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])