/
_stenger.py
487 lines (461 loc) · 14.2 KB
/
_stenger.py
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# TODO According to Stroud, Stenger's original article has data up to n == 20.
from ..helpers import article, fsd, untangle, z
from ._helpers import SnScheme, volume_nball
source = article(
authors=["W.G."],
title="Review: Tabulation of Certain Fully Symmetric Numerical Integration Formulas of Degree 7, 9 and 11 by Frank Stenger",
journal="Mathematics of Computation",
volume="25",
number="116",
month="oct",
year="1971",
pages="935+s58-s125",
url="https://www.jstor.org/stable/2004361",
)
def stenger_7a(n):
u, v, B = {
3: (
0.285231516480645,
0.765055323929465,
[
-0.542565297000466e01,
+0.490769827031175e01,
+0.318402337562406,
-0.391335495408044e01,
+0.436675887074467e-01,
+0.308676098900275e01,
],
),
4: (
0.266216481931920,
0.727412389740367,
[
-0.678226800331914e02,
+0.299222999307746e02,
+0.240390523860780,
-0.118821423811283e02,
+0.459003238309213e-01,
+0.361018024911844e01,
],
),
5: (
# ERR Stroud lists 0.25056280_8_085732
0.250562807085732,
0.694746590606866,
[
-0.220221371883822e03,
+0.730167125339176e02,
+0.143281369027706,
-0.203714128400494e02,
+0.448293291677155e-01,
+0.383685702879441e01,
],
),
6: (
0.237383303308445,
0.666069941755647,
[
-0.472474075695084e03,
+0.127114265148486e03,
+0.477511403723473e-01,
-0.275405124688594e02,
+0.410240990676946e-01,
+0.376041809497708e01,
],
),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (u, 2))),
(B[4], fsd(n, (v, 2))),
(B[5], fsd(n, (u, 3))),
]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 7a", n, weights, points, 7, source, 5.878e-13)
def stenger_7b(n):
u, v, B = {
3: (
0.765055323929465,
0.285231516480645,
[
+0.192021185324132e02,
+0.351560542364457,
-0.743934568569924e01,
+0.270884863064213e-01,
+0.226016702392506e01,
+0.828955120051268e-02,
],
),
4: (
0.727412389740367,
0.266216481931920,
[
+0.474254976042737e02,
+0.344486899232533,
-0.133998630586466e02,
+0.112015320403370e-01,
+0.255857861534543e01,
+0.867469794764608e-02,
],
),
5: (
0.694746590606866,
0.250562807085732,
[
+0.860517194425776e02,
+0.345922662173719,
-0.190678561571482e02,
-0.583099411878751e-02,
+0.264972933271699e01,
+0.844338721441717e-02,
],
),
6: (
0.666069941755647,
0.237383303308445,
[
+0.127959890358419e03,
+0.355983426079872,
-0.233024586505972e02,
-0.206223580738102e-01,
+0.254283229095729e01,
+0.770580714268811e-02,
],
),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (u, 2))),
(B[4], fsd(n, (v, 2))),
(B[5], fsd(n, (u, 3))),
]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 7b", n, weights, points, 7, source, 8.052e-14)
def stenger_9a(n):
u, v, B = {
3: (
0.468848793470714,
0.830223896278566,
[
-0.287231328328341e-01,
+0.199630268052470,
+0.636651162483951e-01,
+0.214860470062486e-01,
-0.323488968539688e-02,
+0.543046730120527e-01,
+0.138854806789114,
+0.572067170204320e-03,
],
),
4: (
0.442930458136056,
0.798214220988774,
[
-0.713264931069213,
+0.399661978095933,
-0.399833383634508e-01,
-0.114245162120668,
-0.380033830313039e-02,
+0.528728807201216e-01,
+0.771286345600799e-01,
+0.606328581574664e-03,
+0.361445580516503e-01,
],
),
5: (
0.420914805023811,
0.769455324331787,
[
-0.148607372859000e01,
+0.517337302000060,
-0.128693800819958,
-0.972921046543696e-01,
-0.434640813962413e-02,
+0.485386100042993e-01,
+0.418811699326262e-03,
+0.591022411210733e-03,
+0.370636589159518e-01,
],
),
6: (
0.401905036089210,
0.743477004521219,
[
-0.141522888827140e01,
+0.225283928934273,
-0.189925600936916,
+0.767542859780860e-01,
-0.467837989649319e-02,
+0.422029877037414e-01,
-0.736785106177554e-01,
+0.537264443490880e-03,
+0.353013676287120e-01,
],
),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (u, 2))),
(B[4], fsd(n, (v, 2))),
(B[5], fsd(n, (u, 1), (v, 1))),
(B[6], fsd(n, (u, 3))),
(B[7], fsd(n, (v, 3))),
]
if n > 3:
data += [(B[8], fsd(n, (u, 4)))]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 9a", n, weights, points, 9, source)
def stenger_9b(n):
u, v, B = {
4: (
0.798214220988774,
0.442930458136056,
[
-0.128637919793059e01,
-0.425826693459664e-01,
+0.688818442509136,
-0.250067281187269e-02,
-0.258823394327268,
+0.528728807201216e-01,
-0.435041640541910e-04,
+0.149417750663380,
+0.324916372814428e-03,
],
),
5: (
0.769455324331787,
0.420914805023811,
[
-0.442739121785452e01,
-0.138203890424606,
+0.170337438731052e01,
-0.780124537881237e-03,
-0.542056011645791,
+0.485386100042993e-01,
-0.597738789370232e-03,
+0.148673447363133,
+0.297190300145242e-03,
],
),
6: (
0.743477004521219,
0.401905036089210,
[
-0.982577681614207e01,
-0.210519035276982,
+0.304939333923123e01,
+0.149965040552639e-02,
-0.770478537111000,
+0.422029877037414e-01,
-0.100724313201402e-02,
+0.138129695154516,
+0.257417929250816e-03,
],
),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (u, 2))),
(B[4], fsd(n, (v, 2))),
(B[5], fsd(n, (u, 1), (v, 1))),
(B[6], fsd(n, (u, 3))),
(B[7], fsd(n, (v, 3))),
]
if n > 3:
data += [(B[8], fsd(n, (u, 4)))]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 9b", n, weights, points, 9, source)
def stenger_11a(n):
u, v, w, B = {
3: (
0.871740148509601,
0.209299217902484,
0.591700181433148,
[
-0.241858949226757e02,
-0.260615043902870,
+0.135920710866830e02,
+0.128644782328402,
-0.827362457508920e-02,
-0.773259998046230e01,
+0.121529267823348,
+0.107430092990906,
-0.457147160368296e-02,
-0.454573469083632e-03,
+0.453152994236346e01,
+0.249979156876400e-01,
+0.537579305958677e-02,
],
),
4: (
0.844750603518454,
0.198187323325469,
0.564399481003361,
[
+0.142841140985092e03,
-0.409969170254506,
-0.833355966329160e02,
+0.477233536818062e-01,
-0.167713384337120e-01,
+0.486046977166435e02,
+0.762265878588936e-01,
+0.945800683336929e-01,
-0.500622516633870e-02,
-0.498867819405501e-03,
-0.283869034949532e02,
+0.255516059013563e-01,
+0.504633074872994e-02,
+0.535903211027872e-04,
+0.166581940110341e02,
],
),
# TODO find typo
# 5: (
# 0.819845995463488,
# 0.540604637387361,
# 0.188677422490785,
# [
# -0.455346412352218e+03,
# -0.643198057179463,
# -0.313723910937508,
# +0.142863899851242e+03,
# -0.115044196304602e-02,
# +0.149484688898586,
# -0.373831314185824e+02,
# -0.141469445049282e-01,
# +0.104863970436266,
# -0.973070178977534e-03,
# -0.862234640073899e-02,
# +0.654476925250512e+01,
# +0.153312717360660e-02,
# -0.611928443128898e-04,
# +0.622126777947090e-02,
# +0.887274197302674e-05,
# ],
# ),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (w, 1))),
(B[4], fsd(n, (u, 2))),
(B[5], fsd(n, (v, 2))),
(B[6], fsd(n, (w, 2))),
(B[7], fsd(n, (u, 1), (v, 1))),
(B[8], fsd(n, (u, 1), (w, 1))),
(B[9], fsd(n, (u, 3))),
(B[10], fsd(n, (v, 3))),
(B[11], fsd(n, (w, 3))),
(B[12], fsd(n, (u, 2), (v, 1))),
]
if n > 3:
data += [(B[13], fsd(n, (u, 4))), (B[14], fsd(n, (v, 4)))]
if n > 4:
data += [(B[15], fsd(n, (u, 5)))]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 11a", n, weights, points, 11, source, 1.171e-13)
def stenger_11b(n):
u, v, w, B = {
# TODO find typo
# 3: (
# 0.871740148509601,
# 0.591700181433148,
# 0.209299217902484,
# [
# -0.369608422804220e+02,
# -0.292281498761429,
# +0.981554121166593e-01,
# +0.200417005906264e+02,
# +0.915649460852702e-03,
# +0.140881585655056,
# -0.109681663185532e+02,
# -0.867910432951898e-02,
# +0.118181679110079,
# -0.172723379038583e-02,
# +0.153215767717862e-01,
# +0.614931311140891e+01,
# +0.205381636291801e-02,
# ],
# ),
4: (
0.844750603518454,
0.564399481003361,
0.198187323325469,
[
-0.176252077581424e03,
-0.499462881047293,
-0.922793735873844e-01,
+0.698573153138448e02,
-0.179342748736783e-04,
+0.138909533618563,
-0.246907169366506e02,
-0.122578300082603e-01,
+0.114765391328613,
-0.134570335654617e-02,
+0.330804559042674e-02,
+0.659514417940443e01,
+0.181290121048040e-02,
-0.544956432165929e-04,
+0.657282387101229e-02,
],
),
5: (
0.819845995463488,
0.188677422490785,
0.540604637387361,
[
+0.101979474263003e04,
-0.483876622763145,
-0.435066365787229e03,
+0.431869862745281e-01,
-0.229234716520547e-01,
+0.174586702318035e03,
+0.261731304360525e-01,
+0.779664996582579e-01,
-0.494818146328864e-02,
-0.473327076222103e-03,
-0.625137212983254e02,
+0.243721155686252e-01,
+0.448291181300128e-02,
+0.284624253627758e-04,
+0.168650924640304e02,
+0.887274197302674e-05,
],
),
}[n]
data = [
(B[0], z(n)),
(B[1], fsd(n, (u, 1))),
(B[2], fsd(n, (v, 1))),
(B[3], fsd(n, (w, 1))),
(B[4], fsd(n, (u, 2))),
(B[5], fsd(n, (v, 2))),
(B[6], fsd(n, (w, 2))),
(B[7], fsd(n, (u, 1), (v, 1))),
(B[8], fsd(n, (u, 1), (w, 1))),
(B[9], fsd(n, (u, 3))),
(B[10], fsd(n, (v, 3))),
(B[11], fsd(n, (w, 3))),
(B[12], fsd(n, (u, 2), (v, 1))),
]
if n > 3:
data += [(B[13], fsd(n, (u, 4))), (B[14], fsd(n, (v, 4)))]
if n > 4:
data += [(B[15], fsd(n, (u, 5)))]
points, weights = untangle(data)
weights /= volume_nball(n, symbolic=False)
return SnScheme("Stenger 11b", n, weights, points, 11, source, 2.721e-12)