodes, weights)
def __spgrGetSeq(self,d,norm):
"""
__spgrGetSeq(d,norm): function for generating matrix of all rowvectors in N^D_{norm}
Syntax:
out = __spgrGetSeq(d,norm)
Input:
d = dimension, will be #columns in output
norm = row sum of elements each of the rows has to have
Output:
out = matrix with d columns. Each row represents one vector with all elements >=1
and the sum of elements == norm
"""
seq = np.zeros(d,dtype=int)
a = norm - d
seq[0] = a
fs = np.vstack( (np.empty((0,d),dtype=int), seq.copy()) )
c = 1
while seq[d-1] < a:
if (c == d):
for i in range(c-1,0,-1):
c = i
if seq[i-1] != 0:
break
seq[c-1] = seq[c-1] - 1
c = c + 1
seq[c-1] = a - sum(seq[0:(c-1)])
if (c < d):
seq[c:d] = 0
fs = np.vstack( (fs,seq) )
return fs
def __spgrKronProd(self,n1D,w1D):
"""
SpGrKronProd(): function for generating tensor product quadrature rule
Syntax:
(nodes,weights) = __spgrKronProd(n1d,w1D)
Input:
n1D = array of 1D nodes
w1D = array of 1D weights
Output:
nodes = matrix of tensor product nodes
weights = vector of tensor product weights
"""
x = [n1D[0,]]
weights = w1D[0,]
for j in range(1,len(n1D)):
x.append(n1D[j,])
weights = np.kron(weights, w1D[j,])
nodes = np.asarray(list(itertools.product(*x)))
return (nodes,weights)
[docs]def GQU(l):
""" GQU(): function for generating 1D Gaussian quadrature rules for unweighted integral over [0,1] (Gauss-Legendre)
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [5.0000000000000000e-001];
w = [1.0000000000000000e+000];
elif l == 2:
n = [7.8867513459481287e-001];
w = [5.0000000000000000e-001];
elif l == 3:
n = [5.0000000000000000e-001, 8.8729833462074170e-001];
w = [4.4444444444444570e-001, 2.7777777777777712e-001];
elif l == 4:
n = [6.6999052179242813e-001, 9.3056815579702623e-001];
w = [3.2607257743127516e-001, 1.7392742256872484e-001];
elif l == 5:
n = [5.0000000000000000e-001, 7.6923465505284150e-001, 9.5308992296933193e-001];
w = [2.8444444444444655e-001, 2.3931433524968501e-001, 1.1846344252809174e-001];
elif l == 6:
n = [6.1930959304159849e-001, 8.3060469323313235e-001, 9.6623475710157603e-001];
w = [2.3395696728634746e-001, 1.8038078652407072e-001, 8.5662246189581834e-002];
elif l == 7:
n = [5.0000000000000000e-001, 7.0292257568869854e-001, 8.7076559279969723e-001, 9.7455395617137919e-001];
w = [2.0897959183673620e-001, 1.9091502525256090e-001, 1.3985269574463935e-001, 6.4742483084431701e-002];
elif l == 8:
n = [5.9171732124782495e-001, 7.6276620495816450e-001, 8.9833323870681348e-001, 9.8014492824876809e-001];
w = [1.8134189168918213e-001, 1.5685332293894469e-001, 1.1119051722668793e-001, 5.0614268145185180e-002];
elif l == 9:
n = [5.0000000000000000e-001, 6.6212671170190451e-001, 8.0668571635029518e-001, 9.1801555366331788e-001, 9.8408011975381304e-001];
w = [1.6511967750063075e-001, 1.5617353852000226e-001, 1.3030534820146844e-001, 9.0324080347429253e-002, 4.0637194180784583e-002];
elif l == 10:
n = [5.7443716949081558e-001, 7.1669769706462361e-001, 8.3970478414951222e-001, 9.3253168334449232e-001, 9.8695326425858587e-001];
w = [1.4776211235737713e-001, 1.3463335965499873e-001, 1.0954318125799158e-001, 7.4725674575290599e-002, 3.3335672154342001e-002];
elif l == 11:
n = [5.0000000000000000e-001, 6.3477157797617245e-001, 7.5954806460340585e-001, 8.6507600278702468e-001, 9.4353129988404771e-001, 9.8911432907302843e-001];
w = [1.3646254338895086e-001, 1.3140227225512388e-001, 1.1659688229599563e-001, 9.3145105463867520e-002, 6.2790184732452625e-002, 2.7834283558084916e-002];
elif l == 12:
n = [5.6261670425573451e-001, 6.8391574949909006e-001, 7.9365897714330869e-001, 8.8495133709715235e-001, 9.5205862818523745e-001, 9.9078031712335957e-001];
w = [1.2457352290670189e-001, 1.1674626826917781e-001, 1.0158371336153328e-001, 8.0039164271673444e-002, 5.3469662997659276e-002, 2.3587668193254314e-002];
elif l == 13:
n = [5.0000000000000000e-001, 6.1522915797756739e-001, 7.2424637551822335e-001, 8.2117466972017006e-001, 9.0078904536665494e-001, 9.5879919961148907e-001, 9.9209152735929407e-001];
w = [1.1627577661543741e-001, 1.1314159013144903e-001, 1.0390802376844462e-001, 8.9072990380973202e-002, 6.9436755109893875e-002, 4.6060749918864378e-002, 2.0242002382656228e-002];
elif l == 14:
n = [5.5402747435367183e-001, 6.5955618446394482e-001, 7.5762431817907705e-001, 8.4364645240584268e-001, 9.1360065753488251e-001, 9.6421744183178681e-001, 9.9314190434840621e-001];
w = [1.0763192673157916e-001, 1.0259923186064811e-001, 9.2769198738969161e-002, 7.8601583579096995e-002, 6.0759285343951711e-002, 4.0079043579880291e-002, 1.7559730165874574e-002];
elif l == 15:
n = [5.0000000000000000e-001, 6.0059704699871730e-001, 6.9707567353878175e-001, 7.8548608630426942e-001, 8.6220886568008503e-001, 9.2410329170521366e-001, 9.6863669620035298e-001, 9.9399625901024269e-001];
w = [1.0128912096278091e-001, 9.9215742663556039e-002, 9.3080500007781286e-002, 8.3134602908497196e-002, 6.9785338963077315e-002, 5.3579610233586157e-002, 3.5183023744054159e-002, 1.5376620998057434e-002];
elif l == 16:
n = [5.4750625491881877e-001, 6.4080177538962946e-001, 7.2900838882861363e-001, 8.0893812220132189e-001, 8.7770220417750155e-001, 9.3281560119391593e-001, 9.7228751153661630e-001, 9.9470046749582497e-001];
w = [9.4725305227534431e-002, 9.1301707522462000e-002, 8.4578259697501462e-002, 7.4797994408288562e-002, 6.2314485627767105e-002, 4.7579255841246545e-002, 3.1126761969323954e-002, 1.3576229705875955e-002];
elif l == 17:
n = [5.0000000000000000e-001, 5.8924209074792389e-001, 6.7561588172693821e-001, 7.5634526854323847e-001, 8.2883557960834531e-001, 8.9075700194840068e-001, 9.4011957686349290e-001, 9.7533776088438384e-001, 9.9528773765720868e-001];
w = [8.9723235178103419e-002, 8.8281352683496447e-002, 8.4002051078225143e-002, 7.7022880538405308e-002, 6.7568184234262890e-002, 5.5941923596702053e-002, 4.2518074158589644e-002, 2.7729764686993612e-002, 1.2074151434273140e-002];
elif l == 18:
n = [5.4238750652086765e-001, 6.2594311284575277e-001, 7.0587558073142131e-001, 7.7988541553697377e-001, 8.4584352153017661e-001, 9.0185247948626157e-001, 9.4630123324877791e-001, 9.7791197478569880e-001, 9.9578258421046550e-001];
w = [8.4571191481571939e-002, 8.2138241872916504e-002, 7.7342337563132801e-002, 7.0321457335325452e-002, 6.1277603355739306e-002, 5.0471022053143716e-002, 3.8212865127444665e-002, 2.4857274447484968e-002, 1.0808006763240719e-002];
elif l == 19:
n = [5.0000000000000000e-001, 5.8017932282011264e-001, 6.5828204998181494e-001, 7.3228537068798050e-001, 8.0027265233084055e-001, 8.6048308866761469e-001, 9.1135732826857141e-001, 9.5157795180740901e-001, 9.8010407606741501e-001, 9.9620342192179212e-001];
w = [8.0527224924391946e-002, 7.9484421696977337e-002, 7.6383021032929960e-002, 7.1303351086803413e-002, 6.4376981269668232e-002, 5.5783322773667113e-002, 4.5745010811225124e-002, 3.4522271368820669e-002, 2.2407113382849821e-002, 9.7308941148624341e-003];
elif l == 20:
n = [5.3826326056674867e-001, 6.1389292557082253e-001, 6.8685304435770977e-001, 7.5543350097541362e-001, 8.1802684036325757e-001, 8.7316595323007540e-001, 9.1955848591110945e-001, 9.5611721412566297e-001, 9.8198596363895696e-001, 9.9656429959254744e-001];
w = [7.6376693565363113e-002, 7.4586493236301996e-002, 7.1048054659191187e-002, 6.5844319224588346e-002, 5.9097265980759248e-002, 5.0965059908620318e-002, 4.1638370788352433e-002, 3.1336024167054569e-002, 2.0300714900193556e-002, 8.8070035695753026e-003];
elif l == 21:
n = [5.0000000000000000e-001, 5.7278092708044759e-001, 6.4401065840120053e-001, 7.1217106010371944e-001, 7.7580941794360991e-001, 8.3356940209870611e-001, 8.8421998173783889e-001, 9.2668168229165859e-001, 9.6004966707520034e-001, 9.8361341928315316e-001, 9.9687608531019478e-001];
w = [7.3040566824845346e-002, 7.2262201994985134e-002, 6.9943697395536658e-002, 6.6134469316668845e-002, 6.0915708026864350e-002, 5.4398649583574356e-002, 4.6722211728016994e-002, 3.8050056814189707e-002, 2.8567212713428641e-002, 1.8476894885426285e-002, 8.0086141288864491e-003];
elif l == 22:
n = [5.3486963665986109e-001, 6.0393021334411068e-001, 6.7096791044604209e-001, 7.3467791899337853e-001, 7.9382020175345580e-001, 8.4724363159334137e-001, 8.9390840298960406e-001, 9.3290628886015003e-001, 9.6347838609358694e-001, 9.8503024891771429e-001, 9.9714729274119962e-001];
w = [6.9625936427816129e-002, 6.8270749173007697e-002, 6.5586752393531317e-002, 6.1626188405256251e-002, 5.6466148040269712e-002, 5.0207072221440600e-002, 4.2970803108533975e-002, 3.4898234212260300e-002, 2.6146667576341692e-002, 1.6887450792407110e-002, 7.3139976491353280e-003];
elif l == 23:
n = [5.0000000000000000e-001, 5.6662841214923310e-001, 6.3206784048517251e-001, 6.9515051901514546e-001, 7.5475073892300371e-001, 8.0980493788182306e-001, 8.5933068156597514e-001, 9.0244420080942001e-001, 9.3837617913522076e-001, 9.6648554341300807e-001, 9.8627123560905761e-001, 9.9738466749877608e-001];
w = [6.6827286093053176e-002, 6.6231019702348404e-002, 6.4452861094041150e-002, 6.1524542153364815e-002, 5.7498320111205814e-002, 5.2446045732270824e-002, 4.6457883030017563e-002, 3.9640705888359551e-002, 3.2116210704262994e-002, 2.4018835865542369e-002, 1.5494002928489686e-002, 6.7059297435702412e-003];
elif l == 24:
n = [5.3202844643130276e-001, 5.9555943373680820e-001, 6.5752133984808170e-001, 7.1689675381302254e-001, 7.7271073569441984e-001, 8.2404682596848777e-001, 8.7006209578927718e-001, 9.1000099298695147e-001, 9.4320776350220048e-001, 9.6913727600136634e-001, 9.8736427798565474e-001, 9.9759360999851066e-001];
w = [6.3969097673376246e-002, 6.2918728173414318e-002, 6.0835236463901793e-002, 5.7752834026862883e-002, 5.3722135057982914e-002, 4.8809326052057039e-002, 4.3095080765976693e-002, 3.6673240705540205e-002, 2.9649292457718385e-002, 2.2138719408709880e-002, 1.4265694314466934e-002, 6.1706148999928351e-003];
elif l == 25:
n = [5.0000000000000000e-001, 5.6143234630535521e-001, 6.2193344186049426e-001, 6.8058615290469393e-001, 7.3650136572285752e-001, 7.8883146512061142e-001, 8.3678318423673415e-001, 8.7962963151867890e-001, 9.1672131438041693e-001, 9.4749599893913761e-001, 9.7148728561448716e-001, 9.8833196072975871e-001, 9.9777848489524912e-001];
w = [6.1588026863357799e-002, 6.1121221495155122e-002, 5.9727881767892461e-002, 5.7429129572855862e-002, 5.4259812237131867e-002, 5.0267974533525363e-002, 4.5514130991481903e-002, 4.0070350167500532e-002, 3.4019166906178545e-002, 2.7452347987917691e-002, 2.0469578350653148e-002, 1.3177493307516108e-002, 5.6968992505125535e-003];
return (n,w)
[docs]def GQN(l):
""" GQN(): function for generating 1D Gaussian quadrature for integral with Gaussian weight (Gauss-Hermite)
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [0.0000000000000000e+000];
w = [1.0000000000000000e+000];
elif l == 2:
n = [1.0000000000000002e+000];
w = [5.0000000000000000e-001];
elif l == 3:
n = [0.0000000000000000e+000, 1.7320508075688772e+000];
w = [6.6666666666666663e-001, 1.6666666666666674e-001];
elif l == 4:
n = [7.4196378430272591e-001, 2.3344142183389773e+000];
w = [4.5412414523193145e-001, 4.5875854768068498e-002];
elif l == 5:
n = [0.0000000000000000e+000, 1.3556261799742659e+000, 2.8569700138728056e+000];
w = [5.3333333333333344e-001, 2.2207592200561263e-001, 1.1257411327720691e-002];
elif l == 6:
n = [6.1670659019259422e-001, 1.8891758777537109e+000, 3.3242574335521193e+000];
w = [4.0882846955602919e-001, 8.8615746041914523e-002, 2.5557844020562431e-003];
elif l == 7:
n = [0.0000000000000000e+000, 1.1544053947399682e+000, 2.3667594107345415e+000, 3.7504397177257425e+000];
w = [4.5714285714285757e-001, 2.4012317860501250e-001, 3.0757123967586491e-002, 5.4826885597221875e-004];
elif l == 8:
n = [5.3907981135137517e-001, 1.6365190424351082e+000, 2.8024858612875416e+000, 4.1445471861258945e+000];
w = [3.7301225767907736e-001, 1.1723990766175897e-001, 9.6352201207882630e-003, 1.1261453837536784e-004];
elif l == 9:
n = [0.0000000000000000e+000, 1.0232556637891326e+000, 2.0768479786778302e+000, 3.2054290028564703e+000, 4.5127458633997826e+000];
w = [4.0634920634920685e-001, 2.4409750289493909e-001, 4.9916406765217969e-002, 2.7891413212317675e-003, 2.2345844007746563e-005];
elif l == 10:
n = [4.8493570751549764e-001, 1.4659890943911582e+000, 2.4843258416389546e+000, 3.5818234835519269e+000, 4.8594628283323127e+000];
w = [3.4464233493201940e-001, 1.3548370298026730e-001, 1.9111580500770317e-002, 7.5807093431221972e-004, 4.3106526307183106e-006];
elif l == 11:
n = [0.0000000000000000e+000, 9.2886899738106388e-001, 1.8760350201548459e+000, 2.8651231606436447e+000, 3.9361666071299775e+000, 5.1880012243748714e+000];
w = [3.6940836940836957e-001, 2.4224029987397003e-001, 6.6138746071057644e-002, 6.7202852355372697e-003, 1.9567193027122324e-004, 8.1218497902149036e-007];
elif l == 12:
n = [4.4440300194413901e-001, 1.3403751971516167e+000, 2.2594644510007993e+000, 3.2237098287700974e+000, 4.2718258479322815e+000, 5.5009017044677480e+000];
w = [3.2166436151283007e-001, 1.4696704804532995e-001, 2.9116687912364138e-002, 2.2033806875331849e-003, 4.8371849225906076e-005, 1.4999271676371597e-007];
elif l == 13:
n = [0.0000000000000000e+000, 8.5667949351945005e-001, 1.7254183795882394e+000, 2.6206899734322149e+000, 3.5634443802816347e+000, 4.5913984489365207e+000, 5.8001672523865011e+000];
w = [3.4099234099234149e-001, 2.3787152296413588e-001, 7.9168955860450141e-002, 1.1770560505996543e-002, 6.8123635044292619e-004, 1.1526596527333885e-005, 2.7226276428059039e-008];
elif l == 14:
n = [4.1259045795460181e-001, 1.2426889554854643e+000, 2.0883447457019444e+000, 2.9630365798386675e+000, 3.8869245750597696e+000, 4.8969363973455646e+000, 6.0874095469012914e+000];
w = [3.0263462681301945e-001, 1.5408333984251366e-001, 3.8650108824253432e-002, 4.4289191069474066e-003, 2.0033955376074381e-004, 2.6609913440676334e-006, 4.8681612577483872e-009];
elif l == 15:
n = [0.0000000000000000e+000, 7.9912906832454811e-001, 1.6067100690287301e+000, 2.4324368270097581e+000, 3.2890824243987664e+000, 4.1962077112690155e+000, 5.1900935913047821e+000, 6.3639478888298378e+000];
w = [3.1825951825951820e-001, 2.3246229360973222e-001, 8.9417795399844444e-002, 1.7365774492137616e-002, 1.5673575035499571e-003, 5.6421464051890157e-005, 5.9754195979205961e-007, 8.5896498996331805e-010];
elif l == 16:
n = [3.8676060450055738e-001, 1.1638291005549648e+000, 1.9519803457163336e+000, 2.7602450476307019e+000, 3.6008736241715487e+000, 4.4929553025200120e+000, 5.4722257059493433e+000, 6.6308781983931295e+000];
w = [2.8656852123801241e-001, 1.5833837275094925e-001, 4.7284752354014067e-002, 7.2669376011847411e-003, 5.2598492657390979e-004, 1.5300032162487286e-005, 1.3094732162868203e-007, 1.4978147231618314e-010];
elif l == 17:
n = [0.0000000000000000e+000, 7.5184260070389630e-001, 1.5098833077967408e+000, 2.2810194402529889e+000, 3.0737971753281941e+000, 3.9000657171980104e+000, 4.7785315896299840e+000, 5.7444600786594071e+000, 6.8891224398953330e+000];
w = [2.9953837012660756e-001, 2.2670630846897877e-001, 9.7406371162718081e-002, 2.3086657025711152e-002, 2.8589460622846499e-003, 1.6849143155133945e-004, 4.0126794479798725e-006, 2.8080161179305783e-008, 2.5843149193749151e-011];
elif l == 18:
n = [3.6524575550769767e-001, 1.0983955180915013e+000, 1.8397799215086457e+000, 2.5958336889112403e+000, 3.3747365357780907e+000, 4.1880202316294044e+000, 5.0540726854427405e+000, 6.0077459113595975e+000, 7.1394648491464796e+000];
w = [2.7278323465428789e-001, 1.6068530389351263e-001, 5.4896632480222654e-002, 1.0516517751941352e-002, 1.0654847962916496e-003, 5.1798961441161962e-005, 1.0215523976369816e-006, 5.9054884788365484e-009, 4.4165887693587078e-012];
elif l == 19:
n = [0.0000000000000000e+000, 7.1208504404237993e-001, 1.4288766760783731e+000, 2.1555027613169351e+000, 2.8980512765157536e+000, 3.6644165474506383e+000, 4.4658726268310316e+000, 5.3205363773360386e+000, 6.2628911565132519e+000, 7.3825790240304316e+000];
w = [2.8377319275152108e-001, 2.2094171219914366e-001, 1.0360365727614400e-001, 2.8666691030118496e-002, 4.5072354203420355e-003, 3.7850210941426759e-004, 1.5351145954666744e-005, 2.5322200320928681e-007, 1.2203708484474786e-009, 7.4828300540572308e-013];
elif l == 20:
n = [3.4696415708135592e-001, 1.0429453488027509e+000, 1.7452473208141270e+000, 2.4586636111723679e+000, 3.1890148165533900e+000, 3.9439673506573163e+000, 4.7345813340460552e+000, 5.5787388058932015e+000, 6.5105901570136551e+000, 7.6190485416797591e+000];
w = [2.6079306344955544e-001, 1.6173933398400026e-001, 6.1506372063976029e-002, 1.3997837447101043e-002, 1.8301031310804918e-003, 1.2882627996192898e-004, 4.4021210902308646e-006, 6.1274902599829597e-008, 2.4820623623151838e-010, 1.2578006724379305e-013];
elif l == 21:
n = [0.0000000000000000e+000, 6.7804569244064405e-001, 1.3597658232112304e+000, 2.0491024682571628e+000, 2.7505929810523733e+000, 3.4698466904753764e+000, 4.2143439816884216e+000, 4.9949639447820253e+000, 5.8293820073044706e+000, 6.7514447187174609e+000, 7.8493828951138225e+000];
w = [2.7026018357287707e-001, 2.1533371569505982e-001, 1.0839228562641938e-001, 3.3952729786542839e-002, 6.4396970514087768e-003, 7.0804779548153736e-004, 4.2192347425515866e-005, 1.2253548361482522e-006, 1.4506612844930740e-008, 4.9753686041217464e-011, 2.0989912195656652e-014];
elif l == 22:
n = [3.3117931571527381e-001, 9.9516242227121554e-001, 1.6641248391179071e+000, 2.3417599962877080e+000, 3.0324042278316763e+000, 3.7414963502665177e+000, 4.4763619773108685e+000, 5.2477244337144251e+000, 6.0730749511228979e+000, 6.9859804240188152e+000, 8.0740299840217116e+000];
w = [2.5024359658693501e-001, 1.6190629341367538e-001, 6.7196311428889891e-002, 1.7569072880805774e-002, 2.8087610475772107e-003, 2.6228330325596416e-004, 1.3345977126808712e-005, 3.3198537498140043e-007, 3.3665141594582109e-009, 9.8413789823460105e-012, 3.4794606478771428e-015];
elif l == 23:
n = [0.0000000000000000e+000, 6.4847115353449580e-001, 1.2998764683039790e+000, 1.9573275529334242e+000, 2.6243236340591820e+000, 3.3050400217529652e+000, 4.0047753217333044e+000, 4.7307241974514733e+000, 5.4934739864717947e+000, 6.3103498544483996e+000, 7.2146594350518622e+000, 8.2933860274173536e+000];
w = [2.5850974080883904e-001, 2.0995966957754261e-001, 1.1207338260262091e-001, 3.8867183703480947e-002, 8.5796783914656640e-003, 1.1676286374978613e-003, 9.3408186090312983e-005, 4.0899772449921549e-006, 8.7750624838617161e-008, 7.6708888623999076e-010, 1.9229353115677913e-012, 5.7323831678020873e-016];
elif l == 24:
n = [3.1737009662945231e-001, 9.5342192293210926e-001, 1.5934804298164202e+000, 2.2404678516917524e+000, 2.8977286432233140e+000, 3.5693067640735610e+000, 4.2603836050199053e+000, 4.9780413746391208e+000, 5.7327471752512009e+000, 6.5416750050986341e+000, 7.4378906660216630e+000, 8.5078035191952583e+000];
w = [2.4087011554664056e-001, 1.6145951286700025e-001, 7.2069364017178436e-002, 2.1126344408967029e-002, 3.9766089291813113e-003, 4.6471871877939763e-004, 3.2095005652745989e-005, 1.2176597454425830e-006, 2.2674616734804651e-008, 1.7186649279648690e-010, 3.7149741527624159e-013, 9.3901936890419202e-017];
elif l == 25:
n = [0.0000000000000000e+000, 6.2246227918607611e-001, 1.2473119756167892e+000, 1.8770583699478387e+000, 2.5144733039522058e+000, 3.1627756793881927e+000, 3.8259005699724917e+000, 4.5089299229672850e+000, 5.2188480936442794e+000, 5.9660146906067020e+000, 6.7674649638097168e+000, 7.6560379553930762e+000, 8.7175976783995885e+000];
w = [2.4816935117648548e-001, 2.0485102565034041e-001, 1.1488092430395164e-001, 4.3379970167644971e-002, 1.0856755991462316e-002, 1.7578504052637961e-003, 1.7776690692652660e-004, 1.0672194905202536e-005, 3.5301525602454978e-007, 5.7380238688993763e-009, 3.7911500004771871e-011, 7.1021030370039253e-014, 1.5300389979986825e-017];
return (n,w)
[docs]def KPU(l):
""" KPU(): function for generating 1D Nested rule for unweighted integral over [0,1]
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [5.0000000000000000e-001];
w = [1.0000000000000000e+000];
elif l == 2:
n = [5.0000000000000000e-001, 8.8729829999999998e-001];
w = [4.4444440000000002e-001, 2.7777780000000002e-001];
elif l == 3:
n = [5.0000000000000000e-001, 8.8729829999999998e-001];
w = [4.4444440000000002e-001, 2.7777780000000002e-001];
elif l == 4:
n = [5.0000000000000000e-001, 7.1712189999999998e-001, 8.8729829999999998e-001, 9.8024560000000005e-001];
w = [2.2545832254583223e-001, 2.0069872006987199e-001, 1.3424401342440134e-001, 5.2328105232810521e-002];
elif l == 5:
n = [5.0000000000000000e-001, 7.1712189999999998e-001, 8.8729829999999998e-001, 9.8024560000000005e-001];
w = [2.2545832254583223e-001, 2.0069872006987199e-001, 1.3424401342440134e-001, 5.2328105232810521e-002];
elif l == 6:
n = [5.0000000000000000e-001, 7.1712189999999998e-001, 8.8729829999999998e-001, 9.8024560000000005e-001];
w = [2.2545832254583223e-001, 2.0069872006987199e-001, 1.3424401342440134e-001, 5.2328105232810521e-002];
elif l == 7:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 8:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 9:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 10:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 11:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 12:
n = [5.0000000000000000e-001, 6.1169330000000000e-001, 7.1712189999999998e-001, 8.1055149999999998e-001, 8.8729829999999998e-001, 9.4422960000000000e-001, 9.8024560000000005e-001, 9.9691600000000002e-001];
w = [1.1275520000000000e-001, 1.0957840000000001e-001, 1.0031430000000000e-001, 8.5755999999999999e-002, 6.7207600000000006e-002, 4.6463600000000001e-002, 2.5801600000000001e-002, 8.5009000000000005e-003];
elif l == 13:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 14:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 15:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 16:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 17:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 18:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 19:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 20:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 21:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 22:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 23:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 24:
n = [5.0000000000000000e-001, 5.5624450000000003e-001, 6.1169330000000000e-001, 6.6556769999999998e-001, 7.1712189999999998e-001, 7.6565989999999995e-001, 8.1055149999999998e-001, 8.5124809999999995e-001, 8.8729829999999998e-001, 9.1836300000000004e-001, 9.4422960000000000e-001, 9.6482740000000000e-001, 9.8024560000000005e-001, 9.9076560000000002e-001, 9.9691600000000002e-001, 9.9954909999999997e-001];
w = [5.6377600000000014e-002, 5.5978400000000011e-002, 5.4789200000000017e-002, 5.2834900000000011e-002, 5.0157100000000017e-002, 4.6813600000000004e-002, 4.2878000000000006e-002, 3.8439800000000010e-002, 3.3603900000000006e-002, 2.8489800000000006e-002, 2.3231400000000003e-002, 1.7978600000000004e-002, 1.2903800000000003e-002, 8.2230000000000011e-003, 4.2173000000000011e-003, 1.2724000000000001e-003];
elif l == 25:
n = [5.0000000000000000e-001, 5.2817210000000003e-001, 5.5624450000000003e-001, 5.8411769999999996e-001, 6.1169330000000000e-001, 6.3887490000000002e-001, 6.6556769999999998e-001, 6.9167970000000001e-001, 7.1712189999999998e-001, 7.4180900000000005e-001, 7.6565989999999995e-001, 7.8859789999999996e-001, 8.1055149999999998e-001, 8.3145480000000005e-001, 8.5124809999999995e-001, 8.6987800000000004e-001, 8.8729829999999998e-001, 9.0347029999999995e-001, 9.1836300000000004e-001, 9.3195399999999995e-001, 9.4422960000000000e-001, 9.5518559999999997e-001, 9.6482740000000000e-001, 9.7317140000000002e-001, 9.8024560000000005e-001, 9.8609139999999995e-001, 9.9076560000000002e-001, 9.9434239999999996e-001, 9.9691600000000002e-001, 9.9860309999999997e-001, 9.9954909999999997e-001, 9.9993650000000001e-001];
w = [2.8188799999999993e-002, 2.8138799999999992e-002, 2.7989199999999992e-002, 2.7740699999999993e-002, 2.7394599999999995e-002, 2.6952699999999993e-002, 2.6417499999999993e-002, 2.5791599999999994e-002, 2.5078599999999993e-002, 2.4282199999999993e-002, 2.3406799999999995e-002, 2.2457299999999996e-002, 2.1438999999999996e-002, 2.0357799999999995e-002, 1.9219899999999998e-002, 1.8032199999999998e-002, 1.6801899999999998e-002, 1.5536799999999996e-002, 1.4244899999999996e-002, 1.2934799999999996e-002, 1.1615699999999998e-002, 1.0297099999999998e-002, 8.9892999999999987e-003, 7.7033999999999983e-003, 6.4518999999999983e-003, 5.2490999999999987e-003, 4.1114999999999988e-003, 3.0577999999999990e-003, 2.1087999999999997e-003, 1.2894999999999998e-003, 6.3259999999999987e-004, 1.8159999999999997e-004];
return (n,w)
[docs]def KPN(l):
""" KPN(): function for generating 1D Nested rule for integral with Gaussian weight
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [0.0000000000000000e+000];
w = [1.0000000000000000e+000];
elif l == 2:
n = [0.0000000000000000e+000, 1.7320508075688772e+000];
w = [6.6666666666666663e-001, 1.6666666666666666e-001];
elif l == 3:
n = [0.0000000000000000e+000, 1.7320508075688772e+000];
w = [6.6666666666666674e-001, 1.6666666666666666e-001];
elif l == 4:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.7320508075688772e+000, 4.1849560176727323e+000];
w = [4.5874486825749189e-001, 1.3137860698313561e-001, 1.3855327472974924e-001, 6.9568415836913987e-004];
elif l == 5:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.7320508075688772e+000, 2.8612795760570582e+000, 4.1849560176727323e+000];
w = [2.5396825396825407e-001, 2.7007432957793776e-001, 9.4850948509485125e-002, 7.9963254708935293e-003, 9.4269457556517470e-005];
elif l == 6:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.7320508075688772e+000, 2.8612795760570582e+000, 4.1849560176727323e+000];
w = [2.5396825396825429e-001, 2.7007432957793776e-001, 9.4850948509485070e-002, 7.9963254708935293e-003, 9.4269457556517551e-005];
elif l == 7:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.7320508075688772e+000, 2.8612795760570582e+000, 4.1849560176727323e+000];
w = [2.5396825396825418e-001, 2.7007432957793781e-001, 9.4850948509485014e-002, 7.9963254708935311e-003, 9.4269457556517592e-005];
elif l == 8:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.7320508075688772e+000, 2.8612795760570582e+000, 4.1849560176727323e+000];
w = [2.5396825396825418e-001, 2.7007432957793781e-001, 9.4850948509485042e-002, 7.9963254708935276e-003, 9.4269457556517375e-005];
elif l == 9:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [2.6692223033505302e-001, 2.5456123204171222e-001, 1.4192654826449365e-002, 8.8681002152028010e-002, 1.9656770938777492e-003, 7.0334802378279075e-003, 1.0563783615416941e-004, -8.2049207541509217e-007, 2.1136499505424257e-008];
elif l == 10:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420623e-001, 2.0832499164960877e-001, 6.1151730125247716e-002, 6.4096054686807610e-002, 1.8085234254798462e-002, -6.3372247933737571e-003, 2.8848804365067559e-003, 6.0123369459847997e-005, 6.0948087314689840e-007, 8.6296846022298632e-010];
elif l == 11:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420623e-001, 2.0832499164960872e-001, 6.1151730125247709e-002, 6.4096054686807541e-002, 1.8085234254798459e-002, -6.3372247933737545e-003, 2.8848804365067555e-003, 6.0123369459847922e-005, 6.0948087314689830e-007, 8.6296846022298839e-010];
elif l == 12:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420623e-001, 2.0832499164960872e-001, 6.1151730125247716e-002, 6.4096054686807624e-002, 1.8085234254798466e-002, -6.3372247933737545e-003, 2.8848804365067559e-003, 6.0123369459847841e-005, 6.0948087314689830e-007, 8.6296846022298963e-010];
elif l == 13:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420600e-001, 2.0832499164960883e-001, 6.1151730125247730e-002, 6.4096054686807638e-002, 1.8085234254798459e-002, -6.3372247933737580e-003, 2.8848804365067555e-003, 6.0123369459847868e-005, 6.0948087314689830e-007, 8.6296846022298756e-010];
elif l == 14:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420617e-001, 2.0832499164960874e-001, 6.1151730125247702e-002, 6.4096054686807596e-002, 1.8085234254798459e-002, -6.3372247933737563e-003, 2.8848804365067555e-003, 6.0123369459847936e-005, 6.0948087314689851e-007, 8.6296846022298322e-010];
elif l == 15:
n = [0.0000000000000000e+000, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000];
w = [3.0346719985420612e-001, 2.0832499164960874e-001, 6.1151730125247723e-002, 6.4096054686807652e-002, 1.8085234254798459e-002, -6.3372247933737597e-003, 2.8848804365067563e-003, 6.0123369459848091e-005, 6.0948087314689851e-007, 8.6296846022298983e-010];
elif l == 16:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [2.5890005324151566e-001, 2.8128101540033167e-002, 1.9968863511734550e-001, 6.5417392836092561e-002, 6.1718532565867179e-002, 1.7608475581318002e-003, 1.6592492698936010e-002, -5.5610063068358157e-003, 2.7298430467334002e-003, 1.5044205390914219e-005, 5.9474961163931621e-005, 6.1435843232617913e-007, 7.9298267864869338e-010, 5.1158053105504208e-012, -1.4840835740298868e-013, 1.2618464280815118e-015];
elif l == 17:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [1.3911022236338039e-001, 1.0387687125574284e-001, 1.7607598741571459e-001, 7.7443602746299481e-002, 5.4677556143463042e-002, 7.3530110204955076e-003, 1.1529247065398790e-002, -2.7712189007789243e-003, 2.1202259559596325e-003, 8.3236045295766745e-005, 5.5691158981081479e-005, 6.9086261179113738e-007, -1.3486017348542930e-008, 1.5542195992782658e-009, -1.9341305000880955e-011, 2.6640625166231651e-013, -9.9313913286822465e-016];
elif l == 18:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806921377e-004, 1.9176011588804434e-001, 1.4807083115521585e-001, 9.2364726716986353e-002, 4.5273685465150391e-002, 1.5673473751851151e-002, 3.1554462691875513e-003, 2.3113452403522071e-003, 8.1895392750226735e-004, 2.7524214116785131e-004, 3.5729348198975332e-005, 2.7342206801187888e-006, 2.4676421345798140e-007, 2.1394194479561062e-008, 4.6011760348655917e-010, 3.0972223576062995e-012, 5.4500412650638128e-015, 1.0541326582334014e-018];
elif l == 19:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806921377e-004, 1.9176011588804437e-001, 1.4807083115521585e-001, 9.2364726716986353e-002, 4.5273685465150523e-002, 1.5673473751851151e-002, 3.1554462691875604e-003, 2.3113452403522050e-003, 8.1895392750226670e-004, 2.7524214116785131e-004, 3.5729348198975447e-005, 2.7342206801187884e-006, 2.4676421345798140e-007, 2.1394194479561056e-008, 4.6011760348656077e-010, 3.0972223576063011e-012, 5.4500412650637663e-015, 1.0541326582337958e-018];
elif l == 20:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806925551e-004, 1.9176011588804440e-001, 1.4807083115521585e-001, 9.2364726716986298e-002, 4.5273685465150537e-002, 1.5673473751851155e-002, 3.1554462691875573e-003, 2.3113452403522080e-003, 8.1895392750226724e-004, 2.7524214116785137e-004, 3.5729348198975352e-005, 2.7342206801187888e-006, 2.4676421345798124e-007, 2.1394194479561056e-008, 4.6011760348656144e-010, 3.0972223576062963e-012, 5.4500412650638365e-015, 1.0541326582335402e-018];
elif l == 21:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806913744e-004, 1.9176011588804429e-001, 1.4807083115521594e-001, 9.2364726716986312e-002, 4.5273685465150391e-002, 1.5673473751851151e-002, 3.1554462691875565e-003, 2.3113452403522089e-003, 8.1895392750226670e-004, 2.7524214116785142e-004, 3.5729348198975285e-005, 2.7342206801187888e-006, 2.4676421345798119e-007, 2.1394194479561059e-008, 4.6011760348656594e-010, 3.0972223576062950e-012, 5.4500412650638696e-015, 1.0541326582332041e-018];
elif l == 22:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806903368e-004, 1.9176011588804448e-001, 1.4807083115521574e-001, 9.2364726716986423e-002, 4.5273685465150516e-002, 1.5673473751851161e-002, 3.1554462691875543e-003, 2.3113452403522063e-003, 8.1895392750226713e-004, 2.7524214116785164e-004, 3.5729348198975319e-005, 2.7342206801187905e-006, 2.4676421345798151e-007, 2.1394194479561082e-008, 4.6011760348656005e-010, 3.0972223576063043e-012, 5.4500412650637592e-015, 1.0541326582339926e-018];
elif l == 23:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806913755e-004, 1.9176011588804442e-001, 1.4807083115521577e-001, 9.2364726716986381e-002, 4.5273685465150468e-002, 1.5673473751851155e-002, 3.1554462691875560e-003, 2.3113452403522045e-003, 8.1895392750226572e-004, 2.7524214116785158e-004, 3.5729348198975298e-005, 2.7342206801187892e-006, 2.4676421345798129e-007, 2.1394194479561072e-008, 4.6011760348656103e-010, 3.0972223576062963e-012, 5.4500412650638207e-015, 1.0541326582338368e-018];
elif l == 24:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806914438e-004, 1.9176011588804442e-001, 1.4807083115521577e-001, 9.2364726716986340e-002, 4.5273685465150509e-002, 1.5673473751851155e-002, 3.1554462691875586e-003, 2.3113452403522058e-003, 8.1895392750226551e-004, 2.7524214116785142e-004, 3.5729348198975386e-005, 2.7342206801187884e-006, 2.4676421345798082e-007, 2.1394194479561059e-008, 4.6011760348656382e-010, 3.0972223576062942e-012, 5.4500412650638381e-015, 1.0541326582336941e-018];
elif l == 25:
n = [0.0000000000000000e+000, 2.4899229757996061e-001, 7.4109534999454085e-001, 1.2304236340273060e+000, 1.7320508075688772e+000, 2.2336260616769419e+000, 2.5960831150492023e+000, 2.8612795760570582e+000, 3.2053337944991944e+000, 3.6353185190372783e+000, 4.1849560176727323e+000, 4.7364330859522967e+000, 5.1870160399136562e+000, 5.6981777684881099e+000, 6.3633944943363696e+000, 7.1221067008046166e+000, 7.9807717985905606e+000, 9.0169397898903032e+000];
w = [5.1489450806919989e-004, 1.9176011588804437e-001, 1.4807083115521580e-001, 9.2364726716986395e-002, 4.5273685465150426e-002, 1.5673473751851158e-002, 3.1554462691875539e-003, 2.3113452403522054e-003, 8.1895392750226681e-004, 2.7524214116785142e-004, 3.5729348198975292e-005, 2.7342206801187884e-006, 2.4676421345798108e-007, 2.1394194479561056e-008, 4.6011760348655901e-010, 3.0972223576062975e-012, 5.4500412650638412e-015, 1.0541326582337527e-018];
return (n,w)
[docs]def CC(l):
order = 2**l
n = np.cos(np.arange(order+1)*np.pi/float(order))
n = n[::-1]
N = np.arange(1,order,2,dtype=np.float64)
v = np.hstack([ 2./N/(N-2.), 1./N[-1], np.zeros(order/2)])
v = -v[0:-1]-v[-1:0:-1]
g = -np.ones(order)
g[order/2] += 2*order
g /= (order**2. - 1.)
w = np.real(ifft(v+g))
w = np.hstack((w,w[0]))
return (n[order/2:],w[order/2:])
[docs]def FEJ(l):
order = 2**l
n = np.cos(np.arange(1,order)*np.pi/float(order))
n = n[::-1]
N = np.arange(1,order,2,dtype=np.float64)
v = np.hstack([ 2./N/(N-2.), 1./N[-1], np.zeros(order/2)])
v = -v[0:-1]-v[-1:0:-1]
w = np.real(ifft(v))
w = np.hstack((w,w[0]))
return (n[order/2-1:],w[order/2:-1])
[docs]def NESTEDLOBATTO(l):
""" NESTEDLOBATTO(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [-1.0, 1.0]
w = [0.5, 0.5]
elif l == 2:
n = [-1.0, -0.44721272127212713, 0.44721272127212713, 1.0]
w = [0.16666748117876523, 0.83333251882123494, 0.8333325188212346, 0.16666748117876518]
elif l == 3:
n = [-1.0, -0.89695369536953695, -0.68022802280228023, -0.44721272127212713, 0.0823682368236823, 0.44721272127212713, 0.84341234123412356, 1.0]
w = [-0.035321584219883032, 0.34288943233190972, -0.04943101085623889, 0.59049992464565337, 0.41004633390082912, 0.40613362450704776, 0.30588054242725765, 0.029302737263424635]
elif l == 4:
n = [-1.0, -0.96954895489548953, -0.89695369536953695, -0.79737973797379735, -0.68022802280228023, -0.44721272127212713, -0.29384938493849383, -0.10815081508150812, 0.0823682368236823, 0.27471947194719482, 0.44721272127212713, 0.61916191619161931, 0.75493149314931507, 0.84341234123412356, 0.96147214721472152, 1.0]
w = [0.012880993962665394, 0.039235848072476882, 0.11312279561296756, 0.065516259253133466, 0.19882930957892514, 0.23372315608946556, 0.10725252763821631, 0.2359861850621798, 0.15680520743896817, 0.21670835259576374, 0.13824278459128811, 0.19937807292590454, 0.053593964124930303, 0.14326136226871977, 0.078686615848127042, 0.0067765649362683091]
elif l == 5:
n = [-1.0, -0.99147514751475152, -0.96954895489548953, -0.94147414741474145, -0.89695369536953695, -0.8548614861486149, -0.79737973797379735, -0.74148214821482139, -0.68022802280228023, -0.59896789678967899, -0.5183118311831183, -0.44721272127212713, -0.29384938493849383, -0.21347334733473347, -0.10815081508150812, -0.018413841384138423, 0.0823682368236823, 0.17586158615861594, 0.27471947194719482, 0.36210021002100212, 0.44721272127212713, 0.53676967696769684, 0.61916191619161931, 0.69272927292729269, 0.75493149314931507, 0.84341234123412356, 0.88271627162716282, 0.92374037403740383, 0.96147214721472152, 0.97978597859785987, 0.99382338233823386, 1.0]
w = [0.00069261643462690528, 0.018972124110117467, 0.020164489012164052, 0.042183691702499068, 0.03702982526937177, 0.05850383170496181, 0.044648086209400285, 0.076655610128514545, 0.04597388951236852, 0.11850648916972194, 0.020852443579229361, 0.14446888543445324, 0.14678818072008709, 0.041977353440731141, 0.1437785990320968, 0.050360517127466711, 0.13777821558315428, 0.058229325770147326, 0.13235762998874034, 0.045977820342724768, 0.12030620126783849, 0.059951765511677184, 0.1060213203486242, 0.035943709230963607, 0.095540382861925552, 0.06946500088757733, 0.026898687766224318, 0.044452920102861747, 0.032051370300259091, 0.0072343363816158189, 0.017405951327067817, -0.0011712702592127115]
elif l == 6:
n = [-1.0, -0.99790379037903787, -0.99147514751475152, -0.9863586358635863, -0.97849384938493855, -0.96954895489548953, -0.95529552955295527, -0.94147414741474145, -0.92054805480548052, -0.89695369536953695, -0.87718371837183717, -0.8548614861486149, -0.82713871387138715, -0.79737973797379735, -0.76993299329932985, -0.74148214821482139, -0.70829082908290819, -0.68022802280228023, -0.63538753875387544, -0.59896789678967899, -0.55373137313731369, -0.5183118311831183, -0.44721272127212713, -0.40502450245024496, -0.35556355635563552, -0.29384938493849383, -0.25727772777277724, -0.21347334733473347, -0.16300030003000293, -0.10815081508150812, -0.065150515051505126, -0.018413841384138423, 0.031327132713271352, 0.0823682368236823, 0.12863286328632878, 0.17586158615861594, 0.22473047304730476, 0.27471947194719482, 0.31810381038103819, 0.36210021002100212, 0.40660066006600665, 0.44721272127212713, 0.49418941894189428, 0.53676967696769684, 0.57869386938693879, 0.61916191619161931, 0.65642164216421639, 0.69272927292729269, 0.72586058605860604, 0.75493149314931507, 0.79027102710271036, 0.81985398539854004, 0.84341234123412356, 0.88271627162716282, 0.90231023102310237, 0.92374037403740383, 0.94168216821682171, 0.96147214721472152, 0.97036903690369047, 0.97978597859785987, 0.98797079707970792, 0.99382338233823386, 0.9982318231823184, 1.0]
w = [-0.00076001797083634218, 0.007177092376060205, -0.00084020462089522531, 0.018919901983671949, -0.0076331566564058146, 0.026127424389895942, 0.0020942627008014312, 0.026859364553197755, 0.014929860524220006, 0.034845001425998223, 0.0018566855394130091, 0.043262494075569445, 0.011165637218201496, 0.052988970959693134, -0.0042891316399548641, 0.064419965209842209, -0.0047396747908790863, 0.064275413096505501, 0.02249493789855753, 0.058850735318281125, 0.019401989574245982, 0.065448230083195516, 0.068000154832394577, 0.02515323487739396, 0.069879073161157101, 0.043013359242201503, 0.04444733985608261, 0.039142072393034598, 0.062322736214533686, 0.041151848109587913, 0.051645668564774282, 0.040375082165479054, 0.059155199404838814, 0.040581414146499004, 0.054277389267548236, 0.040120121560343926, 0.057635977096813153, 0.039622737711463442, 0.049790174841377315, 0.038479108339575636, 0.048276644873290045, 0.03754915297663932, 0.052094364244809017, 0.034733464116533171, 0.048537323511088679, 0.031684948073202046, 0.04389488253211507, 0.027795661174888429, 0.03821195036646538, 0.024059364198076694, 0.043462450664033071, 0.012039751177106978, 0.040228072297464147, 0.03351447745612423, 0.012540177666197234, 0.025009387570112441, 0.014324942852594245, 0.025306075929847092, -0.0062433170733265392, 0.020091251188895359, -0.00098217047630146884, 0.010527758161597038, 4.7682258817950418e-05, 0.0016472312262526621]
else:
raise ValueError("No rule defined for this level")
return (n,w)
[docs]def NESTEDGAUSS(l):
""" NESTEDGAUSS(): function for generating 1D Nested rule for integral with Uniform weight with 2**l scaling
Args:
l (int): level of accuracy of the quadrature rule
Returns:
(:class:`tuple<tuple>` [2] of :class:`ndarray<numpy.ndarray>`) -- nodes and weights
"""
if l == 1:
n = [-0.99930504173577206, 0.99930504173577217]
w = [1.0000000000000002, 1.0]
elif l == 2:
n = [-0.99930504173577206, -0.44636601725346409, 0.4463660172534642, 0.99930504173577217]
w = [0.16774592076146522, 0.83225407923853489, 0.83225407923853467, 0.16774592076146508]
elif l == 3:
n = [-0.99930504173577206, -0.88931544599511403, -0.68523631305423316, -0.44636601725346409, 0.07299312178779889, 0.4463660172534642, 0.84062929625258032, 0.99930504173577217]
w = [-0.021326091693306938, 0.33182966567279193, -0.042726043868955413, 0.57478665153729425, 0.4104779797395684, 0.41070103722154711, 0.30395574582859697, 0.032301055562463869]
elif l == 4:
n = [-0.99930504173577206, -0.9733268277899112, -0.88931544599511403, -0.78397235894334139, -0.68523631305423316, -0.44636601725346409, -0.31132287199021103, -0.12146281929612052, 0.07299312178779889, 0.26468716220876737, 0.4463660172534642, 0.61115535517239306, 0.75281990726053183, 0.84062929625258032, 0.96100879965205377, 0.99930504173577217]
w = [0.0092881582779950067, 0.044562252598558401, 0.12641297142822408, 0.044012734173664692, 0.20274726485677144, 0.23998352281589516, 0.082287548924930343, 0.25002799113654905, 0.15048559185159413, 0.22695733650172306, 0.13545686431191237, 0.19848758224900984, 0.058534919753589217, 0.14362515647026902, 0.080426259522854329, 0.0067038451264598954]
elif l == 5:
n = [-0.99930504173577206, -0.99101337147674418, -0.9733268277899112, -0.94641137485840265, -0.88931544599511403, -0.84062929625258032, -0.78397235894334139, -0.75281990726053194, -0.68523631305423316, -0.61115535517239317, -0.53127946401989457, -0.44636601725346409, -0.31132287199021103, -0.21742364374000711, -0.12146281929612052, -0.024350292663424474, 0.07299312178779889, 0.16964442042399269, 0.26468716220876737, 0.35722015833766813, 0.4463660172534642, 0.53127946401989445, 0.61115535517239306, 0.68523631305423327, 0.75281990726053183, 0.84062929625258032, 0.88931544599511414, 0.92956917213193957, 0.96100879965205377, 0.97332682778991098, 0.99101337147674418, 0.99930504173577217]
w = [0.0010412807320163862, 0.019182354480081498, 0.012319396214323486, 0.047496698150077364, 0.058482842326154021, 0.047271750901184739, 0.043252599146568267, 0.052922837868900167, 0.062861173008646867, 0.09208339739673406, 0.060311288519508822, 0.12219874127349017, 0.13184299605802213, 0.065869168651629534, 0.12138531095845896, 0.075329692206879822, 0.11744614767380487, 0.07694292434181807, 0.11211303383876528, 0.073384832223261726, 0.10449618054823878, 0.065073018439360253, 0.095299108486491924, 0.050740646968442583, 0.089943153622184496, 0.075961607135799553, 0.028556858848048594, 0.050226418070033627, 0.0036001258615282203, 0.028014357952487703, 0.010306966571815224, 0.004043091525242697]
elif l == 6:
n = [-0.99930504173577206, -0.99634011677195511, -0.99101337147674418, -0.98333625388462598, -0.9733268277899112, -0.96100879965205366, -0.94641137485840265, -0.92956917213193957, -0.91052213707850271, -0.88931544599511403, -0.86599939815409277, -0.84062929625258032, -0.81326531512279754, -0.78397235894334139, -0.75281990726053194, -0.71988185017161077, -0.68523631305423316, -0.64896547125465753, -0.61115535517239317, -0.57189564620263411, -0.53127946401989457, -0.48940314570705301, -0.44636601725346409, -0.40227015796399163, -0.35722015833766818, -0.31132287199021103, -0.26468716220876737, -0.21742364374000711, -0.16964442042399294, -0.12146281929612052, -0.072993121787799112, -0.024350292663424474, 0.024350292663424523, 0.07299312178779889, 0.12146281929612059, 0.16964442042399269, 0.21742364374000694, 0.26468716220876737, 0.31132287199021086, 0.35722015833766813, 0.40227015796399163, 0.4463660172534642, 0.48940314570705284, 0.53127946401989445, 0.571895646202634, 0.61115535517239306, 0.6489654712546572, 0.68523631305423327, 0.71988185017161077, 0.75281990726053183, 0.78397235894334105, 0.81326531512279754, 0.84062929625258032, 0.86599939815409288, 0.88931544599511414, 0.91052213707850271, 0.92956917213193957, 0.94641137485840277, 0.96100879965205377, 0.97332682778991098, 0.98333625388462598, 0.99101337147674418, 0.99634011677195533, 0.99930504173577217]
w = [0.0017832807216962951, 0.0041470332605624176, 0.0065044579689787505, 0.0088467598263641846, 0.011168139460130812, 0.013463047896718342, 0.01572603047602484, 0.01795171577569735, 0.020134823153530226, 0.022270173808383451, 0.024352702568711467, 0.026377469715054881, 0.02833967261425906, 0.030234657072402842, 0.032057928354851439, 0.033805161837141641, 0.035472213256882726, 0.037055128540240248, 0.038550153178615175, 0.039953741132720481, 0.041262563242623368, 0.042473515123653036, 0.043583724529323513, 0.04459055816375658, 0.045491627927418239, 0.046284796581314569, 0.046968182816209847, 0.047540165714830218, 0.047999388596458324, 0.048344762234802663, 0.04857546744150356, 0.048690957009139564, 0.048690957009139467, 0.048575467441503539, 0.048344762234802947, 0.047999388596458928, 0.047540165714830426, 0.046968182816209812, 0.046284796581314139, 0.045491627927417601, 0.044590558163756365, 0.043583724529323714, 0.042473515123653535, 0.041262563242623639, 0.039953741132720322, 0.038550153178615404, 0.037055128540239582, 0.035472213256882476, 0.033805161837141912, 0.032057928354852314, 0.030234657072403241, 0.028339672614259612, 0.026377469715054849, 0.024352702568710961, 0.022270173808383, 0.020134823153530226, 0.017951715775696989, 0.015726030476024437, 0.013463047896718641, 0.011168139460131107, 0.0088467598263638481, 0.0065044579689779629, 0.0041470332605625026, 0.0017832807216963517]
else:
raise ValueError("No rule defined for this level")
return (n,w)