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Traktor/myenv/Lib/site-packages/mpmath/tests/test_functions2.py

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losowanie zdjec
2024-05-26 05:12:46 +02:00
import math
import pytest
from mpmath import *
def test_bessel():
mp.dps = 15
assert j0(1).ae(0.765197686557966551)
assert j0(pi).ae(-0.304242177644093864)
assert j0(1000).ae(0.0247866861524201746)
assert j0(-25).ae(0.0962667832759581162)
assert j1(1).ae(0.440050585744933516)
assert j1(pi).ae(0.284615343179752757)
assert j1(1000).ae(0.00472831190708952392)
assert j1(-25).ae(0.125350249580289905)
assert besselj(5,1).ae(0.000249757730211234431)
assert besselj(5+0j,1).ae(0.000249757730211234431)
assert besselj(5,pi).ae(0.0521411843671184747)
assert besselj(5,1000).ae(0.00502540694523318607)
assert besselj(5,-25).ae(0.0660079953984229934)
assert besselj(-3,2).ae(-0.128943249474402051)
assert besselj(-4,2).ae(0.0339957198075684341)
assert besselj(3,3+2j).ae(0.424718794929639595942 + 0.625665327745785804812j)
assert besselj(0.25,4).ae(-0.374760630804249715)
assert besselj(1+2j,3+4j).ae(0.319247428741872131 - 0.669557748880365678j)
assert (besselj(3, 10**10) * 10**5).ae(0.76765081748139204023)
assert bessely(-0.5, 0) == 0
assert bessely(0.5, 0) == -inf
assert bessely(1.5, 0) == -inf
assert bessely(0,0) == -inf
assert bessely(-0.4, 0) == -inf
assert bessely(-0.6, 0) == inf
assert bessely(-1, 0) == inf
assert bessely(-1.4, 0) == inf
assert bessely(-1.6, 0) == -inf
assert bessely(-1, 0) == inf
assert bessely(-2, 0) == -inf
assert bessely(-3, 0) == inf
assert bessely(0.5, 0) == -inf
assert bessely(1, 0) == -inf
assert bessely(1.5, 0) == -inf
assert bessely(2, 0) == -inf
assert bessely(2.5, 0) == -inf
assert bessely(3, 0) == -inf
assert bessely(0,0.5).ae(-0.44451873350670655715)
assert bessely(1,0.5).ae(-1.4714723926702430692)
assert bessely(-1,0.5).ae(1.4714723926702430692)
assert bessely(3.5,0.5).ae(-138.86400867242488443)
assert bessely(0,3+4j).ae(4.6047596915010138655-8.8110771408232264208j)
assert bessely(0,j).ae(-0.26803248203398854876+1.26606587775200833560j)
assert (bessely(3, 10**10) * 10**5).ae(0.21755917537013204058)
assert besseli(0,0) == 1
assert besseli(1,0) == 0
assert besseli(2,0) == 0
assert besseli(-1,0) == 0
assert besseli(-2,0) == 0
assert besseli(0,0.5).ae(1.0634833707413235193)
assert besseli(1,0.5).ae(0.25789430539089631636)
assert besseli(-1,0.5).ae(0.25789430539089631636)
assert besseli(3.5,0.5).ae(0.00068103597085793815863)
assert besseli(0,3+4j).ae(-3.3924877882755196097-1.3239458916287264815j)
assert besseli(0,j).ae(besselj(0,1))
assert (besseli(3, 10**10) * mpf(10)**(-4342944813)).ae(4.2996028505491271875)
assert besselk(0,0) == inf
assert besselk(1,0) == inf
assert besselk(2,0) == inf
assert besselk(-1,0) == inf
assert besselk(-2,0) == inf
assert besselk(0,0.5).ae(0.92441907122766586178)
assert besselk(1,0.5).ae(1.6564411200033008937)
assert besselk(-1,0.5).ae(1.6564411200033008937)
assert besselk(3.5,0.5).ae(207.48418747548460607)
assert besselk(0,3+4j).ae(-0.007239051213570155013+0.026510418350267677215j)
assert besselk(0,j).ae(-0.13863371520405399968-1.20196971531720649914j)
assert (besselk(3, 10**10) * mpf(10)**4342944824).ae(1.1628981033356187851)
# test for issue 331, bug reported by Michael Hartmann
for n in range(10,100,10):
mp.dps = n
assert besseli(91.5,24.7708).ae("4.00830632138673963619656140653537080438462342928377020695738635559218797348548092636896796324190271316137982810144874264e-41")
def test_bessel_zeros():
mp.dps = 15
assert besseljzero(0,1).ae(2.40482555769577276869)
assert besseljzero(2,1).ae(5.1356223018406825563)
assert besseljzero(1,50).ae(157.86265540193029781)
assert besseljzero(10,1).ae(14.475500686554541220)
assert besseljzero(0.5,3).ae(9.4247779607693797153)
assert besseljzero(2,1,1).ae(3.0542369282271403228)
assert besselyzero(0,1).ae(0.89357696627916752158)
assert besselyzero(2,1).ae(3.3842417671495934727)
assert besselyzero(1,50).ae(156.29183520147840108)
assert besselyzero(10,1).ae(12.128927704415439387)
assert besselyzero(0.5,3).ae(7.8539816339744830962)
assert besselyzero(2,1,1).ae(5.0025829314460639452)
def test_hankel():
mp.dps = 15
assert hankel1(0,0.5).ae(0.93846980724081290423-0.44451873350670655715j)
assert hankel1(1,0.5).ae(0.2422684576748738864-1.4714723926702430692j)
assert hankel1(-1,0.5).ae(-0.2422684576748738864+1.4714723926702430692j)
assert hankel1(1.5,0.5).ae(0.0917016996256513026-2.5214655504213378514j)
assert hankel1(1.5,3+4j).ae(0.0066806866476728165382-0.0036684231610839127106j)
assert hankel2(0,0.5).ae(0.93846980724081290423+0.44451873350670655715j)
assert hankel2(1,0.5).ae(0.2422684576748738864+1.4714723926702430692j)
assert hankel2(-1,0.5).ae(-0.2422684576748738864-1.4714723926702430692j)
assert hankel2(1.5,0.5).ae(0.0917016996256513026+2.5214655504213378514j)
assert hankel2(1.5,3+4j).ae(14.783528526098567526-7.397390270853446512j)
def test_struve():
mp.dps = 15
assert struveh(2,3).ae(0.74238666967748318564)
assert struveh(-2.5,3).ae(0.41271003220971599344)
assert struvel(2,3).ae(1.7476573277362782744)
assert struvel(-2.5,3).ae(1.5153394466819651377)
def test_whittaker():
mp.dps = 15
assert whitm(2,3,4).ae(49.753745589025246591)
assert whitw(2,3,4).ae(14.111656223052932215)
def test_kelvin():
mp.dps = 15
assert ber(2,3).ae(0.80836846563726819091)
assert ber(3,4).ae(-0.28262680167242600233)
assert ber(-3,2).ae(-0.085611448496796363669)
assert bei(2,3).ae(-0.89102236377977331571)
assert bei(-3,2).ae(-0.14420994155731828415)
assert ker(2,3).ae(0.12839126695733458928)
assert ker(-3,2).ae(-0.29802153400559142783)
assert ker(0.5,3).ae(-0.085662378535217097524)
assert kei(2,3).ae(0.036804426134164634000)
assert kei(-3,2).ae(0.88682069845786731114)
assert kei(0.5,3).ae(0.013633041571314302948)
def test_hyper_misc():
mp.dps = 15
assert hyp0f1(1,0) == 1
assert hyp1f1(1,2,0) == 1
assert hyp1f2(1,2,3,0) == 1
assert hyp2f1(1,2,3,0) == 1
assert hyp2f2(1,2,3,4,0) == 1
assert hyp2f3(1,2,3,4,5,0) == 1
# Degenerate case: 0F0
assert hyper([],[],0) == 1
assert hyper([],[],-2).ae(exp(-2))
# Degenerate case: 1F0
assert hyper([2],[],1.5) == 4
#
assert hyp2f1((1,3),(2,3),(5,6),mpf(27)/32).ae(1.6)
assert hyp2f1((1,4),(1,2),(3,4),mpf(80)/81).ae(1.8)
assert hyp2f1((2,3),(1,1),(3,2),(2+j)/3).ae(1.327531603558679093+0.439585080092769253j)
mp.dps = 25
v = mpc('1.2282306665029814734863026', '-0.1225033830118305184672133')
assert hyper([(3,4),2+j,1],[1,5,j/3],mpf(1)/5+j/8).ae(v)
mp.dps = 15
def test_elliptic_integrals():
mp.dps = 15
assert ellipk(0).ae(pi/2)
assert ellipk(0.5).ae(gamma(0.25)**2/(4*sqrt(pi)))
assert ellipk(1) == inf
assert ellipk(1+0j) == inf
assert ellipk(-1).ae('1.3110287771460599052')
assert ellipk(-2).ae('1.1714200841467698589')
assert isinstance(ellipk(-2), mpf)
assert isinstance(ellipe(-2), mpf)
assert ellipk(-50).ae('0.47103424540873331679')
mp.dps = 30
n1 = +fraction(99999,100000)
n2 = +fraction(100001,100000)
mp.dps = 15
assert ellipk(n1).ae('7.1427724505817781901')
assert ellipk(n2).ae(mpc('7.1427417367963090109', '-1.5707923998261688019'))
assert ellipe(n1).ae('1.0000332138990829170')
v = ellipe(n2)
assert v.real.ae('0.999966786328145474069137')
assert (v.imag*10**6).ae('7.853952181727432')
assert ellipk(2).ae(mpc('1.3110287771460599052', '-1.3110287771460599052'))
assert ellipk(50).ae(mpc('0.22326753950210985451', '-0.47434723226254522087'))
assert ellipk(3+4j).ae(mpc('0.91119556380496500866', '0.63133428324134524388'))
assert ellipk(3-4j).ae(mpc('0.91119556380496500866', '-0.63133428324134524388'))
assert ellipk(-3+4j).ae(mpc('0.95357894880405122483', '0.23093044503746114444'))
assert ellipk(-3-4j).ae(mpc('0.95357894880405122483', '-0.23093044503746114444'))
assert isnan(ellipk(nan))
assert isnan(ellipe(nan))
assert ellipk(inf) == 0
assert isinstance(ellipk(inf), mpc)
assert ellipk(-inf) == 0
assert ellipk(1+0j) == inf
assert ellipe(0).ae(pi/2)
assert ellipe(0.5).ae(pi**(mpf(3)/2)/gamma(0.25)**2 +gamma(0.25)**2/(8*sqrt(pi)))
assert ellipe(1) == 1
assert ellipe(1+0j) == 1
assert ellipe(inf) == mpc(0,inf)
assert ellipe(-inf) == inf
assert ellipe(3+4j).ae(1.4995535209333469543-1.5778790079127582745j)
assert ellipe(3-4j).ae(1.4995535209333469543+1.5778790079127582745j)
assert ellipe(-3+4j).ae(2.5804237855343377803-0.8306096791000413778j)
assert ellipe(-3-4j).ae(2.5804237855343377803+0.8306096791000413778j)
assert ellipe(2).ae(0.59907011736779610372+0.59907011736779610372j)
assert ellipe('1e-1000000000').ae(pi/2)
assert ellipk('1e-1000000000').ae(pi/2)
assert ellipe(-pi).ae(2.4535865983838923)
mp.dps = 50
assert ellipk(1/pi).ae('1.724756270009501831744438120951614673874904182624739673')
assert ellipe(1/pi).ae('1.437129808135123030101542922290970050337425479058225712')
assert ellipk(-10*pi).ae('0.5519067523886233967683646782286965823151896970015484512')
assert ellipe(-10*pi).ae('5.926192483740483797854383268707108012328213431657645509')
v = ellipk(pi)
assert v.real.ae('0.973089521698042334840454592642137667227167622330325225')
assert v.imag.ae('-1.156151296372835303836814390793087600271609993858798016')
v = ellipe(pi)
assert v.real.ae('0.4632848917264710404078033487934663562998345622611263332')
assert v.imag.ae('1.0637961621753130852473300451583414489944099504180510966')
mp.dps = 15
def test_exp_integrals():
mp.dps = 15
x = +e
z = e + sqrt(3)*j
assert ei(x).ae(8.21168165538361560)
assert li(x).ae(1.89511781635593676)
assert si(x).ae(1.82104026914756705)
assert ci(x).ae(0.213958001340379779)
assert shi(x).ae(4.11520706247846193)
assert chi(x).ae(4.09647459290515367)
assert fresnels(x).ae(0.437189718149787643)
assert fresnelc(x).ae(0.401777759590243012)
assert airyai(x).ae(0.0108502401568586681)
assert airybi(x).ae(8.98245748585468627)
assert ei(z).ae(3.72597969491314951 + 7.34213212314224421j)
assert li(z).ae(2.28662658112562502 + 1.50427225297269364j)
assert si(z).ae(2.48122029237669054 + 0.12684703275254834j)
assert ci(z).ae(0.169255590269456633 - 0.892020751420780353j)
assert shi(z).ae(1.85810366559344468 + 3.66435842914920263j)
assert chi(z).ae(1.86787602931970484 + 3.67777369399304159j)
assert fresnels(z/3).ae(0.034534397197008182 + 0.754859844188218737j)
assert fresnelc(z/3).ae(1.261581645990027372 + 0.417949198775061893j)
assert airyai(z).ae(-0.0162552579839056062 - 0.0018045715700210556j)
assert airybi(z).ae(-4.98856113282883371 + 2.08558537872180623j)
assert li(0) == 0.0
assert li(1) == -inf
assert li(inf) == inf
assert isinstance(li(0.7), mpf)
assert si(inf).ae(pi/2)
assert si(-inf).ae(-pi/2)
assert ci(inf) == 0
assert ci(0) == -inf
assert isinstance(ei(-0.7), mpf)
assert airyai(inf) == 0
assert airybi(inf) == inf
assert airyai(-inf) == 0
assert airybi(-inf) == 0
assert fresnels(inf) == 0.5
assert fresnelc(inf) == 0.5
assert fresnels(-inf) == -0.5
assert fresnelc(-inf) == -0.5
assert shi(0) == 0
assert shi(inf) == inf
assert shi(-inf) == -inf
assert chi(0) == -inf
assert chi(inf) == inf
def test_ei():
mp.dps = 15
assert ei(0) == -inf
assert ei(inf) == inf
assert ei(-inf) == -0.0
assert ei(20+70j).ae(6.1041351911152984397e6 - 2.7324109310519928872e6j)
# tests for the asymptotic expansion
# values checked with Mathematica ExpIntegralEi
mp.dps = 50
r = ei(20000)
s = '3.8781962825045010930273870085501819470698476975019e+8681'
assert str(r) == s
r = ei(-200)
s = '-6.8852261063076355977108174824557929738368086933303e-90'
assert str(r) == s
r =ei(20000 + 10*j)
sre = '-3.255138234032069402493850638874410725961401274106e+8681'
sim = '-2.1081929993474403520785942429469187647767369645423e+8681'
assert str(r.real) == sre and str(r.imag) == sim
mp.dps = 15
# More asymptotic expansions
assert chi(-10**6+100j).ae('1.3077239389562548386e+434288 + 7.6808956999707408158e+434287j')
assert shi(-10**6+100j).ae('-1.3077239389562548386e+434288 - 7.6808956999707408158e+434287j')
mp.dps = 15
assert ei(10j).ae(-0.0454564330044553726+3.2291439210137706686j)
assert ei(100j).ae(-0.0051488251426104921+3.1330217936839529126j)
u = ei(fmul(10**20, j, exact=True))
assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps)
assert u.imag.ae(pi)
assert ei(-10j).ae(-0.0454564330044553726-3.2291439210137706686j)
assert ei(-100j).ae(-0.0051488251426104921-3.1330217936839529126j)
u = ei(fmul(-10**20, j, exact=True))
assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps)
assert u.imag.ae(-pi)
assert ei(10+10j).ae(-1576.1504265768517448+436.9192317011328140j)
u = ei(-10+10j)
assert u.real.ae(7.6698978415553488362543e-7, abs_eps=0, rel_eps=8*eps)
assert u.imag.ae(3.141595611735621062025)
def test_e1():
mp.dps = 15
assert e1(0) == inf
assert e1(inf) == 0
assert e1(-inf) == mpc(-inf, -pi)
assert e1(10j).ae(0.045456433004455372635 + 0.087551267423977430100j)
assert e1(100j).ae(0.0051488251426104921444 - 0.0085708599058403258790j)
assert e1(fmul(10**20, j, exact=True)).ae(6.4525128526578084421e-21 - 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps)
assert e1(-10j).ae(0.045456433004455372635 - 0.087551267423977430100j)
assert e1(-100j).ae(0.0051488251426104921444 + 0.0085708599058403258790j)
assert e1(fmul(-10**20, j, exact=True)).ae(6.4525128526578084421e-21 + 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps)
def test_expint():
mp.dps = 15
assert expint(0,0) == inf
assert expint(0,1).ae(1/e)
assert expint(0,1.5).ae(2/exp(1.5)/3)
assert expint(1,1).ae(-ei(-1))
assert expint(2,0).ae(1)
assert expint(3,0).ae(1/2.)
assert expint(4,0).ae(1/3.)
assert expint(-2, 0.5).ae(26/sqrt(e))
assert expint(-1,-1) == 0
assert expint(-2,-1).ae(-e)
assert expint(5.5, 0).ae(2/9.)
assert expint(2.00000001,0).ae(100000000./100000001)
assert expint(2+3j,4-j).ae(0.0023461179581675065414+0.0020395540604713669262j)
assert expint('1.01', '1e-1000').ae(99.9999999899412802)
assert expint('1.000000000001', 3.5).ae(0.00697013985754701819446)
assert expint(2,3).ae(3*ei(-3)+exp(-3))
assert (expint(10,20)*10**10).ae(0.694439055541231353)
assert expint(3,inf) == 0
assert expint(3.2,inf) == 0
assert expint(3.2+2j,inf) == 0
assert expint(1,3j).ae(-0.11962978600800032763 + 0.27785620120457163717j)
assert expint(1,3).ae(0.013048381094197037413)
assert expint(1,-3).ae(-ei(3)-pi*j)
#assert expint(3) == expint(1,3)
assert expint(1,-20).ae(-25615652.66405658882 - 3.1415926535897932385j)
assert expint(1000000,0).ae(1./999999)
assert expint(0,2+3j).ae(-0.025019798357114678171 + 0.027980439405104419040j)
assert expint(-1,2+3j).ae(-0.022411973626262070419 + 0.038058922011377716932j)
assert expint(-1.5,0) == inf
def test_trig_integrals():
mp.dps = 30
assert si(mpf(1)/1000000).ae('0.000000999999999999944444444444446111')
assert ci(mpf(1)/1000000).ae('-13.2382948930629912435014366276')
assert si(10**10).ae('1.5707963267075846569685111517747537')
assert ci(10**10).ae('-4.87506025174822653785729773959e-11')
assert si(10**100).ae(pi/2)
assert (ci(10**100)*10**100).ae('-0.372376123661276688262086695553')
assert si(-3) == -si(3)
assert ci(-3).ae(ci(3) + pi*j)
# Test complex structure
mp.dps = 15
assert mp.ci(50).ae(-0.0056283863241163054402)
assert mp.ci(50+2j).ae(-0.018378282946133067149+0.070352808023688336193j)
assert mp.ci(20j).ae(1.28078263320282943611e7+1.5707963267949j)
assert mp.ci(-2+20j).ae(-4.050116856873293505e6+1.207476188206989909e7j)
assert mp.ci(-50+2j).ae(-0.0183782829461330671+3.0712398455661049023j)
assert mp.ci(-50).ae(-0.0056283863241163054+3.1415926535897932385j)
assert mp.ci(-50-2j).ae(-0.0183782829461330671-3.0712398455661049023j)
assert mp.ci(-2-20j).ae(-4.050116856873293505e6-1.207476188206989909e7j)
assert mp.ci(-20j).ae(1.28078263320282943611e7-1.5707963267949j)
assert mp.ci(50-2j).ae(-0.018378282946133067149-0.070352808023688336193j)
assert mp.si(50).ae(1.5516170724859358947)
assert mp.si(50+2j).ae(1.497884414277228461-0.017515007378437448j)
assert mp.si(20j).ae(1.2807826332028294459e7j)
assert mp.si(-2+20j).ae(-1.20747603112735722103e7-4.050116856873293554e6j)
assert mp.si(-50+2j).ae(-1.497884414277228461-0.017515007378437448j)
assert mp.si(-50).ae(-1.5516170724859358947)
assert mp.si(-50-2j).ae(-1.497884414277228461+0.017515007378437448j)
assert mp.si(-2-20j).ae(-1.20747603112735722103e7+4.050116856873293554e6j)
assert mp.si(-20j).ae(-1.2807826332028294459e7j)
assert mp.si(50-2j).ae(1.497884414277228461+0.017515007378437448j)
assert mp.chi(50j).ae(-0.0056283863241163054+1.5707963267948966192j)
assert mp.chi(-2+50j).ae(-0.0183782829461330671+1.6411491348185849554j)
assert mp.chi(-20).ae(1.28078263320282943611e7+3.1415926535898j)
assert mp.chi(-20-2j).ae(-4.050116856873293505e6+1.20747571696809187053e7j)
assert mp.chi(-2-50j).ae(-0.0183782829461330671-1.6411491348185849554j)
assert mp.chi(-50j).ae(-0.0056283863241163054-1.5707963267948966192j)
assert mp.chi(2-50j).ae(-0.0183782829461330671-1.500443518771208283j)
assert mp.chi(20-2j).ae(-4.050116856873293505e6-1.20747603112735722951e7j)
assert mp.chi(20).ae(1.2807826332028294361e7)
assert mp.chi(2+50j).ae(-0.0183782829461330671+1.500443518771208283j)
assert mp.shi(50j).ae(1.5516170724859358947j)
assert mp.shi(-2+50j).ae(0.017515007378437448+1.497884414277228461j)
assert mp.shi(-20).ae(-1.2807826332028294459e7)
assert mp.shi(-20-2j).ae(4.050116856873293554e6-1.20747603112735722103e7j)
assert mp.shi(-2-50j).ae(0.017515007378437448-1.497884414277228461j)
assert mp.shi(-50j).ae(-1.5516170724859358947j)
assert mp.shi(2-50j).ae(-0.017515007378437448-1.497884414277228461j)
assert mp.shi(20-2j).ae(-4.050116856873293554e6-1.20747603112735722103e7j)
assert mp.shi(20).ae(1.2807826332028294459e7)
assert mp.shi(2+50j).ae(-0.017515007378437448+1.497884414277228461j)
def ae(x,y,tol=1e-12):
return abs(x-y) <= abs(y)*tol
assert fp.ci(fp.inf) == 0
assert ae(fp.ci(fp.ninf), fp.pi*1j)
assert ae(fp.si(fp.inf), fp.pi/2)
assert ae(fp.si(fp.ninf), -fp.pi/2)
assert fp.si(0) == 0
assert ae(fp.ci(50), -0.0056283863241163054402)
assert ae(fp.ci(50+2j), -0.018378282946133067149+0.070352808023688336193j)
assert ae(fp.ci(20j), 1.28078263320282943611e7+1.5707963267949j)
assert ae(fp.ci(-2+20j), -4.050116856873293505e6+1.207476188206989909e7j)
assert ae(fp.ci(-50+2j), -0.0183782829461330671+3.0712398455661049023j)
assert ae(fp.ci(-50), -0.0056283863241163054+3.1415926535897932385j)
assert ae(fp.ci(-50-2j), -0.0183782829461330671-3.0712398455661049023j)
assert ae(fp.ci(-2-20j), -4.050116856873293505e6-1.207476188206989909e7j)
assert ae(fp.ci(-20j), 1.28078263320282943611e7-1.5707963267949j)
assert ae(fp.ci(50-2j), -0.018378282946133067149-0.070352808023688336193j)
assert ae(fp.si(50), 1.5516170724859358947)
assert ae(fp.si(50+2j), 1.497884414277228461-0.017515007378437448j)
assert ae(fp.si(20j), 1.2807826332028294459e7j)
assert ae(fp.si(-2+20j), -1.20747603112735722103e7-4.050116856873293554e6j)
assert ae(fp.si(-50+2j), -1.497884414277228461-0.017515007378437448j)
assert ae(fp.si(-50), -1.5516170724859358947)
assert ae(fp.si(-50-2j), -1.497884414277228461+0.017515007378437448j)
assert ae(fp.si(-2-20j), -1.20747603112735722103e7+4.050116856873293554e6j)
assert ae(fp.si(-20j), -1.2807826332028294459e7j)
assert ae(fp.si(50-2j), 1.497884414277228461+0.017515007378437448j)
assert ae(fp.chi(50j), -0.0056283863241163054+1.5707963267948966192j)
assert ae(fp.chi(-2+50j), -0.0183782829461330671+1.6411491348185849554j)
assert ae(fp.chi(-20), 1.28078263320282943611e7+3.1415926535898j)
assert ae(fp.chi(-20-2j), -4.050116856873293505e6+1.20747571696809187053e7j)
assert ae(fp.chi(-2-50j), -0.0183782829461330671-1.6411491348185849554j)
assert ae(fp.chi(-50j), -0.0056283863241163054-1.5707963267948966192j)
assert ae(fp.chi(2-50j), -0.0183782829461330671-1.500443518771208283j)
assert ae(fp.chi(20-2j), -4.050116856873293505e6-1.20747603112735722951e7j)
assert ae(fp.chi(20), 1.2807826332028294361e7)
assert ae(fp.chi(2+50j), -0.0183782829461330671+1.500443518771208283j)
assert ae(fp.shi(50j), 1.5516170724859358947j)
assert ae(fp.shi(-2+50j), 0.017515007378437448+1.497884414277228461j)
assert ae(fp.shi(-20), -1.2807826332028294459e7)
assert ae(fp.shi(-20-2j), 4.050116856873293554e6-1.20747603112735722103e7j)
assert ae(fp.shi(-2-50j), 0.017515007378437448-1.497884414277228461j)
assert ae(fp.shi(-50j), -1.5516170724859358947j)
assert ae(fp.shi(2-50j), -0.017515007378437448-1.497884414277228461j)
assert ae(fp.shi(20-2j), -4.050116856873293554e6-1.20747603112735722103e7j)
assert ae(fp.shi(20), 1.2807826332028294459e7)
assert ae(fp.shi(2+50j), -0.017515007378437448+1.497884414277228461j)
def test_airy():
mp.dps = 15
assert (airyai(10)*10**10).ae(1.1047532552898687)
assert (airybi(10)/10**9).ae(0.45564115354822515)
assert (airyai(1000)*10**9158).ae(9.306933063179556004)
assert (airybi(1000)/10**9154).ae(5.4077118391949465477)
assert airyai(-1000).ae(0.055971895773019918842)
assert airybi(-1000).ae(-0.083264574117080633012)
assert (airyai(100+100j)*10**188).ae(2.9099582462207032076 + 2.353013591706178756j)
assert (airybi(100+100j)/10**185).ae(1.7086751714463652039 - 3.1416590020830804578j)
def test_hyper_0f1():
mp.dps = 15
v = 8.63911136507950465
assert hyper([],[(1,3)],1.5).ae(v)
assert hyper([],[1/3.],1.5).ae(v)
assert hyp0f1(1/3.,1.5).ae(v)
assert hyp0f1((1,3),1.5).ae(v)
# Asymptotic expansion
assert hyp0f1(3,1e9).ae('4.9679055380347771271e+27455')
assert hyp0f1(3,1e9j).ae('-2.1222788784457702157e+19410 + 5.0840597555401854116e+19410j')
def test_hyper_1f1():
mp.dps = 15
v = 1.2917526488617656673
assert hyper([(1,2)],[(3,2)],0.7).ae(v)
assert hyper([(1,2)],[(3,2)],0.7+0j).ae(v)
assert hyper([0.5],[(3,2)],0.7).ae(v)
assert hyper([0.5],[1.5],0.7).ae(v)
assert hyper([0.5],[(3,2)],0.7+0j).ae(v)
assert hyper([0.5],[1.5],0.7+0j).ae(v)
assert hyper([(1,2)],[1.5+0j],0.7).ae(v)
assert hyper([0.5+0j],[1.5],0.7).ae(v)
assert hyper([0.5+0j],[1.5+0j],0.7+0j).ae(v)
assert hyp1f1(0.5,1.5,0.7).ae(v)
assert hyp1f1((1,2),1.5,0.7).ae(v)
# Asymptotic expansion
assert hyp1f1(2,3,1e10).ae('2.1555012157015796988e+4342944809')
assert (hyp1f1(2,3,1e10j)*10**10).ae(-0.97501205020039745852 - 1.7462392454512132074j)
# Shouldn't use asymptotic expansion
assert hyp1f1(-2, 1, 10000).ae(49980001)
# Bug
assert hyp1f1(1j,fraction(1,3),0.415-69.739j).ae(25.857588206024346592 + 15.738060264515292063j)
def test_hyper_2f1():
mp.dps = 15
v = 1.0652207633823291032
assert hyper([(1,2), (3,4)], [2], 0.3).ae(v)
assert hyper([(1,2), 0.75], [2], 0.3).ae(v)
assert hyper([0.5, 0.75], [2.0], 0.3).ae(v)
assert hyper([0.5, 0.75], [2.0], 0.3+0j).ae(v)
assert hyper([0.5+0j, (3,4)], [2.0], 0.3+0j).ae(v)
assert hyper([0.5+0j, (3,4)], [2.0], 0.3).ae(v)
assert hyper([0.5, (3,4)], [2.0+0j], 0.3).ae(v)
assert hyper([0.5+0j, 0.75+0j], [2.0+0j], 0.3+0j).ae(v)
v = 1.09234681096223231717 + 0.18104859169479360380j
assert hyper([(1,2),0.75+j], [2], 0.5).ae(v)
assert hyper([0.5,0.75+j], [2.0], 0.5).ae(v)
assert hyper([0.5,0.75+j], [2.0], 0.5+0j).ae(v)
assert hyper([0.5,0.75+j], [2.0+0j], 0.5+0j).ae(v)
v = 0.9625 - 0.125j
assert hyper([(3,2),-1],[4], 0.1+j/3).ae(v)
assert hyper([1.5,-1.0],[4], 0.1+j/3).ae(v)
assert hyper([1.5,-1.0],[4+0j], 0.1+j/3).ae(v)
assert hyper([1.5+0j,-1.0+0j],[4+0j], 0.1+j/3).ae(v)
v = 1.02111069501693445001 - 0.50402252613466859521j
assert hyper([(2,10),(3,10)],[(4,10)],1.5).ae(v)
assert hyper([0.2,(3,10)],[0.4+0j],1.5).ae(v)
assert hyper([0.2,(3,10)],[0.4+0j],1.5+0j).ae(v)
v = 0.76922501362865848528 + 0.32640579593235886194j
assert hyper([(2,10),(3,10)],[(4,10)],4+2j).ae(v)
assert hyper([0.2,(3,10)],[0.4+0j],4+2j).ae(v)
assert hyper([0.2,(3,10)],[(4,10)],4+2j).ae(v)
def test_hyper_2f1_hard():
mp.dps = 15
# Singular cases
assert hyp2f1(2,-1,-1,3).ae(7)
assert hyp2f1(2,-1,-1,3,eliminate_all=True).ae(0.25)
assert hyp2f1(2,-2,-2,3).ae(34)
assert hyp2f1(2,-2,-2,3,eliminate_all=True).ae(0.25)
assert hyp2f1(2,-2,-3,3) == 14
assert hyp2f1(2,-3,-2,3) == inf
assert hyp2f1(2,-1.5,-1.5,3) == 0.25
assert hyp2f1(1,2,3,0) == 1
assert hyp2f1(0,1,0,0) == 1
assert hyp2f1(0,0,0,0) == 1
assert isnan(hyp2f1(1,1,0,0))
assert hyp2f1(2,-1,-5, 0.25+0.25j).ae(1.1+0.1j)
assert hyp2f1(2,-5,-5, 0.25+0.25j, eliminate=False).ae(163./128 + 125./128*j)
assert hyp2f1(0.7235, -1, -5, 0.3).ae(1.04341)
assert hyp2f1(0.7235, -5, -5, 0.3, eliminate=False).ae(1.2939225017815903812)
assert hyp2f1(-1,-2,4,1) == 1.5
assert hyp2f1(1,2,-3,1) == inf
assert hyp2f1(-2,-2,1,1) == 6
assert hyp2f1(1,-2,-4,1).ae(5./3)
assert hyp2f1(0,-6,-4,1) == 1
assert hyp2f1(0,-3,-4,1) == 1
assert hyp2f1(0,0,0,1) == 1
assert hyp2f1(1,0,0,1,eliminate=False) == 1
assert hyp2f1(1,1,0,1) == inf
assert hyp2f1(1,-6,-4,1) == inf
assert hyp2f1(-7.2,-0.5,-4.5,1) == 0
assert hyp2f1(-7.2,-1,-2,1).ae(-2.6)
assert hyp2f1(1,-0.5,-4.5, 1) == inf
assert hyp2f1(1,0.5,-4.5, 1) == -inf
# Check evaluation on / close to unit circle
z = exp(j*pi/3)
w = (nthroot(2,3)+1)*exp(j*pi/12)/nthroot(3,4)**3
assert hyp2f1('1/2','1/6','1/3', z).ae(w)
assert hyp2f1('1/2','1/6','1/3', z.conjugate()).ae(w.conjugate())
assert hyp2f1(0.25, (1,3), 2, '0.999').ae(1.06826449496030635)
assert hyp2f1(0.25, (1,3), 2, '1.001').ae(1.06867299254830309446-0.00001446586793975874j)
assert hyp2f1(0.25, (1,3), 2, -1).ae(0.96656584492524351673)
assert hyp2f1(0.25, (1,3), 2, j).ae(0.99041766248982072266+0.03777135604180735522j)
assert hyp2f1(2,3,5,'0.99').ae(27.699347904322690602)
assert hyp2f1((3,2),-0.5,3,'0.99').ae(0.68403036843911661388)
assert hyp2f1(2,3,5,1j).ae(0.37290667145974386127+0.59210004902748285917j)
assert fsum([hyp2f1((7,10),(2,3),(-1,2), 0.95*exp(j*k)) for k in range(1,15)]).ae(52.851400204289452922+6.244285013912953225j)
assert fsum([hyp2f1((7,10),(2,3),(-1,2), 1.05*exp(j*k)) for k in range(1,15)]).ae(54.506013786220655330-3.000118813413217097j)
assert fsum([hyp2f1((7,10),(2,3),(-1,2), exp(j*k)) for k in range(1,15)]).ae(55.792077935955314887+1.731986485778500241j)
assert hyp2f1(2,2.5,-3.25,0.999).ae(218373932801217082543180041.33)
# Branches
assert hyp2f1(1,1,2,1.01).ae(4.5595744415723676911-3.1104877758314784539j)
assert hyp2f1(1,1,2,1.01+0.1j).ae(2.4149427480552782484+1.4148224796836938829j)
assert hyp2f1(1,1,2,3+4j).ae(0.14576709331407297807+0.48379185417980360773j)
assert hyp2f1(1,1,2,4).ae(-0.27465307216702742285 - 0.78539816339744830962j)
assert hyp2f1(1,1,2,-4).ae(0.40235947810852509365)
# Other:
# Cancellation with a large parameter involved (bug reported on sage-devel)
assert hyp2f1(112, (51,10), (-9,10), -0.99999).ae(-1.6241361047970862961e-24, abs_eps=0, rel_eps=eps*16)
def test_hyper_3f2_etc():
assert hyper([1,2,3],[1.5,8],-1).ae(0.67108992351533333030)
assert hyper([1,2,3,4],[5,6,7], -1).ae(0.90232988035425506008)
assert hyper([1,2,3],[1.25,5], 1).ae(28.924181329701905701)
assert hyper([1,2,3,4],[5,6,7],5).ae(1.5192307344006649499-1.1529845225075537461j)
assert hyper([1,2,3,4,5],[6,7,8,9],-1).ae(0.96288759462882357253)
assert hyper([1,2,3,4,5],[6,7,8,9],1).ae(1.0428697385885855841)
assert hyper([1,2,3,4,5],[6,7,8,9],5).ae(1.33980653631074769423-0.07143405251029226699j)
assert hyper([1,2.79,3.08,4.37],[5.2,6.1,7.3],5).ae(1.0996321464692607231-1.7748052293979985001j)
assert hyper([1,1,1],[1,2],1) == inf
assert hyper([1,1,1],[2,(101,100)],1).ae(100.01621213528313220)
# slow -- covered by doctests
#assert hyper([1,1,1],[2,3],0.9999).ae(1.2897972005319693905)
def test_hyper_u():
mp.dps = 15
assert hyperu(2,-3,0).ae(0.05)
assert hyperu(2,-3.5,0).ae(4./99)
assert hyperu(2,0,0) == 0.5
assert hyperu(-5,1,0) == -120
assert hyperu(-5,2,0) == inf
assert hyperu(-5,-2,0) == 0
assert hyperu(7,7,3).ae(0.00014681269365593503986) #exp(3)*gammainc(-6,3)
assert hyperu(2,-3,4).ae(0.011836478100271995559)
assert hyperu(3,4,5).ae(1./125)
assert hyperu(2,3,0.0625) == 256
assert hyperu(-1,2,0.25+0.5j) == -1.75+0.5j
assert hyperu(0.5,1.5,7.25).ae(2/sqrt(29))
assert hyperu(2,6,pi).ae(0.55804439825913399130)
assert (hyperu((3,2),8,100+201j)*10**4).ae(-0.3797318333856738798 - 2.9974928453561707782j)
assert (hyperu((5,2),(-1,2),-5000)*10**10).ae(-5.6681877926881664678j)
# XXX: fails because of undetected cancellation in low level series code
# Alternatively: could use asymptotic series here, if convergence test
# tweaked back to recognize this one
#assert (hyperu((5,2),(-1,2),-500)*10**7).ae(-1.82526906001593252847j)
def test_hyper_2f0():
mp.dps = 15
assert hyper([1,2],[],3) == hyp2f0(1,2,3)
assert hyp2f0(2,3,7).ae(0.0116108068639728714668 - 0.0073727413865865802130j)
assert hyp2f0(2,3,0) == 1
assert hyp2f0(0,0,0) == 1
assert hyp2f0(-1,-1,1).ae(2)
assert hyp2f0(-4,1,1.5).ae(62.5)
assert hyp2f0(-4,1,50).ae(147029801)
assert hyp2f0(-4,1,0.0001).ae(0.99960011997600240000)
assert hyp2f0(0.5,0.25,0.001).ae(1.0001251174078538115)
assert hyp2f0(0.5,0.25,3+4j).ae(0.85548875824755163518 + 0.21636041283392292973j)
# Important: cancellation check
assert hyp2f0((1,6),(5,6),-0.02371708245126284498).ae(0.996785723120804309)
# Should be exact; polynomial case
assert hyp2f0(-2,1,0.5+0.5j,zeroprec=200) == 0
assert hyp2f0(1,-2,0.5+0.5j,zeroprec=200) == 0
# There used to be a bug in thresholds that made one of the following hang
for d in [15, 50, 80]:
mp.dps = d
assert hyp2f0(1.5, 0.5, 0.009).ae('1.006867007239309717945323585695344927904000945829843527398772456281301440034218290443367270629519483 + 1.238277162240704919639384945859073461954721356062919829456053965502443570466701567100438048602352623e-46j')
def test_hyper_1f2():
mp.dps = 15
assert hyper([1],[2,3],4) == hyp1f2(1,2,3,4)
a1,b1,b2 = (1,10),(2,3),1./16
assert hyp1f2(a1,b1,b2,10).ae(298.7482725554557568)
assert hyp1f2(a1,b1,b2,100).ae(224128961.48602947604)
assert hyp1f2(a1,b1,b2,1000).ae(1.1669528298622675109e+27)
assert hyp1f2(a1,b1,b2,10000).ae(2.4780514622487212192e+86)
assert hyp1f2(a1,b1,b2,100000).ae(1.3885391458871523997e+274)
assert hyp1f2(a1,b1,b2,1000000).ae('9.8851796978960318255e+867')
assert hyp1f2(a1,b1,b2,10**7).ae('1.1505659189516303646e+2746')
assert hyp1f2(a1,b1,b2,10**8).ae('1.4672005404314334081e+8685')
assert hyp1f2(a1,b1,b2,10**20).ae('3.6888217332150976493e+8685889636')
assert hyp1f2(a1,b1,b2,10*j).ae(-16.163252524618572878 - 44.321567896480184312j)
assert hyp1f2(a1,b1,b2,100*j).ae(61938.155294517848171 + 637349.45215942348739j)
assert hyp1f2(a1,b1,b2,1000*j).ae(8455057657257695958.7 + 6261969266997571510.6j)
assert hyp1f2(a1,b1,b2,10000*j).ae(-8.9771211184008593089e+60 + 4.6550528111731631456e+59j)
assert hyp1f2(a1,b1,b2,100000*j).ae(2.6398091437239324225e+193 + 4.1658080666870618332e+193j)
assert hyp1f2(a1,b1,b2,1000000*j).ae('3.5999042951925965458e+613 + 1.5026014707128947992e+613j')
assert hyp1f2(a1,b1,b2,10**7*j).ae('-8.3208715051623234801e+1939 - 3.6752883490851869429e+1941j')
assert hyp1f2(a1,b1,b2,10**8*j).ae('2.0724195707891484454e+6140 - 1.3276619482724266387e+6141j')
assert hyp1f2(a1,b1,b2,10**20*j).ae('-1.1734497974795488504e+6141851462 + 1.1498106965385471542e+6141851462j')
def test_hyper_2f3():
mp.dps = 15
assert hyper([1,2],[3,4,5],6) == hyp2f3(1,2,3,4,5,6)
a1,a2,b1,b2,b3 = (1,10),(2,3),(3,10), 2, 1./16
# Check asymptotic expansion
assert hyp2f3(a1,a2,b1,b2,b3,10).ae(128.98207160698659976)
assert hyp2f3(a1,a2,b1,b2,b3,1000).ae(6.6309632883131273141e25)
assert hyp2f3(a1,a2,b1,b2,b3,10000).ae(4.6863639362713340539e84)
assert hyp2f3(a1,a2,b1,b2,b3,100000).ae(8.6632451236103084119e271)
assert hyp2f3(a1,a2,b1,b2,b3,10**6).ae('2.0291718386574980641e865')
assert hyp2f3(a1,a2,b1,b2,b3,10**7).ae('7.7639836665710030977e2742')
assert hyp2f3(a1,a2,b1,b2,b3,10**8).ae('3.2537462584071268759e8681')
assert hyp2f3(a1,a2,b1,b2,b3,10**20).ae('1.2966030542911614163e+8685889627')
assert hyp2f3(a1,a2,b1,b2,b3,10*j).ae(-18.551602185587547854 - 13.348031097874113552j)
assert hyp2f3(a1,a2,b1,b2,b3,100*j).ae(78634.359124504488695 + 74459.535945281973996j)
assert hyp2f3(a1,a2,b1,b2,b3,1000*j).ae(597682550276527901.59 - 65136194809352613.078j)
assert hyp2f3(a1,a2,b1,b2,b3,10000*j).ae(-1.1779696326238582496e+59 + 1.2297607505213133872e+59j)
assert hyp2f3(a1,a2,b1,b2,b3,100000*j).ae(2.9844228969804380301e+191 + 7.5587163231490273296e+190j)
assert hyp2f3(a1,a2,b1,b2,b3,1000000*j).ae('7.4859161049322370311e+610 - 2.8467477015940090189e+610j')
assert hyp2f3(a1,a2,b1,b2,b3,10**7*j).ae('-1.7477645579418800826e+1938 - 1.7606522995808116405e+1938j')
assert hyp2f3(a1,a2,b1,b2,b3,10**8*j).ae('-1.6932731942958401784e+6137 - 2.4521909113114629368e+6137j')
assert hyp2f3(a1,a2,b1,b2,b3,10**20*j).ae('-2.0988815677627225449e+6141851451 + 5.7708223542739208681e+6141851452j')
def test_hyper_2f2():
mp.dps = 15
assert hyper([1,2],[3,4],5) == hyp2f2(1,2,3,4,5)
a1,a2,b1,b2 = (3,10),4,(1,2),1./16
assert hyp2f2(a1,a2,b1,b2,10).ae(448225936.3377556696)
assert hyp2f2(a1,a2,b1,b2,10000).ae('1.2012553712966636711e+4358')
assert hyp2f2(a1,a2,b1,b2,-20000).ae(-0.04182343755661214626)
assert hyp2f2(a1,a2,b1,b2,10**20).ae('1.1148680024303263661e+43429448190325182840')
def test_orthpoly():
mp.dps = 15
assert jacobi(-4,2,3,0.7).ae(22800./4913)
assert jacobi(3,2,4,5.5) == 4133.125
assert jacobi(1.5,5/6.,4,0).ae(-1.0851951434075508417)
assert jacobi(-2, 1, 2, 4).ae(-0.16)
assert jacobi(2, -1, 2.5, 4).ae(34.59375)
#assert jacobi(2, -1, 2, 4) == 28.5
assert legendre(5, 7) == 129367
assert legendre(0.5,0).ae(0.53935260118837935667)
assert legendre(-1,-1) == 1
assert legendre(0,-1) == 1
assert legendre(0, 1) == 1
assert legendre(1, -1) == -1
assert legendre(7, 1) == 1
assert legendre(7, -1) == -1
assert legendre(8,1.5).ae(15457523./32768)
assert legendre(j,-j).ae(2.4448182735671431011 + 0.6928881737669934843j)
assert chebyu(5,1) == 6
assert chebyt(3,2) == 26
assert legendre(3.5,-1) == inf
assert legendre(4.5,-1) == -inf
assert legendre(3.5+1j,-1) == mpc(inf,inf)
assert legendre(4.5+1j,-1) == mpc(-inf,-inf)
assert laguerre(4, -2, 3).ae(-1.125)
assert laguerre(3, 1+j, 0.5).ae(0.2291666666666666667 + 2.5416666666666666667j)
def test_hermite():
mp.dps = 15
assert hermite(-2, 0).ae(0.5)
assert hermite(-1, 0).ae(0.88622692545275801365)
assert hermite(0, 0).ae(1)
assert hermite(1, 0) == 0
assert hermite(2, 0).ae(-2)
assert hermite(0, 2).ae(1)
assert hermite(1, 2).ae(4)
assert hermite(1, -2).ae(-4)
assert hermite(2, -2).ae(14)
assert hermite(0.5, 0).ae(0.69136733903629335053)
assert hermite(9, 0) == 0
assert hermite(4,4).ae(3340)
assert hermite(3,4).ae(464)
assert hermite(-4,4).ae(0.00018623860287512396181)
assert hermite(-3,4).ae(0.0016540169879668766270)
assert hermite(9, 2.5j).ae(13638725j)
assert hermite(9, -2.5j).ae(-13638725j)
assert hermite(9, 100).ae(511078883759363024000)
assert hermite(9, -100).ae(-511078883759363024000)
assert hermite(9, 100j).ae(512922083920643024000j)
assert hermite(9, -100j).ae(-512922083920643024000j)
assert hermite(-9.5, 2.5j).ae(-2.9004951258126778174e-6 + 1.7601372934039951100e-6j)
assert hermite(-9.5, -2.5j).ae(-2.9004951258126778174e-6 - 1.7601372934039951100e-6j)
assert hermite(-9.5, 100).ae(1.3776300722767084162e-22, abs_eps=0, rel_eps=eps)
assert hermite(-9.5, -100).ae('1.3106082028470671626e4355')
assert hermite(-9.5, 100j).ae(-9.7900218581864768430e-23 - 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps)
assert hermite(-9.5, -100j).ae(-9.7900218581864768430e-23 + 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps)
assert hermite(2+3j, -1-j).ae(851.3677063883687676 - 1496.4373467871007997j)
def test_gegenbauer():
mp.dps = 15
assert gegenbauer(1,2,3).ae(12)
assert gegenbauer(2,3,4).ae(381)
assert gegenbauer(0,0,0) == 0
assert gegenbauer(2,-1,3) == 0
assert gegenbauer(-7, 0.5, 3).ae(8989)
assert gegenbauer(1, -0.5, 3).ae(-3)
assert gegenbauer(1, -1.5, 3).ae(-9)
assert gegenbauer(1, -0.5, 3).ae(-3)
assert gegenbauer(-0.5, -0.5, 3).ae(-2.6383553159023906245)
assert gegenbauer(2+3j, 1-j, 3+4j).ae(14.880536623203696780 + 20.022029711598032898j)
#assert gegenbauer(-2, -0.5, 3).ae(-12)
def test_legenp():
mp.dps = 15
assert legenp(2,0,4) == legendre(2,4)
assert legenp(-2, -1, 0.5).ae(0.43301270189221932338)
assert legenp(-2, -1, 0.5, type=3).ae(0.43301270189221932338j)
assert legenp(-2, 1, 0.5).ae(-0.86602540378443864676)
assert legenp(2+j, 3+4j, -j).ae(134742.98773236786148 + 429782.72924463851745j)
assert legenp(2+j, 3+4j, -j, type=3).ae(802.59463394152268507 - 251.62481308942906447j)
assert legenp(2,4,3).ae(0)
assert legenp(2,4,3,type=3).ae(0)
assert legenp(2,1,0.5).ae(-1.2990381056766579701)
assert legenp(2,1,0.5,type=3).ae(1.2990381056766579701j)
assert legenp(3,2,3).ae(-360)
assert legenp(3,3,3).ae(240j*2**0.5)
assert legenp(3,4,3).ae(0)
assert legenp(0,0.5,2).ae(0.52503756790433198939 - 0.52503756790433198939j)
assert legenp(-1,-0.5,2).ae(0.60626116232846498110 + 0.60626116232846498110j)
assert legenp(-2,0.5,2).ae(1.5751127037129959682 - 1.5751127037129959682j)
assert legenp(-2,0.5,-0.5).ae(-0.85738275810499171286)
def test_legenq():
mp.dps = 15
f = legenq
# Evaluation at poles
assert isnan(f(3,2,1))
assert isnan(f(3,2,-1))
assert isnan(f(3,2,1,type=3))
assert isnan(f(3,2,-1,type=3))
# Evaluation at 0
assert f(0,1,0,type=2).ae(-1)
assert f(-2,2,0,type=2,zeroprec=200).ae(0)
assert f(1.5,3,0,type=2).ae(-2.2239343475841951023)
assert f(0,1,0,type=3).ae(j)
assert f(-2,2,0,type=3,zeroprec=200).ae(0)
assert f(1.5,3,0,type=3).ae(2.2239343475841951022*(1-1j))
# Standard case, degree 0
assert f(0,0,-1.5).ae(-0.8047189562170501873 + 1.5707963267948966192j)
assert f(0,0,-0.5).ae(-0.54930614433405484570)
assert f(0,0,0,zeroprec=200).ae(0)
assert f(0,0,0.5).ae(0.54930614433405484570)
assert f(0,0,1.5).ae(0.8047189562170501873 - 1.5707963267948966192j)
assert f(0,0,-1.5,type=3).ae(-0.80471895621705018730)
assert f(0,0,-0.5,type=3).ae(-0.5493061443340548457 - 1.5707963267948966192j)
assert f(0,0,0,type=3).ae(-1.5707963267948966192j)
assert f(0,0,0.5,type=3).ae(0.5493061443340548457 - 1.5707963267948966192j)
assert f(0,0,1.5,type=3).ae(0.80471895621705018730)
# Standard case, degree 1
assert f(1,0,-1.5).ae(0.2070784343255752810 - 2.3561944901923449288j)
assert f(1,0,-0.5).ae(-0.72534692783297257715)
assert f(1,0,0).ae(-1)
assert f(1,0,0.5).ae(-0.72534692783297257715)
assert f(1,0,1.5).ae(0.2070784343255752810 - 2.3561944901923449288j)
# Standard case, degree 2
assert f(2,0,-1.5).ae(-0.0635669991240192885 + 4.5160394395353277803j)
assert f(2,0,-0.5).ae(0.81866326804175685571)
assert f(2,0,0,zeroprec=200).ae(0)
assert f(2,0,0.5).ae(-0.81866326804175685571)
assert f(2,0,1.5).ae(0.0635669991240192885 - 4.5160394395353277803j)
# Misc orders and degrees
assert f(2,3,1.5,type=2).ae(-5.7243340223994616228j)
assert f(2,3,1.5,type=3).ae(-5.7243340223994616228)
assert f(2,3,0.5,type=2).ae(-12.316805742712016310)
assert f(2,3,0.5,type=3).ae(-12.316805742712016310j)
assert f(2,3,-1.5,type=2).ae(-5.7243340223994616228j)
assert f(2,3,-1.5,type=3).ae(5.7243340223994616228)
assert f(2,3,-0.5,type=2).ae(-12.316805742712016310)
assert f(2,3,-0.5,type=3).ae(-12.316805742712016310j)
assert f(2+3j, 3+4j, 0.5, type=3).ae(0.0016119404873235186807 - 0.0005885900510718119836j)
assert f(2+3j, 3+4j, -1.5, type=3).ae(0.008451400254138808670 + 0.020645193304593235298j)
assert f(-2.5,1,-1.5).ae(3.9553395527435335749j)
assert f(-2.5,1,-0.5).ae(1.9290561746445456908)
assert f(-2.5,1,0).ae(1.2708196271909686299)
assert f(-2.5,1,0.5).ae(-0.31584812990742202869)
assert f(-2.5,1,1.5).ae(-3.9553395527435335742 + 0.2993235655044701706j)
assert f(-2.5,1,-1.5,type=3).ae(0.29932356550447017254j)
assert f(-2.5,1,-0.5,type=3).ae(-0.3158481299074220287 - 1.9290561746445456908j)
assert f(-2.5,1,0,type=3).ae(1.2708196271909686292 - 1.2708196271909686299j)
assert f(-2.5,1,0.5,type=3).ae(1.9290561746445456907 + 0.3158481299074220287j)
assert f(-2.5,1,1.5,type=3).ae(-0.29932356550447017254)
def test_agm():
mp.dps = 15
assert agm(0,0) == 0
assert agm(0,1) == 0
assert agm(1,1) == 1
assert agm(7,7) == 7
assert agm(j,j) == j
assert (1/agm(1,sqrt(2))).ae(0.834626841674073186)
assert agm(1,2).ae(1.4567910310469068692)
assert agm(1,3).ae(1.8636167832448965424)
assert agm(1,j).ae(0.599070117367796104+0.599070117367796104j)
assert agm(2) == agm(1,2)
assert agm(-3,4).ae(0.63468509766550907+1.3443087080896272j)
def test_gammainc():
mp.dps = 15
assert gammainc(2,5).ae(6*exp(-5))
assert gammainc(2,0,5).ae(1-6*exp(-5))
assert gammainc(2,3,5).ae(-6*exp(-5)+4*exp(-3))
assert gammainc(-2.5,-0.5).ae(-0.9453087204829418812-5.3164237738936178621j)
assert gammainc(0,2,4).ae(0.045121158298212213088)
assert gammainc(0,3).ae(0.013048381094197037413)
assert gammainc(0,2+j,1-j).ae(0.00910653685850304839-0.22378752918074432574j)
assert gammainc(0,1-j).ae(0.00028162445198141833+0.17932453503935894015j)
assert gammainc(3,4,5,True).ae(0.11345128607046320253)
assert gammainc(3.5,0,inf).ae(gamma(3.5))
assert gammainc(-150.5,500).ae('6.9825435345798951153e-627')
assert gammainc(-150.5,800).ae('4.6885137549474089431e-788')
assert gammainc(-3.5, -20.5).ae(0.27008820585226911 - 1310.31447140574997636j)
assert gammainc(-3.5, -200.5).ae(0.27008820585226911 - 5.3264597096208368435e76j) # XXX real part
assert gammainc(0,0,2) == inf
assert gammainc(1,b=1).ae(0.6321205588285576784)
assert gammainc(3,2,2) == 0
assert gammainc(2,3+j,3-j).ae(-0.28135485191849314194j)
assert gammainc(4+0j,1).ae(5.8860710587430771455)
# GH issue #301
assert gammainc(-1,-1).ae(-0.8231640121031084799 + 3.1415926535897932385j)
assert gammainc(-2,-1).ae(1.7707229202810768576 - 1.5707963267948966192j)
assert gammainc(-3,-1).ae(-1.4963349162467073643 + 0.5235987755982988731j)
assert gammainc(-4,-1).ae(1.05365418617643814992 - 0.13089969389957471827j)
# Regularized upper gamma
assert isnan(gammainc(0, 0, regularized=True))
assert gammainc(-1, 0, regularized=True) == inf
assert gammainc(1, 0, regularized=True) == 1
assert gammainc(0, 5, regularized=True) == 0
assert gammainc(0, 2+3j, regularized=True) == 0
assert gammainc(0, 5000, regularized=True) == 0
assert gammainc(0, 10**30, regularized=True) == 0
assert gammainc(-1, 5, regularized=True) == 0
assert gammainc(-1, 5000, regularized=True) == 0
assert gammainc(-1, 10**30, regularized=True) == 0
assert gammainc(-1, -5, regularized=True) == 0
assert gammainc(-1, -5000, regularized=True) == 0
assert gammainc(-1, -10**30, regularized=True) == 0
assert gammainc(-1, 3+4j, regularized=True) == 0
assert gammainc(1, 5, regularized=True).ae(exp(-5))
assert gammainc(1, 5000, regularized=True).ae(exp(-5000))
assert gammainc(1, 10**30, regularized=True).ae(exp(-10**30))
assert gammainc(1, 3+4j, regularized=True).ae(exp(-3-4j))
assert gammainc(-1000000,2).ae('1.3669297209397347754e-301037', abs_eps=0, rel_eps=8*eps)
assert gammainc(-1000000,2,regularized=True) == 0
assert gammainc(-1000000,3+4j).ae('-1.322575609404222361e-698979 - 4.9274570591854533273e-698978j', abs_eps=0, rel_eps=8*eps)
assert gammainc(-1000000,3+4j,regularized=True) == 0
assert gammainc(2+3j, 4+5j, regularized=True).ae(0.085422013530993285774-0.052595379150390078503j)
assert gammainc(1000j, 1000j, regularized=True).ae(0.49702647628921131761 + 0.00297355675013575341j)
# Generalized
assert gammainc(3,4,2) == -gammainc(3,2,4)
assert gammainc(4, 2, 3).ae(1.2593494302978947396)
assert gammainc(4, 2, 3, regularized=True).ae(0.20989157171631578993)
assert gammainc(0, 2, 3).ae(0.035852129613864082155)
assert gammainc(0, 2, 3, regularized=True) == 0
assert gammainc(-1, 2, 3).ae(0.015219822548487616132)
assert gammainc(-1, 2, 3, regularized=True) == 0
assert gammainc(0, 2, 3).ae(0.035852129613864082155)
assert gammainc(0, 2, 3, regularized=True) == 0
# Should use upper gammas
assert gammainc(5, 10000, 12000).ae('1.1359381951461801687e-4327', abs_eps=0, rel_eps=8*eps)
# Should use lower gammas
assert gammainc(10000, 2, 3).ae('8.1244514125995785934e4765')
# GH issue 306
assert gammainc(3,-1-1j) == 0
assert gammainc(3,-1+1j) == 0
assert gammainc(2,-1) == 0
assert gammainc(2,-1+0j) == 0
assert gammainc(2+0j,-1) == 0
def test_gammainc_expint_n():
# These tests are intended to check all cases of the low-level code
# for upper gamma and expint with small integer index.
# Need to cover positive/negative arguments; small/large/huge arguments
# for both positive and negative indices, as well as indices 0 and 1
# which may be special-cased
mp.dps = 15
assert expint(-3,3.5).ae(0.021456366563296693987)
assert expint(-2,3.5).ae(0.014966633183073309405)
assert expint(-1,3.5).ae(0.011092916359219041088)
assert expint(0,3.5).ae(0.0086278238349481430685)
assert expint(1,3.5).ae(0.0069701398575483929193)
assert expint(2,3.5).ae(0.0058018939208991255223)
assert expint(3,3.5).ae(0.0049453773495857807058)
assert expint(-3,-3.5).ae(-4.6618170604073311319)
assert expint(-2,-3.5).ae(-5.5996974157555515963)
assert expint(-1,-3.5).ae(-6.7582555017739415818)
assert expint(0,-3.5).ae(-9.4615577024835182145)
assert expint(1,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j)
assert expint(2,-3.5).ae(-15.62328702434085977 - 10.995574287564276335j)
assert expint(3,-3.5).ae(-10.783026313250347722 - 19.242255003237483586j)
assert expint(-3,350).ae(2.8614825451252838069e-155, abs_eps=0, rel_eps=8*eps)
assert expint(-2,350).ae(2.8532837224504675901e-155, abs_eps=0, rel_eps=8*eps)
assert expint(-1,350).ae(2.8451316155828634555e-155, abs_eps=0, rel_eps=8*eps)
assert expint(0,350).ae(2.8370258275042797989e-155, abs_eps=0, rel_eps=8*eps)
assert expint(1,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps)
assert expint(2,350).ae(2.8209516419468505006e-155, abs_eps=0, rel_eps=8*eps)
assert expint(3,350).ae(2.8129824725501272171e-155, abs_eps=0, rel_eps=8*eps)
assert expint(-3,-350).ae(-2.8528796154044839443e+149)
assert expint(-2,-350).ae(-2.8610072121701264351e+149)
assert expint(-1,-350).ae(-2.8691813842677537647e+149)
assert expint(0,-350).ae(-2.8774025343659421709e+149)
u = expint(1,-350)
assert u.ae(-2.8856710698020863568e+149)
assert u.imag.ae(-3.1415926535897932385)
u = expint(2,-350)
assert u.ae(-2.8939874026504650534e+149)
assert u.imag.ae(-1099.5574287564276335)
u = expint(3,-350)
assert u.ae(-2.9023519497915044349e+149)
assert u.imag.ae(-192422.55003237483586)
assert expint(-3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(-2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(-1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert expint(-3,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871')
assert expint(-2,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871')
assert expint(-1,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871')
assert expint(0,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871')
u = expint(1,-350000000000000000000000)
assert u.ae('-3.7805306852415755699e+152003068666138139677871')
assert u.imag.ae(-3.1415926535897932385)
u = expint(2,-350000000000000000000000)
assert u.imag.ae(-1.0995574287564276335e+24)
assert u.ae('-3.7805306852415755699e+152003068666138139677871')
u = expint(3,-350000000000000000000000)
assert u.imag.ae(-1.9242255003237483586e+47)
assert u.ae('-3.7805306852415755699e+152003068666138139677871')
# Small case; no branch cut
assert gammainc(-3,3.5).ae(0.00010020262545203707109)
assert gammainc(-2,3.5).ae(0.00040370427343557393517)
assert gammainc(-1,3.5).ae(0.0016576839773997501492)
assert gammainc(0,3.5).ae(0.0069701398575483929193)
assert gammainc(1,3.5).ae(0.03019738342231850074)
assert gammainc(2,3.5).ae(0.13588822540043325333)
assert gammainc(3,3.5).ae(0.64169439772426814072)
# Small case; with branch cut
assert gammainc(-3,-3.5).ae(0.03595832954467563286 + 0.52359877559829887308j)
assert gammainc(-2,-3.5).ae(-0.88024704597962022221 - 1.5707963267948966192j)
assert gammainc(-1,-3.5).ae(4.4637962926688170771 + 3.1415926535897932385j)
assert gammainc(0,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j)
assert gammainc(1,-3.5).ae(33.115451958692313751)
assert gammainc(2,-3.5).ae(-82.788629896730784377)
assert gammainc(3,-3.5).ae(240.08702670051927469)
# Asymptotic case; no branch cut
assert gammainc(-3,350).ae(6.5424095113340358813e-163, abs_eps=0, rel_eps=8*eps)
assert gammainc(-2,350).ae(2.296312222489899769e-160, abs_eps=0, rel_eps=8*eps)
assert gammainc(-1,350).ae(8.059861834133858573e-158, abs_eps=0, rel_eps=8*eps)
assert gammainc(0,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps)
assert gammainc(1,350).ae(9.9295903962649792963e-153, abs_eps=0, rel_eps=8*eps)
assert gammainc(2,350).ae(3.485286229089007733e-150, abs_eps=0, rel_eps=8*eps)
assert gammainc(3,350).ae(1.2233453960006379793e-147, abs_eps=0, rel_eps=8*eps)
# Asymptotic case; branch cut
u = gammainc(-3,-350)
assert u.ae(6.7889565783842895085e+141)
assert u.imag.ae(0.52359877559829887308)
u = gammainc(-2,-350)
assert u.ae(-2.3692668977889832121e+144)
assert u.imag.ae(-1.5707963267948966192)
u = gammainc(-1,-350)
assert u.ae(8.2685354361441858669e+146)
assert u.imag.ae(3.1415926535897932385)
u = gammainc(0,-350)
assert u.ae(-2.8856710698020863568e+149)
assert u.imag.ae(-3.1415926535897932385)
u = gammainc(1,-350)
assert u.ae(1.0070908870280797598e+152)
assert u.imag == 0
u = gammainc(2,-350)
assert u.ae(-3.5147471957279983618e+154)
assert u.imag == 0
u = gammainc(3,-350)
assert u.ae(1.2266568422179417091e+157)
assert u.imag == 0
# Extreme asymptotic case
assert gammainc(-3,350000000000000000000000).ae('5.0362468738874738859e-152003068666138139677990', abs_eps=0, rel_eps=8*eps)
assert gammainc(-2,350000000000000000000000).ae('1.7626864058606158601e-152003068666138139677966', abs_eps=0, rel_eps=8*eps)
assert gammainc(-1,350000000000000000000000).ae('6.1694024205121555102e-152003068666138139677943', abs_eps=0, rel_eps=8*eps)
assert gammainc(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps)
assert gammainc(1,350000000000000000000000).ae('7.5575179651273905e-152003068666138139677896', abs_eps=0, rel_eps=8*eps)
assert gammainc(2,350000000000000000000000).ae('2.645131287794586675e-152003068666138139677872', abs_eps=0, rel_eps=8*eps)
assert gammainc(3,350000000000000000000000).ae('9.2579595072810533625e-152003068666138139677849', abs_eps=0, rel_eps=8*eps)
u = gammainc(-3,-350000000000000000000000)
assert u.ae('8.8175642804468234866e+152003068666138139677800')
assert u.imag.ae(0.52359877559829887308)
u = gammainc(-2,-350000000000000000000000)
assert u.ae('-3.0861474981563882203e+152003068666138139677824')
assert u.imag.ae(-1.5707963267948966192)
u = gammainc(-1,-350000000000000000000000)
assert u.ae('1.0801516243547358771e+152003068666138139677848')
assert u.imag.ae(3.1415926535897932385)
u = gammainc(0,-350000000000000000000000)
assert u.ae('-3.7805306852415755699e+152003068666138139677871')
assert u.imag.ae(-3.1415926535897932385)
assert gammainc(1,-350000000000000000000000).ae('1.3231857398345514495e+152003068666138139677895')
assert gammainc(2,-350000000000000000000000).ae('-4.6311500894209300731e+152003068666138139677918')
assert gammainc(3,-350000000000000000000000).ae('1.6209025312973255256e+152003068666138139677942')
def test_incomplete_beta():
mp.dps = 15
assert betainc(-2,-3,0.5,0.75).ae(63.4305673311255413583969)
assert betainc(4.5,0.5+2j,2.5,6).ae(0.2628801146130621387903065 + 0.5162565234467020592855378j)
assert betainc(4,5,0,6).ae(90747.77142857142857142857)
def test_erf():
mp.dps = 15
assert erf(0) == 0
assert erf(1).ae(0.84270079294971486934)
assert erf(3+4j).ae(-120.186991395079444098 - 27.750337293623902498j)
assert erf(-4-3j).ae(-0.99991066178539168236 + 0.00004972026054496604j)
assert erf(pi).ae(0.99999112385363235839)
assert erf(1j).ae(1.6504257587975428760j)
assert erf(-1j).ae(-1.6504257587975428760j)
assert isinstance(erf(1), mpf)
assert isinstance(erf(-1), mpf)
assert isinstance(erf(0), mpf)
assert isinstance(erf(0j), mpc)
assert erf(inf) == 1
assert erf(-inf) == -1
assert erfi(0) == 0
assert erfi(1/pi).ae(0.371682698493894314)
assert erfi(inf) == inf
assert erfi(-inf) == -inf
assert erf(1+0j) == erf(1)
assert erfc(1+0j) == erfc(1)
assert erf(0.2+0.5j).ae(1 - erfc(0.2+0.5j))
assert erfc(0) == 1
assert erfc(1).ae(1-erf(1))
assert erfc(-1).ae(1-erf(-1))
assert erfc(1/pi).ae(1-erf(1/pi))
assert erfc(-10) == 2
assert erfc(-1000000) == 2
assert erfc(-inf) == 2
assert erfc(inf) == 0
assert isnan(erfc(nan))
assert (erfc(10**4)*mpf(10)**43429453).ae('3.63998738656420')
assert erf(8+9j).ae(-1072004.2525062051158 + 364149.91954310255423j)
assert erfc(8+9j).ae(1072005.2525062051158 - 364149.91954310255423j)
assert erfc(-8-9j).ae(-1072003.2525062051158 + 364149.91954310255423j)
mp.dps = 50
# This one does not use the asymptotic series
assert (erfc(10)*10**45).ae('2.0884875837625447570007862949577886115608181193212')
# This one does
assert (erfc(50)*10**1088).ae('2.0709207788416560484484478751657887929322509209954')
mp.dps = 15
assert str(erfc(10**50)) == '3.66744826532555e-4342944819032518276511289189166050822943970058036665661144537831658646492088707747292249493384317534'
assert erfinv(0) == 0
assert erfinv(0.5).ae(0.47693627620446987338)
assert erfinv(-0.5).ae(-0.47693627620446987338)
assert erfinv(1) == inf
assert erfinv(-1) == -inf
assert erf(erfinv(0.95)).ae(0.95)
assert erf(erfinv(0.999999999995)).ae(0.999999999995)
assert erf(erfinv(-0.999999999995)).ae(-0.999999999995)
mp.dps = 50
assert erf(erfinv('0.99999999999999999999999999999995')).ae('0.99999999999999999999999999999995')
assert erf(erfinv('0.999999999999999999999999999999995')).ae('0.999999999999999999999999999999995')
assert erf(erfinv('-0.999999999999999999999999999999995')).ae('-0.999999999999999999999999999999995')
mp.dps = 15
# Complex asymptotic expansions
v = erfc(50j)
assert v.real == 1
assert v.imag.ae('-6.1481820666053078736e+1083')
assert erfc(-100+5j).ae(2)
assert (erfc(100+5j)*10**4335).ae(2.3973567853824133572 - 3.9339259530609420597j)
assert erfc(100+100j).ae(0.00065234366376857698698 - 0.0039357263629214118437j)
def test_pdf():
mp.dps = 15
assert npdf(-inf) == 0
assert npdf(inf) == 0
assert npdf(5,0,2).ae(npdf(5+4,4,2))
assert quadts(lambda x: npdf(x,-0.5,0.8), [-inf, inf]) == 1
assert ncdf(0) == 0.5
assert ncdf(3,3) == 0.5
assert ncdf(-inf) == 0
assert ncdf(inf) == 1
assert ncdf(10) == 1
# Verify that this is computed accurately
assert (ncdf(-10)*10**24).ae(7.619853024160526)
def test_lambertw():
mp.dps = 15
assert lambertw(0) == 0
assert lambertw(0+0j) == 0
assert lambertw(inf) == inf
assert isnan(lambertw(nan))
assert lambertw(inf,1).real == inf
assert lambertw(inf,1).imag.ae(2*pi)
assert lambertw(-inf,1).real == inf
assert lambertw(-inf,1).imag.ae(3*pi)
assert lambertw(0,-1) == -inf
assert lambertw(0,1) == -inf
assert lambertw(0,3) == -inf
assert lambertw(e).ae(1)
assert lambertw(1).ae(0.567143290409783873)
assert lambertw(-pi/2).ae(j*pi/2)
assert lambertw(-log(2)/2).ae(-log(2))
assert lambertw(0.25).ae(0.203888354702240164)
assert lambertw(-0.25).ae(-0.357402956181388903)
assert lambertw(-1./10000,0).ae(-0.000100010001500266719)
assert lambertw(-0.25,-1).ae(-2.15329236411034965)
assert lambertw(0.25,-1).ae(-3.00899800997004620-4.07652978899159763j)
assert lambertw(-0.25,-1).ae(-2.15329236411034965)
assert lambertw(0.25,1).ae(-3.00899800997004620+4.07652978899159763j)
assert lambertw(-0.25,1).ae(-3.48973228422959210+7.41405453009603664j)
assert lambertw(-4).ae(0.67881197132094523+1.91195078174339937j)
assert lambertw(-4,1).ae(-0.66743107129800988+7.76827456802783084j)
assert lambertw(-4,-1).ae(0.67881197132094523-1.91195078174339937j)
assert lambertw(1000).ae(5.24960285240159623)
assert lambertw(1000,1).ae(4.91492239981054535+5.44652615979447070j)
assert lambertw(1000,-1).ae(4.91492239981054535-5.44652615979447070j)
assert lambertw(1000,5).ae(3.5010625305312892+29.9614548941181328j)
assert lambertw(3+4j).ae(1.281561806123775878+0.533095222020971071j)
assert lambertw(-0.4+0.4j).ae(-0.10396515323290657+0.61899273315171632j)
assert lambertw(3+4j,1).ae(-0.11691092896595324+5.61888039871282334j)
assert lambertw(3+4j,-1).ae(0.25856740686699742-3.85211668616143559j)
assert lambertw(-0.5,-1).ae(-0.794023632344689368-0.770111750510379110j)
assert lambertw(-1./10000,1).ae(-11.82350837248724344+6.80546081842002101j)
assert lambertw(-1./10000,-1).ae(-11.6671145325663544)
assert lambertw(-1./10000,-2).ae(-11.82350837248724344-6.80546081842002101j)
assert lambertw(-1./100000,4).ae(-14.9186890769540539+26.1856750178782046j)
assert lambertw(-1./100000,5).ae(-15.0931437726379218666+32.5525721210262290086j)
assert lambertw((2+j)/10).ae(0.173704503762911669+0.071781336752835511j)
assert lambertw((2+j)/10,1).ae(-3.21746028349820063+4.56175438896292539j)
assert lambertw((2+j)/10,-1).ae(-3.03781405002993088-3.53946629633505737j)
assert lambertw((2+j)/10,4).ae(-4.6878509692773249+23.8313630697683291j)
assert lambertw(-(2+j)/10).ae(-0.226933772515757933-0.164986470020154580j)
assert lambertw(-(2+j)/10,1).ae(-2.43569517046110001+0.76974067544756289j)
assert lambertw(-(2+j)/10,-1).ae(-3.54858738151989450-6.91627921869943589j)
assert lambertw(-(2+j)/10,4).ae(-4.5500846928118151+20.6672982215434637j)
mp.dps = 50
assert lambertw(pi).ae('1.073658194796149172092178407024821347547745350410314531')
mp.dps = 15
# Former bug in generated branch
assert lambertw(-0.5+0.002j).ae(-0.78917138132659918344 + 0.76743539379990327749j)
assert lambertw(-0.5-0.002j).ae(-0.78917138132659918344 - 0.76743539379990327749j)
assert lambertw(-0.448+0.4j).ae(-0.11855133765652382241 + 0.66570534313583423116j)
assert lambertw(-0.448-0.4j).ae(-0.11855133765652382241 - 0.66570534313583423116j)
assert lambertw(-0.65475+0.0001j).ae(-0.61053421111385310898+1.0396534993944097723803j)
# Huge branch index
w = lambertw(1,10**20)
assert w.real.ae(-47.889578926290259164)
assert w.imag.ae(6.2831853071795864769e+20)
def test_lambertw_hard():
def check(x,y):
y = convert(y)
type_ok = True
if isinstance(y, mpf):
type_ok = isinstance(x, mpf)
real_ok = abs(x.real-y.real) <= abs(y.real)*8*eps
imag_ok = abs(x.imag-y.imag) <= abs(y.imag)*8*eps
#print x, y, abs(x.real-y.real), abs(x.imag-y.imag)
return real_ok and imag_ok
# Evaluation near 0
mp.dps = 15
assert check(lambertw(1e-10), 9.999999999000000000e-11)
assert check(lambertw(-1e-10), -1.000000000100000000e-10)
assert check(lambertw(1e-10j), 9.999999999999999999733e-21 + 9.99999999999999999985e-11j)
assert check(lambertw(-1e-10j), 9.999999999999999999733e-21 - 9.99999999999999999985e-11j)
assert check(lambertw(1e-10,1), -26.303186778379041559 + 3.265093911703828397j)
assert check(lambertw(-1e-10,1), -26.326236166739163892 + 6.526183280686333315j)
assert check(lambertw(1e-10j,1), -26.312931726911421551 + 4.896366881798013421j)
assert check(lambertw(-1e-10j,1), -26.297238779529035066 + 1.632807161345576513j)
assert check(lambertw(1e-10,-1), -26.303186778379041559 - 3.265093911703828397j)
assert check(lambertw(-1e-10,-1), -26.295238819246925694)
assert check(lambertw(1e-10j,-1), -26.297238779529035028 - 1.6328071613455765135j)
assert check(lambertw(-1e-10j,-1), -26.312931726911421551 - 4.896366881798013421j)
# Test evaluation very close to the branch point -1/e
# on the -1, 0, and 1 branches
add = lambda x, y: fadd(x,y,exact=True)
sub = lambda x, y: fsub(x,y,exact=True)
addj = lambda x, y: fadd(x,fmul(y,1j,exact=True),exact=True)
subj = lambda x, y: fadd(x,fmul(y,-1j,exact=True),exact=True)
mp.dps = 1500
a = -1/e + 10*eps
d3 = mpf('1e-3')
d10 = mpf('1e-10')
d20 = mpf('1e-20')
d40 = mpf('1e-40')
d80 = mpf('1e-80')
d300 = mpf('1e-300')
d1000 = mpf('1e-1000')
mp.dps = 15
# ---- Branch 0 ----
# -1/e + eps
assert check(lambertw(add(a,d3)), -0.92802015005456704876)
assert check(lambertw(add(a,d10)), -0.99997668374140088071)
assert check(lambertw(add(a,d20)), -0.99999999976683560186)
assert lambertw(add(a,d40)) == -1
assert lambertw(add(a,d80)) == -1
assert lambertw(add(a,d300)) == -1
assert lambertw(add(a,d1000)) == -1
# -1/e - eps
assert check(lambertw(sub(a,d3)), -0.99819016149860989001+0.07367191188934638577j)
assert check(lambertw(sub(a,d10)), -0.9999999998187812114595992+0.0000233164398140346109194j)
assert check(lambertw(sub(a,d20)), -0.99999999999999999998187+2.331643981597124203344e-10j)
assert check(lambertw(sub(a,d40)), -1.0+2.33164398159712420336e-20j)
assert check(lambertw(sub(a,d80)), -1.0+2.33164398159712420336e-40j)
assert check(lambertw(sub(a,d300)), -1.0+2.33164398159712420336e-150j)
assert check(lambertw(sub(a,d1000)), mpc(-1,'2.33164398159712420336e-500'))
# -1/e + eps*j
assert check(lambertw(addj(a,d3)), -0.94790387486938526634+0.05036819639190132490j)
assert check(lambertw(addj(a,d10)), -0.9999835127872943680999899+0.0000164870314895821225256j)
assert check(lambertw(addj(a,d20)), -0.999999999835127872929987+1.64872127051890935830e-10j)
assert check(lambertw(addj(a,d40)), -0.9999999999999999999835+1.6487212707001281468305e-20j)
assert check(lambertw(addj(a,d80)), -1.0 + 1.64872127070012814684865e-40j)
assert check(lambertw(addj(a,d300)), -1.0 + 1.64872127070012814684865e-150j)
assert check(lambertw(addj(a,d1000)), mpc(-1.0,'1.64872127070012814684865e-500'))
# -1/e - eps*j
assert check(lambertw(subj(a,d3)), -0.94790387486938526634-0.05036819639190132490j)
assert check(lambertw(subj(a,d10)), -0.9999835127872943680999899-0.0000164870314895821225256j)
assert check(lambertw(subj(a,d20)), -0.999999999835127872929987-1.64872127051890935830e-10j)
assert check(lambertw(subj(a,d40)), -0.9999999999999999999835-1.6487212707001281468305e-20j)
assert check(lambertw(subj(a,d80)), -1.0 - 1.64872127070012814684865e-40j)
assert check(lambertw(subj(a,d300)), -1.0 - 1.64872127070012814684865e-150j)
assert check(lambertw(subj(a,d1000)), mpc(-1.0,'-1.64872127070012814684865e-500'))
# ---- Branch 1 ----
assert check(lambertw(addj(a,d3),1), -3.088501303219933378005990 + 7.458676867597474813950098j)
assert check(lambertw(addj(a,d80),1), -3.088843015613043855957087 + 7.461489285654254556906117j)
assert check(lambertw(addj(a,d300),1), -3.088843015613043855957087 + 7.461489285654254556906117j)
assert check(lambertw(addj(a,d1000),1), -3.088843015613043855957087 + 7.461489285654254556906117j)
assert check(lambertw(subj(a,d3),1), -1.0520914180450129534365906 + 0.0539925638125450525673175j)
assert check(lambertw(subj(a,d10),1), -1.0000164872127056318529390 + 0.000016487393927159250398333077j)
assert check(lambertw(subj(a,d20),1), -1.0000000001648721270700128 + 1.64872127088134693542628e-10j)
assert check(lambertw(subj(a,d40),1), -1.000000000000000000016487 + 1.64872127070012814686677e-20j)
assert check(lambertw(subj(a,d80),1), -1.0 + 1.64872127070012814684865e-40j)
assert check(lambertw(subj(a,d300),1), -1.0 + 1.64872127070012814684865e-150j)
assert check(lambertw(subj(a,d1000),1), mpc(-1.0, '1.64872127070012814684865e-500'))
# ---- Branch -1 ----
# -1/e + eps
assert check(lambertw(add(a,d3),-1), -1.075608941186624989414945)
assert check(lambertw(add(a,d10),-1), -1.000023316621036696460620)
assert check(lambertw(add(a,d20),-1), -1.000000000233164398177834)
assert lambertw(add(a,d40),-1) == -1
assert lambertw(add(a,d80),-1) == -1
assert lambertw(add(a,d300),-1) == -1
assert lambertw(add(a,d1000),-1) == -1
# -1/e - eps
assert check(lambertw(sub(a,d3),-1), -0.99819016149860989001-0.07367191188934638577j)
assert check(lambertw(sub(a,d10),-1), -0.9999999998187812114595992-0.0000233164398140346109194j)
assert check(lambertw(sub(a,d20),-1), -0.99999999999999999998187-2.331643981597124203344e-10j)
assert check(lambertw(sub(a,d40),-1), -1.0-2.33164398159712420336e-20j)
assert check(lambertw(sub(a,d80),-1), -1.0-2.33164398159712420336e-40j)
assert check(lambertw(sub(a,d300),-1), -1.0-2.33164398159712420336e-150j)
assert check(lambertw(sub(a,d1000),-1), mpc(-1,'-2.33164398159712420336e-500'))
# -1/e + eps*j
assert check(lambertw(addj(a,d3),-1), -1.0520914180450129534365906 - 0.0539925638125450525673175j)
assert check(lambertw(addj(a,d10),-1), -1.0000164872127056318529390 - 0.0000164873939271592503983j)
assert check(lambertw(addj(a,d20),-1), -1.0000000001648721270700 - 1.64872127088134693542628e-10j)
assert check(lambertw(addj(a,d40),-1), -1.00000000000000000001648 - 1.6487212707001281468667726e-20j)
assert check(lambertw(addj(a,d80),-1), -1.0 - 1.64872127070012814684865e-40j)
assert check(lambertw(addj(a,d300),-1), -1.0 - 1.64872127070012814684865e-150j)
assert check(lambertw(addj(a,d1000),-1), mpc(-1.0,'-1.64872127070012814684865e-500'))
# -1/e - eps*j
assert check(lambertw(subj(a,d3),-1), -3.088501303219933378005990-7.458676867597474813950098j)
assert check(lambertw(subj(a,d10),-1), -3.088843015579260686911033-7.461489285372968780020716j)
assert check(lambertw(subj(a,d20),-1), -3.088843015613043855953708-7.461489285654254556877988j)
assert check(lambertw(subj(a,d40),-1), -3.088843015613043855957087-7.461489285654254556906117j)
assert check(lambertw(subj(a,d80),-1), -3.088843015613043855957087 - 7.461489285654254556906117j)
assert check(lambertw(subj(a,d300),-1), -3.088843015613043855957087 - 7.461489285654254556906117j)
assert check(lambertw(subj(a,d1000),-1), -3.088843015613043855957087 - 7.461489285654254556906117j)
# One more case, testing higher precision
mp.dps = 500
x = -1/e + mpf('1e-13')
ans = "-0.99999926266961377166355784455394913638782494543377383"\
"744978844374498153493943725364881490261187530235150668593869563"\
"168276697689459394902153960200361935311512317183678882"
mp.dps = 15
assert lambertw(x).ae(ans)
mp.dps = 50
assert lambertw(x).ae(ans)
mp.dps = 150
assert lambertw(x).ae(ans)
def test_meijerg():
mp.dps = 15
assert meijerg([[2,3],[1]],[[0.5,2],[3,4]], 2.5).ae(4.2181028074787439386)
assert meijerg([[],[1+j]],[[1],[1]], 3+4j).ae(271.46290321152464592 - 703.03330399954820169j)
assert meijerg([[0.25],[1]],[[0.5],[2]],0) == 0
assert meijerg([[0],[]],[[0,0,'1/3','2/3'], []], '2/27').ae(2.2019391389653314120)
# Verify 1/z series being used
assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -0.5).ae(-1.338096165935754898687431)
assert meijerg([[1-(-1)],[1-(-2.5)]], [[1-(-3)],[1-(-0.5)]], -2.0).ae(-1.338096165935754898687431)
assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -1).ae(-(pi+4)/(4*pi))
a = 2.5
b = 1.25
for z in [mpf(0.25), mpf(2)]:
x1 = hyp1f1(a,b,z)
x2 = gamma(b)/gamma(a)*meijerg([[1-a],[]],[[0],[1-b]],-z)
x3 = gamma(b)/gamma(a)*meijerg([[1-0],[1-(1-b)]],[[1-(1-a)],[]],-1/z)
assert x1.ae(x2)
assert x1.ae(x3)
def test_appellf1():
mp.dps = 15
assert appellf1(2,-2,1,1,2,3).ae(-1.75)
assert appellf1(2,1,-2,1,2,3).ae(-8)
assert appellf1(2,1,-2,1,0.5,0.25).ae(1.5)
assert appellf1(-2,1,3,2,3,3).ae(19)
assert appellf1(1,2,3,4,0.5,0.125).ae( 1.53843285792549786518)
def test_coulomb():
# Note: most tests are doctests
# Test for a bug:
mp.dps = 15
assert coulombg(mpc(-5,0),2,3).ae(20.087729487721430394)
def test_hyper_param_accuracy():
mp.dps = 15
As = [n+1e-10 for n in range(-5,-1)]
Bs = [n+1e-10 for n in range(-12,-5)]
assert hyper(As,Bs,10).ae(-381757055858.652671927)
assert legenp(0.5, 100, 0.25).ae(-2.4124576567211311755e+144)
assert (hyp1f1(1000,1,-100)*10**24).ae(5.2589445437370169113)
assert (hyp2f1(10, -900, 10.5, 0.99)*10**24).ae(1.9185370579660768203)
assert (hyp2f1(1000,1.5,-3.5,-1.5)*10**385).ae(-2.7367529051334000764)
assert hyp2f1(-5, 10, 3, 0.5, zeroprec=500) == 0
assert (hyp1f1(-10000, 1000, 100)*10**424).ae(-3.1046080515824859974)
assert (hyp2f1(1000,1.5,-3.5,-0.75,maxterms=100000)*10**231).ae(-4.0534790813913998643)
assert legenp(2, 3, 0.25) == 0
pytest.raises(ValueError, lambda: hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3]))
assert hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3], infprec=200) == inf
assert meijerg([[],[]],[[0,0,0,0],[]],0.1).ae(1.5680822343832351418)
assert (besselk(400,400)*10**94).ae(1.4387057277018550583)
mp.dps = 5
(hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522)
(hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311)
mp.dps = 15
(hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522)
(hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311)
assert hyp0f1(fadd(-20,'1e-100',exact=True), 0.25).ae(1.85014429040102783e+49)
assert hyp0f1((-20*10**100+1, 10**100), 0.25).ae(1.85014429040102783e+49)
def test_hypercomb_zero_pow():
# check that 0^0 = 1
assert hypercomb(lambda a: (([0],[a],[],[],[],[],0),), [0]) == 1
assert meijerg([[-1.5],[]],[[0],[-0.75]],0).ae(1.4464090846320771425)
def test_spherharm():
mp.dps = 15
t = 0.5; r = 0.25
assert spherharm(0,0,t,r).ae(0.28209479177387814347)
assert spherharm(1,-1,t,r).ae(0.16048941205971996369 - 0.04097967481096344271j)
assert spherharm(1,0,t,r).ae(0.42878904414183579379)
assert spherharm(1,1,t,r).ae(-0.16048941205971996369 - 0.04097967481096344271j)
assert spherharm(2,-2,t,r).ae(0.077915886919031181734 - 0.042565643022253962264j)
assert spherharm(2,-1,t,r).ae(0.31493387233497459884 - 0.08041582001959297689j)
assert spherharm(2,0,t,r).ae(0.41330596756220761898)
assert spherharm(2,1,t,r).ae(-0.31493387233497459884 - 0.08041582001959297689j)
assert spherharm(2,2,t,r).ae(0.077915886919031181734 + 0.042565643022253962264j)
assert spherharm(3,-3,t,r).ae(0.033640236589690881646 - 0.031339125318637082197j)
assert spherharm(3,-2,t,r).ae(0.18091018743101461963 - 0.09883168583167010241j)
assert spherharm(3,-1,t,r).ae(0.42796713930907320351 - 0.10927795157064962317j)
assert spherharm(3,0,t,r).ae(0.27861659336351639787)
assert spherharm(3,1,t,r).ae(-0.42796713930907320351 - 0.10927795157064962317j)
assert spherharm(3,2,t,r).ae(0.18091018743101461963 + 0.09883168583167010241j)
assert spherharm(3,3,t,r).ae(-0.033640236589690881646 - 0.031339125318637082197j)
assert spherharm(0,-1,t,r) == 0
assert spherharm(0,-2,t,r) == 0
assert spherharm(0,1,t,r) == 0
assert spherharm(0,2,t,r) == 0
assert spherharm(1,2,t,r) == 0
assert spherharm(1,3,t,r) == 0
assert spherharm(1,-2,t,r) == 0
assert spherharm(1,-3,t,r) == 0
assert spherharm(2,3,t,r) == 0
assert spherharm(2,4,t,r) == 0
assert spherharm(2,-3,t,r) == 0
assert spherharm(2,-4,t,r) == 0
assert spherharm(3,4.5,0.5,0.25).ae(-22.831053442240790148 + 10.910526059510013757j)
assert spherharm(2+3j, 1-j, 1+j, 3+4j).ae(-2.6582752037810116935 - 1.0909214905642160211j)
assert spherharm(-6,2.5,t,r).ae(0.39383644983851448178 + 0.28414687085358299021j)
assert spherharm(-3.5, 3, 0.5, 0.25).ae(0.014516852987544698924 - 0.015582769591477628495j)
assert spherharm(-3, 3, 0.5, 0.25) == 0
assert spherharm(-6, 3, 0.5, 0.25).ae(-0.16544349818782275459 - 0.15412657723253924562j)
assert spherharm(-6, 1.5, 0.5, 0.25).ae(0.032208193499767402477 + 0.012678000924063664921j)
assert spherharm(3,0,0,1).ae(0.74635266518023078283)
assert spherharm(3,-2,0,1) == 0
assert spherharm(3,-2,1,1).ae(-0.16270707338254028971 - 0.35552144137546777097j)
def test_qfunctions():
mp.dps = 15
assert qp(2,3,100).ae('2.7291482267247332183e2391')
def test_issue_239():
mp.prec = 150
x = ldexp(2476979795053773,-52)
assert betainc(206, 385, 0, 0.55, 1).ae('0.99999999999999999999996570910644857895771110649954')
mp.dps = 15
pytest.raises(ValueError, lambda: hyp2f1(-5,5,0.5,0.5))
# Extra stress testing for Bessel functions
# Reference zeros generated with the aid of scipy.special
# jn_zero, jnp_zero, yn_zero, ynp_zero
V = 15
M = 15
jn_small_zeros = \
[[2.4048255576957728,
5.5200781102863106,
8.6537279129110122,
11.791534439014282,
14.930917708487786,
18.071063967910923,
21.211636629879259,
24.352471530749303,
27.493479132040255,
30.634606468431975,
33.775820213573569,
36.917098353664044,
40.058425764628239,
43.19979171317673,
46.341188371661814],
[3.8317059702075123,
7.0155866698156188,
10.173468135062722,
13.323691936314223,
16.470630050877633,
19.615858510468242,
22.760084380592772,
25.903672087618383,
29.046828534916855,
32.189679910974404,
35.332307550083865,
38.474766234771615,
41.617094212814451,
44.759318997652822,
47.901460887185447],
[5.1356223018406826,
8.4172441403998649,
11.619841172149059,
14.795951782351261,
17.959819494987826,
21.116997053021846,
24.270112313573103,
27.420573549984557,
30.569204495516397,
33.7165195092227,
36.86285651128381,
40.008446733478192,
43.153453778371463,
46.297996677236919,
49.442164110416873],
[6.3801618959239835,
9.7610231299816697,
13.015200721698434,
16.223466160318768,
19.409415226435012,
22.582729593104442,
25.748166699294978,
28.908350780921758,
32.064852407097709,
35.218670738610115,
38.370472434756944,
41.520719670406776,
44.669743116617253,
47.817785691533302,
50.965029906205183],
[7.5883424345038044,
11.064709488501185,
14.37253667161759,
17.615966049804833,
20.826932956962388,
24.01901952477111,
27.199087765981251,
30.371007667117247,
33.537137711819223,
36.699001128744649,
39.857627302180889,
43.01373772335443,
46.167853512924375,
49.320360686390272,
52.471551398458023],
[8.771483815959954,
12.338604197466944,
15.700174079711671,
18.980133875179921,
22.217799896561268,
25.430341154222704,
28.626618307291138,
31.811716724047763,
34.988781294559295,
38.159868561967132,
41.326383254047406,
44.489319123219673,
47.649399806697054,
50.80716520300633,
53.963026558378149],
[9.9361095242176849,
13.589290170541217,
17.003819667816014,
20.320789213566506,
23.58608443558139,
26.820151983411405,
30.033722386570469,
33.233041762847123,
36.422019668258457,
39.603239416075404,
42.778481613199507,
45.949015998042603,
49.11577372476426,
52.279453903601052,
55.440592068853149],
[11.086370019245084,
14.821268727013171,
18.287582832481726,
21.641541019848401,
24.934927887673022,
28.191188459483199,
31.42279419226558,
34.637089352069324,
37.838717382853611,
41.030773691585537,
44.21540850526126,
47.394165755570512,
50.568184679795566,
53.738325371963291,
56.905249991978781],
[12.225092264004655,
16.037774190887709,
19.554536430997055,
22.94517313187462,
26.266814641176644,
29.54565967099855,
32.795800037341462,
36.025615063869571,
39.240447995178135,
42.443887743273558,
45.638444182199141,
48.825930381553857,
52.007691456686903,
55.184747939289049,
58.357889025269694],
[13.354300477435331,
17.241220382489128,
20.807047789264107,
24.233885257750552,
27.583748963573006,
30.885378967696675,
34.154377923855096,
37.400099977156589,
40.628553718964528,
43.843801420337347,
47.048700737654032,
50.245326955305383,
53.435227157042058,
56.619580266508436,
59.799301630960228],
[14.475500686554541,
18.433463666966583,
22.046985364697802,
25.509450554182826,
28.887375063530457,
32.211856199712731,
35.499909205373851,
38.761807017881651,
42.004190236671805,
45.231574103535045,
48.447151387269394,
51.653251668165858,
54.851619075963349,
58.043587928232478,
61.230197977292681],
[15.589847884455485,
19.61596690396692,
23.275853726263409,
26.773322545509539,
30.17906117878486,
33.526364075588624,
36.833571341894905,
40.111823270954241,
43.368360947521711,
46.608132676274944,
49.834653510396724,
53.050498959135054,
56.257604715114484,
59.457456908388002,
62.651217388202912],
[16.698249933848246,
20.789906360078443,
24.494885043881354,
28.026709949973129,
31.45996003531804,
34.829986990290238,
38.156377504681354,
41.451092307939681,
44.721943543191147,
47.974293531269048,
51.211967004101068,
54.437776928325074,
57.653844811906946,
60.8618046824805,
64.062937824850136],
[17.801435153282442,
21.95624406783631,
25.705103053924724,
29.270630441874802,
32.731053310978403,
36.123657666448762,
39.469206825243883,
42.780439265447158,
46.06571091157561,
49.330780096443524,
52.579769064383396,
55.815719876305778,
59.040934037249271,
62.257189393731728,
65.465883797232125],
[18.899997953174024,
23.115778347252756,
26.907368976182104,
30.505950163896036,
33.993184984781542,
37.408185128639695,
40.772827853501868,
44.100590565798301,
47.400347780543231,
50.678236946479898,
53.93866620912693,
57.184898598119301,
60.419409852130297,
63.644117508962281,
66.860533012260103]]
jnp_small_zeros = \
[[0.0,
3.8317059702075123,
7.0155866698156188,
10.173468135062722,
13.323691936314223,
16.470630050877633,
19.615858510468242,
22.760084380592772,
25.903672087618383,
29.046828534916855,
32.189679910974404,
35.332307550083865,
38.474766234771615,
41.617094212814451,
44.759318997652822],
[1.8411837813406593,
5.3314427735250326,
8.5363163663462858,
11.706004902592064,
14.863588633909033,
18.015527862681804,
21.16436985918879,
24.311326857210776,
27.457050571059246,
30.601922972669094,
33.746182898667383,
36.889987409236811,
40.033444053350675,
43.176628965448822,
46.319597561173912],
[3.0542369282271403,
6.7061331941584591,
9.9694678230875958,
13.170370856016123,
16.347522318321783,
19.512912782488205,
22.671581772477426,
25.826037141785263,
28.977672772993679,
32.127327020443474,
35.275535050674691,
38.422654817555906,
41.568934936074314,
44.714553532819734,
47.859641607992093],
[4.2011889412105285,
8.0152365983759522,
11.345924310743006,
14.585848286167028,
17.78874786606647,
20.9724769365377,
24.144897432909265,
27.310057930204349,
30.470268806290424,
33.626949182796679,
36.781020675464386,
39.933108623659488,
43.083652662375079,
46.232971081836478,
49.381300092370349],
[5.3175531260839944,
9.2823962852416123,
12.681908442638891,
15.964107037731551,
19.196028800048905,
22.401032267689004,
25.589759681386733,
28.767836217666503,
31.938539340972783,
35.103916677346764,
38.265316987088158,
41.423666498500732,
44.579623137359257,
47.733667523865744,
50.886159153182682],
[6.4156163757002403,
10.519860873772308,
13.9871886301403,
17.312842487884625,
20.575514521386888,
23.803581476593863,
27.01030789777772,
30.20284907898166,
33.385443901010121,
36.560777686880356,
39.730640230067416,
42.896273163494417,
46.058566273567043,
49.218174614666636,
52.375591529563596],
[7.501266144684147,
11.734935953042708,
15.268181461097873,
18.637443009666202,
21.931715017802236,
25.183925599499626,
28.409776362510085,
31.617875716105035,
34.81339298429743,
37.999640897715301,
41.178849474321413,
44.352579199070217,
47.521956905768113,
50.687817781723741,
53.85079463676896],
[8.5778364897140741,
12.932386237089576,
16.529365884366944,
19.941853366527342,
23.268052926457571,
26.545032061823576,
29.790748583196614,
33.015178641375142,
36.224380548787162,
39.422274578939259,
42.611522172286684,
45.793999658055002,
48.971070951900596,
52.143752969301988,
55.312820330403446],
[9.6474216519972168,
14.115518907894618,
17.774012366915256,
21.229062622853124,
24.587197486317681,
27.889269427955092,
31.155326556188325,
34.39662855427218,
37.620078044197086,
40.830178681822041,
44.030010337966153,
47.221758471887113,
50.407020967034367,
53.586995435398319,
56.762598475105272],
[10.711433970699945,
15.28673766733295,
19.004593537946053,
22.501398726777283,
25.891277276839136,
29.218563499936081,
32.505247352375523,
35.763792928808799,
39.001902811514218,
42.224638430753279,
45.435483097475542,
48.636922645305525,
51.830783925834728,
55.01844255063594,
58.200955824859509],
[11.770876674955582,
16.447852748486498,
20.223031412681701,
23.760715860327448,
27.182021527190532,
30.534504754007074,
33.841965775135715,
37.118000423665604,
40.371068905333891,
43.606764901379516,
46.828959446564562,
50.040428970943456,
53.243223214220535,
56.438892058982552,
59.628631306921512],
[12.826491228033465,
17.600266557468326,
21.430854238060294,
25.008518704644261,
28.460857279654847,
31.838424458616998,
35.166714427392629,
38.460388720328256,
41.728625562624312,
44.977526250903469,
48.211333836373288,
51.433105171422278,
54.645106240447105,
57.849056857839799,
61.046288512821078],
[13.878843069697276,
18.745090916814406,
22.629300302835503,
26.246047773946584,
29.72897816891134,
33.131449953571661,
36.480548302231658,
39.791940718940855,
43.075486800191012,
46.337772104541405,
49.583396417633095,
52.815686826850452,
56.037118687012179,
59.249577075517968,
62.454525995970462],
[14.928374492964716,
19.88322436109951,
23.81938909003628,
27.474339750968247,
30.987394331665278,
34.414545662167183,
37.784378506209499,
41.113512376883377,
44.412454519229281,
47.688252845993366,
50.945849245830813,
54.188831071035124,
57.419876154678179,
60.641030026538746,
63.853885828967512],
[15.975438807484321,
21.015404934568315,
25.001971500138194,
28.694271223110755,
32.236969407878118,
35.688544091185301,
39.078998185245057,
42.425854432866141,
45.740236776624833,
49.029635055514276,
52.299319390331728,
55.553127779547459,
58.793933759028134,
62.02393848337554,
65.244860767043859]]
yn_small_zeros = \
[[0.89357696627916752,
3.9576784193148579,
7.0860510603017727,
10.222345043496417,
13.361097473872763,
16.500922441528091,
19.64130970088794,
22.782028047291559,
25.922957653180923,
29.064030252728398,
32.205204116493281,
35.346452305214321,
38.487756653081537,
41.629104466213808,
44.770486607221993],
[2.197141326031017,
5.4296810407941351,
8.5960058683311689,
11.749154830839881,
14.897442128336725,
18.043402276727856,
21.188068934142213,
24.331942571356912,
27.475294980449224,
30.618286491641115,
33.761017796109326,
36.90355531614295,
40.045944640266876,
43.188218097393211,
46.330399250701687],
[3.3842417671495935,
6.7938075132682675,
10.023477979360038,
13.209986710206416,
16.378966558947457,
19.539039990286384,
22.69395593890929,
25.845613720902269,
28.995080395650151,
32.143002257627551,
35.289793869635804,
38.435733485446343,
41.581014867297885,
44.725777117640461,
47.870122696676504],
[4.5270246611496439,
8.0975537628604907,
11.396466739595867,
14.623077742393873,
17.81845523294552,
20.997284754187761,
24.166235758581828,
27.328799850405162,
30.486989604098659,
33.642049384702463,
36.794791029185579,
39.945767226378749,
43.095367507846703,
46.2438744334407,
49.391498015725107],
[5.6451478942208959,
9.3616206152445429,
12.730144474090465,
15.999627085382479,
19.22442895931681,
22.424810599698521,
25.610267054939328,
28.785893657666548,
31.954686680031668,
35.118529525584828,
38.278668089521758,
41.435960629910073,
44.591018225353424,
47.744288086361052,
50.896105199722123],
[6.7471838248710219,
10.597176726782031,
14.033804104911233,
17.347086393228382,
20.602899017175335,
23.826536030287532,
27.030134937138834,
30.220335654231385,
33.401105611047908,
36.574972486670962,
39.743627733020277,
42.908248189569535,
46.069679073215439,
49.228543693445843,
52.385312123112282],
[7.8377378223268716,
11.811037107609447,
15.313615118517857,
18.670704965906724,
21.958290897126571,
25.206207715021249,
28.429037095235496,
31.634879502950644,
34.828638524084437,
38.013473399691765,
41.19151880917741,
44.364272633271975,
47.53281875312084,
50.697961822183806,
53.860312300118388],
[8.919605734873789,
13.007711435388313,
16.573915129085334,
19.974342312352426,
23.293972585596648,
26.5667563757203,
29.809531451608321,
33.031769327150685,
36.239265816598239,
39.435790312675323,
42.623910919472727,
45.805442883111651,
48.981708325514764,
52.153694518185572,
55.322154420959698],
[9.9946283820824834,
14.190361295800141,
17.817887841179873,
21.26093227125945,
24.612576377421522,
27.910524883974868,
31.173701563441602,
34.412862242025045,
37.634648706110989,
40.843415321050884,
44.04214994542435,
47.232978012841169,
50.417456447370186,
53.596753874948731,
56.771765754432457],
[11.064090256031013,
15.361301343575925,
19.047949646361388,
22.532765416313869,
25.91620496332662,
29.2394205079349,
32.523270869465881,
35.779715464475261,
39.016196664616095,
42.237627509803703,
45.4474001519274,
48.647941127433196,
51.841036928216499,
55.028034667184916,
58.209970905250097],
[12.128927704415439,
16.522284394784426,
20.265984501212254,
23.791669719454272,
27.206568881574774,
30.555020011020762,
33.859683872746356,
37.133649760307504,
40.385117593813002,
43.619533085646856,
46.840676630553575,
50.051265851897857,
53.253310556711732,
56.448332488918971,
59.637507005589829],
[13.189846995683845,
17.674674253171487,
21.473493977824902,
25.03913093040942,
28.485081336558058,
31.858644293774859,
35.184165245422787,
38.475796636190897,
41.742455848758449,
44.990096293791186,
48.222870660068338,
51.443777308699826,
54.655042589416311,
57.858358441436511,
61.055036135780528],
[14.247395665073945,
18.819555894710682,
22.671697117872794,
26.276375544903892,
29.752925495549038,
33.151412708998983,
36.497763772987645,
39.807134090704376,
43.089121522203808,
46.350163579538652,
49.594769786270069,
52.82620892320143,
56.046916910756961,
59.258751140598783,
62.463155567737854],
[15.30200785858925,
19.957808654258601,
23.861599172945054,
27.504429642227545,
31.011103429019229,
34.434283425782942,
37.801385632318459,
41.128514139788358,
44.425913324440663,
47.700482714581842,
50.957073905278458,
54.199216028087261,
57.429547607017405,
60.65008661807661,
63.862406280068586],
[16.354034360047551,
21.090156519983806,
25.044040298785627,
28.724161640881914,
32.260472459522644,
35.708083982611664,
39.095820003878235,
42.440684315990936,
45.75353669045622,
49.041718113283529,
52.310408280968073,
55.56338698149062,
58.803488508906895,
62.032886550960831,
65.253280088312461]]
ynp_small_zeros = \
[[2.197141326031017,
5.4296810407941351,
8.5960058683311689,
11.749154830839881,
14.897442128336725,
18.043402276727856,
21.188068934142213,
24.331942571356912,
27.475294980449224,
30.618286491641115,
33.761017796109326,
36.90355531614295,
40.045944640266876,
43.188218097393211,
46.330399250701687],
[3.6830228565851777,
6.9414999536541757,
10.123404655436613,
13.285758156782854,
16.440058007293282,
19.590241756629495,
22.738034717396327,
25.884314618788867,
29.029575819372535,
32.174118233366201,
35.318134458192094,
38.461753870997549,
41.605066618873108,
44.74813744908079,
47.891014070791065],
[5.0025829314460639,
8.3507247014130795,
11.574195465217647,
14.760909306207676,
17.931285939466855,
21.092894504412739,
24.249231678519058,
27.402145837145258,
30.552708880564553,
33.70158627151572,
36.849213419846257,
39.995887376143356,
43.141817835750686,
46.287157097544201,
49.432018469138281],
[6.2536332084598136,
9.6987879841487711,
12.972409052292216,
16.19044719506921,
19.38238844973613,
22.559791857764261,
25.728213194724094,
28.890678419054777,
32.048984005266337,
35.204266606440635,
38.357281675961019,
41.508551443818436,
44.658448731963676,
47.807246956681162,
50.95515126455207],
[7.4649217367571329,
11.005169149809189,
14.3317235192331,
17.58443601710272,
20.801062338411128,
23.997004122902644,
27.179886689853435,
30.353960608554323,
33.521797098666792,
36.685048382072301,
39.844826969405863,
43.001910515625288,
46.15685955107263,
49.310088614282257,
52.461911043685864],
[8.6495562436971983,
12.280868725807848,
15.660799304540377,
18.949739756016503,
22.192841809428241,
25.409072788867674,
28.608039283077593,
31.795195353138159,
34.973890634255288,
38.14630522169358,
41.313923188794905,
44.477791768537617,
47.638672065035628,
50.797131066967842,
53.953600129601663],
[9.8147970120105779,
13.532811875789828,
16.965526446046053,
20.291285512443867,
23.56186260680065,
26.799499736027237,
30.015665481543419,
33.216968050039509,
36.407516858984748,
39.590015243560459,
42.766320595957378,
45.937754257017323,
49.105283450953203,
52.269633324547373,
55.431358715604255],
[10.965152105242974,
14.765687379508912,
18.250123150217555,
21.612750053384621,
24.911310600813573,
28.171051927637585,
31.40518108895689,
34.621401012564177,
37.824552065973114,
41.017847386464902,
44.203512240871601,
47.3831408366063,
50.557907466622796,
53.728697478957026,
56.896191727313342],
[12.103641941939539,
15.982840905145284,
19.517731005559611,
22.916962141504605,
26.243700855690533,
29.525960140695407,
32.778568197561124,
36.010261572392516,
39.226578757802172,
42.43122493258747,
45.626783824134354,
48.815117837929515,
51.997606404328863,
55.175294723956816,
58.348990221754937],
[13.232403808592215,
17.186756572616758,
20.770762917490496,
24.206152448722253,
27.561059462697153,
30.866053571250639,
34.137476603379774,
37.385039772270268,
40.614946085165892,
43.831373184731238,
47.037251786726299,
50.234705848765229,
53.425316228549359,
56.610286079882087,
59.790548623216652],
[14.35301374369987,
18.379337301642568,
22.011118775283494,
25.482116178696707,
28.865046588695164,
32.192853922166294,
35.483296655830277,
38.747005493021857,
41.990815194320955,
45.219355876831731,
48.435892856078888,
51.642803925173029,
54.84186659475857,
58.034439083840155,
61.221578745109862],
[15.466672066554263,
19.562077985759503,
23.240325531101082,
26.746322986645901,
30.157042415639891,
33.507642948240263,
36.817212798512775,
40.097251300178642,
43.355193847719752,
46.596103410173672,
49.823567279972794,
53.040208868780832,
56.247996968470062,
59.448441365714251,
62.642721301357187],
[16.574317035530872,
20.73617763753932,
24.459631728238804,
27.999993668839644,
31.438208790267783,
34.811512070805535,
38.140243708611251,
41.436725143893739,
44.708963264433333,
47.962435051891027,
51.201037321915983,
54.427630745992975,
57.644369734615238,
60.852911791989989,
64.054555435720397],
[17.676697936439624,
21.9026148697762,
25.670073356263225,
29.244155124266438,
32.709534477396028,
36.105399554497548,
39.453272918267025,
42.766255701958017,
46.052899215578358,
49.319076602061401,
52.568982147952547,
55.805705507386287,
59.031580956740466,
62.248409689597653,
65.457606670836759],
[18.774423978290318,
23.06220035979272,
26.872520985976736,
30.479680663499762,
33.971869047372436,
37.390118854896324,
40.757072537673599,
44.086572292170345,
47.387688809191869,
50.66667461073936,
53.928009929563275,
57.175005343085052,
60.410169281219877,
63.635442539153021,
66.85235358587768]]
@pytest.mark.slow
def test_bessel_zeros_extra():
mp.dps = 15
for v in range(V):
for m in range(1,M+1):
print(v, m, "of", V, M)
# Twice to test cache (if used)
assert besseljzero(v,m).ae(jn_small_zeros[v][m-1])
assert besseljzero(v,m).ae(jn_small_zeros[v][m-1])
assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1])
assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1])
assert besselyzero(v,m).ae(yn_small_zeros[v][m-1])
assert besselyzero(v,m).ae(yn_small_zeros[v][m-1])
assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1])
assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1])
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