diff --git a/gamma_inc.go b/gamma_inc.go new file mode 100644 index 00000000..2dd625cb --- /dev/null +++ b/gamma_inc.go @@ -0,0 +1,50 @@ +// Copyright ©2016 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import ( + "github.com/gonum/mathext/internal/cephes" +) + +// GammaInc computes the incomplete Gamma integral. +// GammaInc(a,x) = (1/ Γ(a)) \int_0^x e^{-t} t^{a-1} dt +// The input argument a must be positive and x must be non-negative or GammaInc +// will panic. +// +// See http://mathworld.wolfram.com/IncompleteGammaFunction.html +// or https://en.wikipedia.org/wiki/Incomplete_gamma_function for more detailed +// information. +func GammaInc(a, x float64) float64 { + return cephes.Igam(a, x) +} + +// GammaIncComp computes the complemented incomplete Gamma integral. +// GammaIncComp(a,x) = 1 - GammaInc(a,x) +// = (1/ Γ(a)) \int_0^\infty e^{-t} t^{a-1} dt +// The input argument a must be positive and x must be non-negative or +// GammaIncComp will panic. +func GammaIncComp(a, x float64) float64 { + return cephes.IgamC(a, x) +} + +// GammaIncInv computes the inverse of the incomplete Gamma integral. That is, +// it returns the x such that: +// GammaInc(a, x) = y +// The input argument a must be positive and y must be between 0 and 1 +// inclusive or GammaIncInv will panic. GammaIncInv should return a positive +// number, but can return NaN if there is a failure to converge. +func GammaIncInv(a, y float64) float64 { + return gammaIncInv(a, y) +} + +// GammaIncCompInv computes the inverse of the complemented incomplete Gamma +// integral. That is, it returns the x such that: +// GammaIncComp(a, x) = y +// The input argument a must be positive and y must be between 0 and 1 +// inclusive or GammaIncCompInv will panic. GammaIncCompInv should return a +// positive number, but can return 0 even with non-zero y due to underflow. +func GammaIncCompInv(a, y float64) float64 { + return cephes.IgamI(a, y) +} diff --git a/gamma_inc_inv.go b/gamma_inc_inv.go new file mode 100644 index 00000000..d7f146df --- /dev/null +++ b/gamma_inc_inv.go @@ -0,0 +1,56 @@ +// Derived from SciPy's special/c_misc/gammaincinv.c +// https://github.com/scipy/scipy/blob/master/scipy/special/c_misc/gammaincinv.c + +// Copyright ©2017 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import ( + "math" + + "github.com/gonum/mathext/internal/cephes" +) + +const ( + allowedATol = 1e-306 + allowedRTol = 1e-6 +) + +func gammaInc(x float64, params []float64) float64 { + return cephes.Igam(params[0], x) - params[1] +} + +// gammaIncInv is the inverse of the incomplete Gamma integral. That is, it +// returns x such that: +// Igam(a, x) = y +// The input argument a must be positive and y must be between 0 and 1 +// inclusive or gammaIncInv will panic. gammaIncInv should return a +// positive number, but can return NaN if there is a failure to converge. +func gammaIncInv(a, y float64) float64 { + // For y not small, we just use + // IgamI(a, 1-y) + // (inverse of the complemented incomplete Gamma integral). For y small, + // however, 1-y is about 1, and we lose digits. + if a <= 0 || y <= 0 || y >= 0.25 { + return cephes.IgamI(a, 1-y) + } + + lo := 0.0 + flo := -y + hi := cephes.IgamI(a, 0.75) + fhi := 0.25 - y + + params := []float64{a, y} + + // Also, after we generate a small interval by bisection above, false + // position will do a large step from an interval of width ~1e-4 to ~1e-14 + // in one step (a=10, x=0.05, but similiar for other values). + result, bestX, _, errEst := falsePosition(lo, hi, flo, fhi, 2*machEp, 2*machEp, 1e-2*a, gammaInc, params) + if result == fSolveMaxIterations && errEst > allowedATol+allowedRTol*math.Abs(bestX) { + bestX = math.NaN() + } + + return bestX +} diff --git a/gamma_inc_test.go b/gamma_inc_test.go new file mode 100644 index 00000000..03d5f014 --- /dev/null +++ b/gamma_inc_test.go @@ -0,0 +1,138 @@ +// Copyright ©2016 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import ( + "math" + "testing" +) + +func TestGammaInc(t *testing.T) { + for i, test := range []struct { + a, x, want float64 + }{ + // Results computed using scipy.special.gamminc + {0, 0, 0}, + {0.0001, 1, 0.99997805936186279}, + {0.001, 0.005, 0.99528424172333985}, + {0.01, 10, 0.99999995718295021}, + {0.1, 10, 0.99999944520142825}, + {0.25, 0.75, 0.89993651328449831}, + {0.5, 0.5, 0.68268949213708596}, + {0.5, 2, 0.95449973610364147}, + {0.75, 2.5, 0.95053039734695643}, + {1, 0.5, 0.39346934028736652}, + {1, 1, 0.63212055882855778}, + {1.5, 0.75, 0.31772966966378746}, + {2.5, 1, 0.15085496391539038}, + {3, 0.05, 2.0067493624397931e-05}, + {3, 20, 0.99999954448504946}, + {5, 50, 1}, + {7, 10, 0.86985857911751696}, + {10, 0.9, 4.2519575433351128e-08}, + {10, 5, 0.031828057306204811}, + {25, 10, 4.6949381426799868e-05}, + } { + if got := GammaInc(test.a, test.x); math.Abs(got-test.want) > 1e-10 { + t.Errorf("test %d GammaInc(%g, %g) failed: got %g want %g", i, test.a, test.x, got, test.want) + } + } +} + +func TestGammaIncComp(t *testing.T) { + for i, test := range []struct { + a, x, want float64 + }{ + // Results computed using scipy.special.gammincc + {0.00001, 0.075, 2.0866541002417804e-05}, + {0.0001, 1, 2.1940638138146658e-05}, + {0.001, 0.005, 0.0047157582766601536}, + {0.01, 0.9, 0.0026263432520514662}, + {0.25, 0.75, 0.10006348671550169}, + {0.5, 0.5, 0.31731050786291404}, + {0.75, 0.25, 0.65343980284081038}, + {0.9, 0.01, 0.98359881081593148}, + {1, 0, 1}, + {1, 0.075, 0.92774348632855297}, + {1, 1, 0.36787944117144233}, + {1, 10, 4.5399929762484861e-05}, + {1, math.Inf(1), 0}, + {3, 20, 4.5551495055892125e-07}, + {5, 10, 0.029252688076961127}, + {10, 3, 0.99889751186988451}, + {50, 25, 0.99999304669475242}, + {100, 10, 1}, + {500, 500, 0.49405285382921321}, + {500, 550, 0.014614408126291296}, + } { + if got := GammaIncComp(test.a, test.x); math.Abs(got-test.want) > 1e-10 { + t.Errorf("test %d GammaIncComp(%g, %g) failed: got %g want %g", i, test.a, test.x, got, test.want) + } + } +} + +func TestGammaIncInv(t *testing.T) { + for i, test := range []struct { + a, x, want float64 + }{ + // Results computed using scipy.special.gammincinv + {0.001, 0.99, 2.4259428385570885e-05}, + {0.01, 0.99, 0.26505255025157959}, + {0.1, 0.5, 0.00059339110446022798}, + {0.2, 0.8, 0.26354363204872067}, + {0.25, 0.5, 0.043673802352873381}, + {0.5, 0.25, 0.050765522133810789}, + {0.5, 0.5, 0.22746821155978625}, + {0.75, 0.25, 0.15340752707472377}, + {1, 0, 0}, + {1, 0.075, 0.077961541469711862}, + {1, 1, math.Inf(1)}, + {2.5, 0.99, 7.5431362346944937}, + {10, 0.5, 9.6687146147141299}, + {25, 0.01, 14.853341349420646}, + {25, 0.99, 38.076945624506337}, + {50, 0.75, 54.570620535040511}, + {100, 0.25, 93.08583383712174}, + {1000, 0.01, 927.90815979664251}, + {1000, 0.99, 1075.0328320864389}, + {10000, 0.5, 9999.6666686420485}, + } { + if got := GammaIncInv(test.a, test.x); math.Abs(got-test.want) > 1e-10 { + t.Errorf("test %d GammaIncInv(%g, %g) failed: got %g want %g", i, test.a, test.x, got, test.want) + } + } +} + +func TestGammaIncCompInv(t *testing.T) { + for i, test := range []struct { + a, x, want float64 + }{ + // Results computed using scipy.special.gamminccinv + {0.001, 0.01, 2.4259428385570885e-05}, + {0.01, 0.01, 0.26505255025158292}, + {0.03, 0.4, 2.316980536227699e-08}, + {0.1, 0.5, 0.00059339110446022798}, + {0.1, 0.75, 5.7917132949696076e-07}, + {0.25, 0.25, 0.26062600197823282}, + {0.5, 0.1, 1.3527717270477047}, + {0.5, 0.5, 0.22746821155978625}, + {0.75, 0.25, 1.0340914067758025}, + {1, 0, math.Inf(1)}, + {1, 0.5, 0.69314718055994529}, + {1, 1, 0}, + {3, 0.75, 1.727299417860519}, + {25, 0.4, 25.945791937289371}, + {25, 0.7, 22.156653488661991}, + {10, 0.5, 9.6687146147141299}, + {100, 0.25, 106.5510925269767}, + {1000, 0.01, 1075.0328320864389}, + {1000, 0.99, 927.90815979664251}, + {10000, 0.5, 9999.6666686420485}, + } { + if got := GammaIncCompInv(test.a, test.x); math.Abs(got-test.want) > 1e-10 { + t.Errorf("test %d GammaIncCompInv(%g, %g) failed: got %g want %g", i, test.a, test.x, got, test.want) + } + } +} diff --git a/internal/cephes/cephes.go b/internal/cephes/cephes.go index 4495051a..3dfab9d0 100644 --- a/internal/cephes/cephes.go +++ b/internal/cephes/cephes.go @@ -2,9 +2,11 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. -// package cephes implements functions originally in the Netlib code by Stephen Mosher +// Package cephes implements functions originally in the Netlib code by Stephen Mosher. package cephes +import "math" + /* Additional copyright information: @@ -15,11 +17,13 @@ https://lists.debian.org/debian-legal/2004/12/msg00295.html */ var ( - badParamOutOfBounds = "cephes: parameter out of bounds" + badParamOutOfBounds = "cephes: parameter out of bounds" + badParamFunctionSingularity = "cephes: function singularity" ) const ( - machEp = 1.11022302462515654042e-16 // 2^-53 - maxLog = 7.09782712893383996732e2 // log(2^127) - minLog = -7.451332191019412076235e2 // log(2^-128) + machEp = 1.0 / (1 << 53) + maxLog = 1024 * math.Ln2 + minLog = -1075 * math.Ln2 + maxIter = 2000 ) diff --git a/internal/cephes/igam.go b/internal/cephes/igam.go new file mode 100644 index 00000000..8b9fdf98 --- /dev/null +++ b/internal/cephes/igam.go @@ -0,0 +1,311 @@ +// Derived from SciPy's special/cephes/igam.c and special/cephes/igam.h +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.h +// Made freely available by Stephen L. Moshier without support or guarantee. + +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Copyright ©1985, ©1987 by Stephen L. Moshier +// Portions Copyright ©2016 The gonum Authors. All rights reserved. + +package cephes + +import "math" + +const ( + igamDimK = 25 + igamDimN = 25 + igam = 1 + igamC = 0 + igamSmall = 20 + igamLarge = 200 + igamSmallRatio = 0.3 + igamLargeRatio = 4.5 +) + +var igamCoefs = [igamDimK][igamDimN]float64{ + [igamDimN]float64{-3.3333333333333333e-1, 8.3333333333333333e-2, -1.4814814814814815e-2, 1.1574074074074074e-3, 3.527336860670194e-4, -1.7875514403292181e-4, 3.9192631785224378e-5, -2.1854485106799922e-6, -1.85406221071516e-6, 8.296711340953086e-7, -1.7665952736826079e-7, 6.7078535434014986e-9, 1.0261809784240308e-8, -4.3820360184533532e-9, 9.1476995822367902e-10, -2.551419399494625e-11, -5.8307721325504251e-11, 2.4361948020667416e-11, -5.0276692801141756e-12, 1.1004392031956135e-13, 3.3717632624009854e-13, -1.3923887224181621e-13, 2.8534893807047443e-14, -5.1391118342425726e-16, -1.9752288294349443e-15}, + [igamDimN]float64{-1.8518518518518519e-3, -3.4722222222222222e-3, 2.6455026455026455e-3, -9.9022633744855967e-4, 2.0576131687242798e-4, -4.0187757201646091e-7, -1.8098550334489978e-5, 7.6491609160811101e-6, -1.6120900894563446e-6, 4.6471278028074343e-9, 1.378633446915721e-7, -5.752545603517705e-8, 1.1951628599778147e-8, -1.7543241719747648e-11, -1.0091543710600413e-9, 4.1627929918425826e-10, -8.5639070264929806e-11, 6.0672151016047586e-14, 7.1624989648114854e-12, -2.9331866437714371e-12, 5.9966963656836887e-13, -2.1671786527323314e-16, -4.9783399723692616e-14, 2.0291628823713425e-14, -4.13125571381061e-15}, + [igamDimN]float64{4.1335978835978836e-3, -2.6813271604938272e-3, 7.7160493827160494e-4, 2.0093878600823045e-6, -1.0736653226365161e-4, 5.2923448829120125e-5, -1.2760635188618728e-5, 3.4235787340961381e-8, 1.3721957309062933e-6, -6.298992138380055e-7, 1.4280614206064242e-7, -2.0477098421990866e-10, -1.4092529910867521e-8, 6.228974084922022e-9, -1.3670488396617113e-9, 9.4283561590146782e-13, 1.2872252400089318e-10, -5.5645956134363321e-11, 1.1975935546366981e-11, -4.1689782251838635e-15, -1.0940640427884594e-12, 4.6622399463901357e-13, -9.905105763906906e-14, 1.8931876768373515e-17, 8.8592218725911273e-15}, + [igamDimN]float64{6.4943415637860082e-4, 2.2947209362139918e-4, -4.6918949439525571e-4, 2.6772063206283885e-4, -7.5618016718839764e-5, -2.3965051138672967e-7, 1.1082654115347302e-5, -5.6749528269915966e-6, 1.4230900732435884e-6, -2.7861080291528142e-11, -1.6958404091930277e-7, 8.0994649053880824e-8, -1.9111168485973654e-8, 2.3928620439808118e-12, 2.0620131815488798e-9, -9.4604966618551322e-10, 2.1541049775774908e-10, -1.388823336813903e-14, -2.1894761681963939e-11, 9.7909989511716851e-12, -2.1782191880180962e-12, 6.2088195734079014e-17, 2.126978363279737e-13, -9.3446887915174333e-14, 2.0453671226782849e-14}, + [igamDimN]float64{-8.618882909167117e-4, 7.8403922172006663e-4, -2.9907248030319018e-4, -1.4638452578843418e-6, 6.6414982154651222e-5, -3.9683650471794347e-5, 1.1375726970678419e-5, 2.5074972262375328e-10, -1.6954149536558306e-6, 8.9075075322053097e-7, -2.2929348340008049e-7, 2.956794137544049e-11, 2.8865829742708784e-8, -1.4189739437803219e-8, 3.4463580499464897e-9, -2.3024517174528067e-13, -3.9409233028046405e-10, 1.8602338968504502e-10, -4.356323005056618e-11, 1.2786001016296231e-15, 4.6792750266579195e-12, -2.1492464706134829e-12, 4.9088156148096522e-13, -6.3385914848915603e-18, -5.0453320690800944e-14}, + [igamDimN]float64{-3.3679855336635815e-4, -6.9728137583658578e-5, 2.7727532449593921e-4, -1.9932570516188848e-4, 6.7977804779372078e-5, 1.419062920643967e-7, -1.3594048189768693e-5, 8.0184702563342015e-6, -2.2914811765080952e-6, -3.252473551298454e-10, 3.4652846491085265e-7, -1.8447187191171343e-7, 4.8240967037894181e-8, -1.7989466721743515e-14, -6.3061945000135234e-9, 3.1624176287745679e-9, -7.8409242536974293e-10, 5.1926791652540407e-15, 9.3589442423067836e-11, -4.5134262161632782e-11, 1.0799129993116827e-11, -3.661886712685252e-17, -1.210902069055155e-12, 5.6807435849905643e-13, -1.3249659916340829e-13}, + [igamDimN]float64{5.3130793646399222e-4, -5.9216643735369388e-4, 2.7087820967180448e-4, 7.9023532326603279e-7, -8.1539693675619688e-5, 5.6116827531062497e-5, -1.8329116582843376e-5, -3.0796134506033048e-9, 3.4651553688036091e-6, -2.0291327396058604e-6, 5.7887928631490037e-7, 2.338630673826657e-13, -8.8286007463304835e-8, 4.7435958880408128e-8, -1.2545415020710382e-8, 8.6496488580102925e-14, 1.6846058979264063e-9, -8.5754928235775947e-10, 2.1598224929232125e-10, -7.6132305204761539e-16, -2.6639822008536144e-11, 1.3065700536611057e-11, -3.1799163902367977e-12, 4.7109761213674315e-18, 3.6902800842763467e-13}, + [igamDimN]float64{3.4436760689237767e-4, 5.1717909082605922e-5, -3.3493161081142236e-4, 2.812695154763237e-4, -1.0976582244684731e-4, -1.2741009095484485e-7, 2.7744451511563644e-5, -1.8263488805711333e-5, 5.7876949497350524e-6, 4.9387589339362704e-10, -1.0595367014026043e-6, 6.1667143761104075e-7, -1.7562973359060462e-7, -1.2974473287015439e-12, 2.695423606288966e-8, -1.4578352908731271e-8, 3.887645959386175e-9, -3.8810022510194121e-17, -5.3279941738772867e-10, 2.7437977643314845e-10, -6.9957960920705679e-11, 2.5899863874868481e-17, 8.8566890996696381e-12, -4.403168815871311e-12, 1.0865561947091654e-12}, + [igamDimN]float64{-6.5262391859530942e-4, 8.3949872067208728e-4, -4.3829709854172101e-4, -6.969091458420552e-7, 1.6644846642067548e-4, -1.2783517679769219e-4, 4.6299532636913043e-5, 4.5579098679227077e-9, -1.0595271125805195e-5, 6.7833429048651666e-6, -2.1075476666258804e-6, -1.7213731432817145e-11, 3.7735877416110979e-7, -2.1867506700122867e-7, 6.2202288040189269e-8, 6.5977038267330006e-16, -9.5903864974256858e-9, 5.2132144922808078e-9, -1.3991589583935709e-9, 5.382058999060575e-16, 1.9484714275467745e-10, -1.0127287556389682e-10, 2.6077347197254926e-11, -5.0904186999932993e-18, -3.3721464474854592e-12}, + [igamDimN]float64{-5.9676129019274625e-4, -7.2048954160200106e-5, 6.7823088376673284e-4, -6.4014752602627585e-4, 2.7750107634328704e-4, 1.8197008380465151e-7, -8.4795071170685032e-5, 6.105192082501531e-5, -2.1073920183404862e-5, -8.8585890141255994e-10, 4.5284535953805377e-6, -2.8427815022504408e-6, 8.7082341778646412e-7, 3.6886101871706965e-12, -1.5344695190702061e-7, 8.862466778790695e-8, -2.5184812301826817e-8, -1.0225912098215092e-14, 3.8969470758154777e-9, -2.1267304792235635e-9, 5.7370135528051385e-10, -1.887749850169741e-19, -8.0931538694657866e-11, 4.2382723283449199e-11, -1.1002224534207726e-11}, + [igamDimN]float64{1.3324454494800656e-3, -1.9144384985654775e-3, 1.1089369134596637e-3, 9.932404122642299e-7, -5.0874501293093199e-4, 4.2735056665392884e-4, -1.6858853767910799e-4, -8.1301893922784998e-9, 4.5284402370562147e-5, -3.127053674781734e-5, 1.044986828530338e-5, 4.8435226265680926e-11, -2.1482565873456258e-6, 1.329369701097492e-6, -4.0295693092101029e-7, -1.7567877666323291e-13, 7.0145043163668257e-8, -4.040787734999483e-8, 1.1474026743371963e-8, 3.9642746853563325e-18, -1.7804938269892714e-9, 9.7480262548731646e-10, -2.6405338676507616e-10, 5.794875163403742e-18, 3.7647749553543836e-11}, + [igamDimN]float64{1.579727660730835e-3, 1.6251626278391582e-4, -2.0633421035543276e-3, 2.1389686185689098e-3, -1.0108559391263003e-3, -3.9912705529919201e-7, 3.6235025084764691e-4, -2.8143901463712154e-4, 1.0449513336495887e-4, 2.1211418491830297e-9, -2.5779417251947842e-5, 1.7281818956040463e-5, -5.6413773872904282e-6, -1.1024320105776174e-11, 1.1223224418895175e-6, -6.8693396379526735e-7, 2.0653236975414887e-7, 4.6714772409838506e-14, -3.5609886164949055e-8, 2.0470855345905963e-8, -5.8091738633283358e-9, -1.332821287582869e-16, 9.0354604391335133e-10, -4.9598782517330834e-10, 1.3481607129399749e-10}, + [igamDimN]float64{-4.0725121195140166e-3, 6.4033628338080698e-3, -4.0410161081676618e-3, -2.183732802866233e-6, 2.1740441801254639e-3, -1.9700440518418892e-3, 8.3595469747962458e-4, 1.9445447567109655e-8, -2.5779387120421696e-4, 1.9009987368139304e-4, -6.7696499937438965e-5, -1.4440629666426572e-10, 1.5712512518742269e-5, -1.0304008744776893e-5, 3.304517767401387e-6, 7.9829760242325709e-13, -6.4097794149313004e-7, 3.8894624761300056e-7, -1.1618347644948869e-7, -2.816808630596451e-15, 1.9878012911297093e-8, -1.1407719956357511e-8, 3.2355857064185555e-9, 4.1759468293455945e-20, -5.0423112718105824e-10}, + [igamDimN]float64{-5.9475779383993003e-3, -5.4016476789260452e-4, 8.7910413550767898e-3, -9.8576315587856125e-3, 5.0134695031021538e-3, 1.2807521786221875e-6, -2.0626019342754683e-3, 1.7109128573523058e-3, -6.7695312714133799e-4, -6.9011545676562133e-9, 1.8855128143995902e-4, -1.3395215663491969e-4, 4.6263183033528039e-5, 4.0034230613321351e-11, -1.0255652921494033e-5, 6.612086372797651e-6, -2.0913022027253008e-6, -2.0951775649603837e-13, 3.9756029041993247e-7, -2.3956211978815887e-7, 7.1182883382145864e-8, 8.925574873053455e-16, -1.2101547235064676e-8, 6.9350618248334386e-9, -1.9661464453856102e-9}, + [igamDimN]float64{1.7402027787522711e-2, -2.9527880945699121e-2, 2.0045875571402799e-2, 7.0289515966903407e-6, -1.2375421071343148e-2, 1.1976293444235254e-2, -5.4156038466518525e-3, -6.3290893396418616e-8, 1.8855118129005065e-3, -1.473473274825001e-3, 5.5515810097708387e-4, 5.2406834412550662e-10, -1.4357913535784836e-4, 9.9181293224943297e-5, -3.3460834749478311e-5, -3.5755837291098993e-12, 7.1560851960630076e-6, -4.5516802628155526e-6, 1.4236576649271475e-6, 1.8803149082089664e-14, -2.6623403898929211e-7, 1.5950642189595716e-7, -4.7187514673841102e-8, -6.5107872958755177e-17, 7.9795091026746235e-9}, + [igamDimN]float64{3.0249124160905891e-2, 2.4817436002649977e-3, -4.9939134373457022e-2, 5.9915643009307869e-2, -3.2483207601623391e-2, -5.7212968652103441e-6, 1.5085251778569354e-2, -1.3261324005088445e-2, 5.5515262632426148e-3, 3.0263182257030016e-8, -1.7229548406756723e-3, 1.2893570099929637e-3, -4.6845138348319876e-4, -1.830259937893045e-10, 1.1449739014822654e-4, -7.7378565221244477e-5, 2.5625836246985201e-5, 1.0766165333192814e-12, -5.3246809282422621e-6, 3.349634863064464e-6, -1.0381253128684018e-6, -5.608909920621128e-15, 1.9150821930676591e-7, -1.1418365800203486e-7, 3.3654425209171788e-8}, + [igamDimN]float64{-9.9051020880159045e-2, 1.7954011706123486e-1, -1.2989606383463778e-1, -3.1478872752284357e-5, 9.0510635276848131e-2, -9.2828824411184397e-2, 4.4412112839877808e-2, 2.7779236316835888e-7, -1.7229543805449697e-2, 1.4182925050891573e-2, -5.6214161633747336e-3, -2.39598509186381e-9, 1.6029634366079908e-3, -1.1606784674435773e-3, 4.1001337768153873e-4, 1.8365800754090661e-11, -9.5844256563655903e-5, 6.3643062337764708e-5, -2.076250624489065e-5, -1.1806020912804483e-13, 4.2131808239120649e-6, -2.6262241337012467e-6, 8.0770620494930662e-7, 6.0125912123632725e-16, -1.4729737374018841e-7}, + [igamDimN]float64{-1.9994542198219728e-1, -1.5056113040026424e-2, 3.6470239469348489e-1, -4.6435192311733545e-1, 2.6640934719197893e-1, 3.4038266027147191e-5, -1.3784338709329624e-1, 1.276467178337056e-1, -5.6213828755200985e-2, -1.753150885483011e-7, 1.9235592956768113e-2, -1.5088821281095315e-2, 5.7401854451350123e-3, 1.0622382710310225e-9, -1.5335082692563998e-3, 1.0819320643228214e-3, -3.7372510193945659e-4, -6.6170909729031985e-12, 8.4263617380909628e-5, -5.5150706827483479e-5, 1.7769536448348069e-5, 3.8827923210205533e-14, -3.53513697488768e-6, 2.1865832130045269e-6, -6.6812849447625594e-7}, + [igamDimN]float64{7.2438608504029431e-1, -1.3918010932653375, 1.0654143352413968, 1.876173868950258e-4, -8.2705501176152696e-1, 8.9352433347828414e-1, -4.4971003995291339e-1, -1.6107401567546652e-6, 1.9235590165271091e-1, -1.6597702160042609e-1, 6.8882222681814333e-2, 1.3910091724608687e-8, -2.146911561508663e-2, 1.6228980898865892e-2, -5.9796016172584256e-3, -1.1287469112826745e-10, 1.5167451119784857e-3, -1.0478634293553899e-3, 3.5539072889126421e-4, 8.1704322111801517e-13, -7.7773013442452395e-5, 5.0291413897007722e-5, -1.6035083867000518e-5, 1.2469354315487605e-14, 3.1369106244517615e-6}, + [igamDimN]float64{1.6668949727276811, 1.165462765994632e-1, -3.3288393225018906, 4.4692325482864037, -2.6977693045875807, -2.600667859891061e-4, 1.5389017615694539, -1.4937962361134612, 6.8881964633233148e-1, 1.3077482004552385e-6, -2.5762963325596288e-1, 2.1097676102125449e-1, -8.3714408359219882e-2, -7.7920428881354753e-9, 2.4267923064833599e-2, -1.7813678334552311e-2, 6.3970330388900056e-3, 4.9430807090480523e-11, -1.5554602758465635e-3, 1.0561196919903214e-3, -3.5277184460472902e-4, 9.3002334645022459e-14, 7.5285855026557172e-5, -4.8186515569156351e-5, 1.5227271505597605e-5}, + [igamDimN]float64{-6.6188298861372935, 1.3397985455142589e+1, -1.0789350606845146e+1, -1.4352254537875018e-3, 9.2333694596189809, -1.0456552819547769e+1, 5.5105526029033471, 1.2024439690716742e-5, -2.5762961164755816, 2.3207442745387179, -1.0045728797216284, -1.0207833290021914e-7, 3.3975092171169466e-1, -2.6720517450757468e-1, 1.0235252851562706e-1, 8.4329730484871625e-10, -2.7998284958442595e-2, 2.0066274144976813e-2, -7.0554368915086242e-3, 1.9402238183698188e-12, 1.6562888105449611e-3, -1.1082898580743683e-3, 3.654545161310169e-4, -5.1290032026971794e-11, -7.6340103696869031e-5}, + [igamDimN]float64{-1.7112706061976095e+1, -1.1208044642899116, 3.7131966511885444e+1, -5.2298271025348962e+1, 3.3058589696624618e+1, 2.4791298976200222e-3, -2.061089403411526e+1, 2.088672775145582e+1, -1.0045703956517752e+1, -1.2238783449063012e-5, 4.0770134274221141, -3.473667358470195, 1.4329352617312006, 7.1359914411879712e-8, -4.4797257159115612e-1, 3.4112666080644461e-1, -1.2699786326594923e-1, -2.8953677269081528e-10, 3.3125776278259863e-2, -2.3274087021036101e-2, 8.0399993503648882e-3, -1.177805216235265e-9, -1.8321624891071668e-3, 1.2108282933588665e-3, -3.9479941246822517e-4}, + [igamDimN]float64{7.389033153567425e+1, -1.5680141270402273e+2, 1.322177542759164e+2, 1.3692876877324546e-2, -1.2366496885920151e+2, 1.4620689391062729e+2, -8.0365587724865346e+1, -1.1259851148881298e-4, 4.0770132196179938e+1, -3.8210340013273034e+1, 1.719522294277362e+1, 9.3519707955168356e-7, -6.2716159907747034, 5.1168999071852637, -2.0319658112299095, -4.9507215582761543e-9, 5.9626397294332597e-1, -4.4220765337238094e-1, 1.6079998700166273e-1, -2.4733786203223402e-8, -4.0307574759979762e-2, 2.7849050747097869e-2, -9.4751858992054221e-3, 6.419922235909132e-6, 2.1250180774699461e-3}, + [igamDimN]float64{2.1216837098382522e+2, 1.3107863022633868e+1, -4.9698285932871748e+2, 7.3121595266969204e+2, -4.8213821720890847e+2, -2.8817248692894889e-2, 3.2616720302947102e+2, -3.4389340280087117e+2, 1.7195193870816232e+2, 1.4038077378096158e-4, -7.52594195897599e+1, 6.651969984520934e+1, -2.8447519748152462e+1, -7.613702615875391e-7, 9.5402237105304373, -7.5175301113311376, 2.8943997568871961, -4.6612194999538201e-7, -8.0615149598794088e-1, 5.8483006570631029e-1, -2.0845408972964956e-1, 1.4765818959305817e-4, 5.1000433863753019e-2, -3.3066252141883665e-2, 1.5109265210467774e-2}, + [igamDimN]float64{-9.8959643098322368e+2, 2.1925555360905233e+3, -1.9283586782723356e+3, -1.5925738122215253e-1, 1.9569985945919857e+3, -2.4072514765081556e+3, 1.3756149959336496e+3, 1.2920735237496668e-3, -7.525941715948055e+2, 7.3171668742208716e+2, -3.4137023466220065e+2, -9.9857390260608043e-6, 1.3356313181291573e+2, -1.1276295161252794e+2, 4.6310396098204458e+1, -7.9237387133614756e-6, -1.4510726927018646e+1, 1.1111771248100563e+1, -4.1690817945270892, 3.1008219800117808e-3, 1.1220095449981468, -7.6052379926149916e-1, 3.6262236505085254e-1, 2.216867741940747e-1, 4.8683443692930507e-1}, +} + +// Igam computes the incomplete Gamma integral. +// Igam(a,x) = (1/ Γ(a)) \int_0^x e^{-t} t^{a-1} dt +// The input argument a must be positive and x must be non-negative or Igam +// will panic. +func Igam(a, x float64) float64 { + // The integral is evaluated by either a power series or continued fraction + // expansion, depending on the relative values of a and x. + // Sources: + // [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov + // [2] Maddock et. al., "Incomplete Gamma Functions", + // http://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html + + // Check zero integration limit first + if x == 0 { + return 0 + } + + if x < 0 || a <= 0 { + panic(badParamOutOfBounds) + } + + // Asymptotic regime where a ~ x; see [2]. + absxmaA := math.Abs(x-a) / a + if (igamSmall < a && a < igamLarge && absxmaA < igamSmallRatio) || + (igamLarge < a && absxmaA < igamLargeRatio/math.Sqrt(a)) { + return asymptoticSeries(a, x, igam) + } + + if x > 1 && x > a { + return 1 - IgamC(a, x) + } + + return igamSeries(a, x) +} + +// IgamC computes the complemented incomplete Gamma integral. +// IgamC(a,x) = 1 - Igam(a,x) +// = (1/ Γ(a)) \int_0^\infty e^{-t} t^{a-1} dt +// The input argument a must be positive and x must be non-negative or +// IgamC will panic. +func IgamC(a, x float64) float64 { + // The integral is evaluated by either a power series or continued fraction + // expansion, depending on the relative values of a and x. + // Sources: + // [1] "The Digital Library of Mathematical Functions", dlmf.nist.gov + // [2] Maddock et. al., "Incomplete Gamma Functions", + // http://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/sf_gamma/igamma.html + + switch { + case x < 0, a <= 0: + panic(badParamOutOfBounds) + case x == 0: + return 1 + case math.IsInf(x, 0): + return 0 + } + + // Asymptotic regime where a ~ x; see [2]. + absxmaA := math.Abs(x-a) / a + if (igamSmall < a && a < igamLarge && absxmaA < igamSmallRatio) || + (igamLarge < a && absxmaA < igamLargeRatio/math.Sqrt(a)) { + return asymptoticSeries(a, x, igamC) + } + + // Everywhere else; see [2]. + if x > 1.1 { + if x < a { + return 1 - igamSeries(a, x) + } + return igamCContinuedFraction(a, x) + } else if x <= 0.5 { + if -0.4/math.Log(x) < a { + return 1 - igamSeries(a, x) + } + return igamCSeries(a, x) + } + + if x*1.1 < a { + return 1 - igamSeries(a, x) + } + return igamCSeries(a, x) +} + +// igamFac computes +// x^a * e^{-x} / Γ(a) +// corrected from (15) and (16) in [2] by replacing +// e^{x - a} +// with +// e^{a - x} +func igamFac(a, x float64) float64 { + if math.Abs(a-x) > 0.4*math.Abs(a) { + ax := a*math.Log(x) - x - lgam(a) + return math.Exp(ax) + } + + fac := a + lanczosG - 0.5 + res := math.Sqrt(fac/math.Exp(1)) / lanczosSumExpgScaled(a) + + if a < 200 && x < 200 { + res *= math.Exp(a-x) * math.Pow(x/fac, a) + } else { + num := x - a - lanczosG + 0.5 + res *= math.Exp(a*log1pmx(num/fac) + x*(0.5-lanczosG)/fac) + } + + return res +} + +// igamCContinuedFraction computes IgamC using DLMF 8.9.2. +func igamCContinuedFraction(a, x float64) float64 { + ax := igamFac(a, x) + if ax == 0 { + return 0 + } + + // Continued fraction + y := 1 - a + z := x + y + 1 + c := 0.0 + pkm2 := 1.0 + qkm2 := x + pkm1 := x + 1.0 + qkm1 := z * x + ans := pkm1 / qkm1 + + for i := 0; i < maxIter; i++ { + c += 1.0 + y += 1.0 + z += 2.0 + yc := y * c + pk := pkm1*z - pkm2*yc + qk := qkm1*z - qkm2*yc + var t float64 + if qk != 0 { + r := pk / qk + t = math.Abs((ans - r) / r) + ans = r + } else { + t = 1.0 + } + pkm2 = pkm1 + pkm1 = pk + qkm2 = qkm1 + qkm1 = qk + if math.Abs(pk) > big { + pkm2 *= biginv + pkm1 *= biginv + qkm2 *= biginv + qkm1 *= biginv + } + if t <= machEp { + break + } + } + + return ans * ax +} + +// igamSeries computes Igam using DLMF 8.11.4. +func igamSeries(a, x float64) float64 { + ax := igamFac(a, x) + if ax == 0 { + return 0 + } + + // Power series + r := a + c := 1.0 + ans := 1.0 + + for i := 0; i < maxIter; i++ { + r += 1.0 + c *= x / r + ans += c + if c <= machEp*ans { + break + } + } + + return ans * ax / a +} + +// igamCSeries computes IgamC using DLMF 8.7.3. This is related to the series +// in igamSeries but extra care is taken to avoid cancellation. +func igamCSeries(a, x float64) float64 { + fac := 1.0 + sum := 0.0 + + for n := 1; n < maxIter; n++ { + fac *= -x / float64(n) + term := fac / (a + float64(n)) + sum += term + if math.Abs(term) <= machEp*math.Abs(sum) { + break + } + } + + logx := math.Log(x) + term := -expm1(a*logx - lgam1p(a)) + return term - math.Exp(a*logx-lgam(a))*sum +} + +// asymptoticSeries computes Igam/IgamC using DLMF 8.12.3/8.12.4. +func asymptoticSeries(a, x float64, fun int) float64 { + maxpow := 0 + lambda := x / a + sigma := (x - a) / a + absoldterm := math.MaxFloat64 + etapow := [igamDimN]float64{1} + sum := 0.0 + afac := 1.0 + + var sgn float64 + if fun == igam { + sgn = -1 + } else { + sgn = 1 + } + + var eta float64 + if lambda > 1 { + eta = math.Sqrt(-2 * log1pmx(sigma)) + } else if lambda < 1 { + eta = -math.Sqrt(-2 * log1pmx(sigma)) + } else { + eta = 0 + } + res := 0.5 * math.Erfc(sgn*eta*math.Sqrt(a/2)) + + for k := 0; k < igamDimK; k++ { + ck := igamCoefs[k][0] + for n := 1; n < igamDimN; n++ { + if n > maxpow { + etapow[n] = eta * etapow[n-1] + maxpow++ + } + ckterm := igamCoefs[k][n] * etapow[n] + ck += ckterm + if math.Abs(ckterm) < machEp*math.Abs(ck) { + break + } + } + term := ck * afac + absterm := math.Abs(term) + if absterm > absoldterm { + break + } + sum += term + if absterm < machEp*math.Abs(sum) { + break + } + absoldterm = absterm + afac /= a + } + res += sgn * math.Exp(-0.5*a*eta*eta) * sum / math.Sqrt(2*math.Pi*a) + + return res +} diff --git a/internal/cephes/igami.go b/internal/cephes/igami.go new file mode 100644 index 00000000..5a8d9a91 --- /dev/null +++ b/internal/cephes/igami.go @@ -0,0 +1,153 @@ +// Derived from SciPy's special/cephes/igami.c +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c +// Made freely available by Stephen L. Moshier without support or guarantee. + +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Copyright ©1984, ©1987, ©1995 by Stephen L. Moshier +// Portions Copyright ©2017 The gonum Authors. All rights reserved. + +package cephes + +import "math" + +// IgamI computes the inverse of the incomplete Gamma function. That is, it +// returns the x such that: +// IgamC(a, x) = p +// The input argument a must be positive and p must be between 0 and 1 +// inclusive or IgamI will panic. IgamI should return a positive number, but +// can return 0 even with non-zero y due to underflow. +func IgamI(a, p float64) float64 { + // Bound the solution + x0 := math.MaxFloat64 + yl := 0.0 + x1 := 0.0 + yh := 1.0 + dithresh := 5.0 * machEp + + if p < 0 || p > 1 || a <= 0 { + panic(badParamOutOfBounds) + } + + if p == 0 { + return math.Inf(1) + } + + if p == 1 { + return 0.0 + } + + // Starting with the approximate value + // x = a y^3 + // where + // y = 1 - d - ndtri(p) sqrt(d) + // and + // d = 1/9a + // the routine performs up to 10 Newton iterations to find the root of + // IgamC(a, x) - p = 0 + d := 1.0 / (9.0 * a) + y := 1.0 - d - Ndtri(p)*math.Sqrt(d) + x := a * y * y * y + + lgm := lgam(a) + + for i := 0; i < 10; i++ { + if x > x0 || x < x1 { + break + } + + y = IgamC(a, x) + + if y < yl || y > yh { + break + } + + if y < p { + x0 = x + yl = y + } else { + x1 = x + yh = y + } + + // Compute the derivative of the function at this point + d = (a-1)*math.Log(x) - x - lgm + if d < -maxLog { + break + } + d = -math.Exp(d) + + // Compute the step to the next approximation of x + d = (y - p) / d + if math.Abs(d/x) < machEp { + return x + } + x = x - d + } + + d = 0.0625 + if x0 == math.MaxFloat64 { + if x <= 0 { + x = 1 + } + for x0 == math.MaxFloat64 { + x = (1 + d) * x + y = IgamC(a, x) + if y < p { + x0 = x + yl = y + break + } + d = d + d + } + } + + d = 0.5 + dir := 0 + for i := 0; i < 400; i++ { + x = x1 + d*(x0-x1) + y = IgamC(a, x) + + lgm = (x0 - x1) / (x1 + x0) + if math.Abs(lgm) < dithresh { + break + } + + lgm = (y - p) / p + if math.Abs(lgm) < dithresh { + break + } + + if x <= 0 { + break + } + + if y >= p { + x1 = x + yh = y + if dir < 0 { + dir = 0 + d = 0.5 + } else if dir > 1 { + d = 0.5*d + 0.5 + } else { + d = (p - yl) / (yh - yl) + } + dir++ + } else { + x0 = x + yl = y + if dir > 0 { + dir = 0 + d = 0.5 + } else if dir < -1 { + d = 0.5 * d + } else { + d = (p - yl) / (yh - yl) + } + dir-- + } + } + + return x +} diff --git a/internal/cephes/lanczos.go b/internal/cephes/lanczos.go new file mode 100644 index 00000000..c32909b5 --- /dev/null +++ b/internal/cephes/lanczos.go @@ -0,0 +1,153 @@ +// Derived from SciPy's special/cephes/lanczos.c +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/lanczos.c + +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Copyright ©2006 John Maddock +// Portions Copyright ©2003 Boost +// Portions Copyright ©2016 The gonum Authors. All rights reserved. + +package cephes + +// Optimal values for G for each N are taken from +// http://web.mala.bc.ca/pughg/phdThesis/phdThesis.pdf, +// as are the theoretical error bounds. + +// Constants calculated using the method described by Godfrey +// http://my.fit.edu/~gabdo/gamma.txt and elaborated by Toth at +// http://www.rskey.org/gamma.htm using NTL::RR at 1000 bit precision. + +var lanczosNum = [...]float64{ + 2.506628274631000270164908177133837338626, + 210.8242777515793458725097339207133627117, + 8071.672002365816210638002902272250613822, + 186056.2653952234950402949897160456992822, + 2876370.628935372441225409051620849613599, + 31426415.58540019438061423162831820536287, + 248874557.8620541565114603864132294232163, + 1439720407.311721673663223072794912393972, + 6039542586.35202800506429164430729792107, + 17921034426.03720969991975575445893111267, + 35711959237.35566804944018545154716670596, + 42919803642.64909876895789904700198885093, + 23531376880.41075968857200767445163675473, +} + +var lanczosDenom = [...]float64{ + 1, + 66, + 1925, + 32670, + 357423, + 2637558, + 13339535, + 45995730, + 105258076, + 150917976, + 120543840, + 39916800, + 0, +} + +var lanczosSumExpgScaledNum = [...]float64{ + 0.006061842346248906525783753964555936883222, + 0.5098416655656676188125178644804694509993, + 19.51992788247617482847860966235652136208, + 449.9445569063168119446858607650988409623, + 6955.999602515376140356310115515198987526, + 75999.29304014542649875303443598909137092, + 601859.6171681098786670226533699352302507, + 3481712.15498064590882071018964774556468, + 14605578.08768506808414169982791359218571, + 43338889.32467613834773723740590533316085, + 86363131.28813859145546927288977868422342, + 103794043.1163445451906271053616070238554, + 56906521.91347156388090791033559122686859, +} + +var lanczosSumExpgScaledDenom = [...]float64{ + 1, + 66, + 1925, + 32670, + 357423, + 2637558, + 13339535, + 45995730, + 105258076, + 150917976, + 120543840, + 39916800, + 0, +} + +var lanczosSumNear1D = [...]float64{ + 0.3394643171893132535170101292240837927725e-9, + -0.2499505151487868335680273909354071938387e-8, + 0.8690926181038057039526127422002498960172e-8, + -0.1933117898880828348692541394841204288047e-7, + 0.3075580174791348492737947340039992829546e-7, + -0.2752907702903126466004207345038327818713e-7, + -0.1515973019871092388943437623825208095123e-5, + 0.004785200610085071473880915854204301886437, + -0.1993758927614728757314233026257810172008, + 1.483082862367253753040442933770164111678, + -3.327150580651624233553677113928873034916, + 2.208709979316623790862569924861841433016, +} + +var lanczosSumNear2D = [...]float64{ + 0.1009141566987569892221439918230042368112e-8, + -0.7430396708998719707642735577238449585822e-8, + 0.2583592566524439230844378948704262291927e-7, + -0.5746670642147041587497159649318454348117e-7, + 0.9142922068165324132060550591210267992072e-7, + -0.8183698410724358930823737982119474130069e-7, + -0.4506604409707170077136555010018549819192e-5, + 0.01422519127192419234315002746252160965831, + -0.5926941084905061794445733628891024027949, + 4.408830289125943377923077727900630927902, + -9.8907772644920670589288081640128194231, + 6.565936202082889535528455955485877361223, +} + +const lanczosG = 6.024680040776729583740234375 + +func lanczosSum(x float64) float64 { + return ratevl(x, + lanczosNum[:], + len(lanczosNum)-1, + lanczosDenom[:], + len(lanczosDenom)-1) +} + +func lanczosSumExpgScaled(x float64) float64 { + return ratevl(x, + lanczosSumExpgScaledNum[:], + len(lanczosSumExpgScaledNum)-1, + lanczosSumExpgScaledDenom[:], + len(lanczosSumExpgScaledDenom)-1) +} + +func lanczosSumNear1(dx float64) float64 { + var result float64 + + for i, val := range lanczosSumNear1D { + k := float64(i + 1) + result += (-val * dx) / (k*dx + k*k) + } + + return result +} + +func lanczosSumNear2(dx float64) float64 { + var result float64 + x := dx + 2 + + for i, val := range lanczosSumNear2D { + k := float64(i + 1) + result += (-val * dx) / (x + k*x + k*k - 1) + } + + return result +} diff --git a/internal/cephes/polevl.go b/internal/cephes/polevl.go index 0aa98436..5cbd8cbb 100644 --- a/internal/cephes/polevl.go +++ b/internal/cephes/polevl.go @@ -1,15 +1,16 @@ -// Copyright ©2016 The gonum Authors. All rights reserved. +// Derived from SciPy's special/cephes/polevl.h +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/polevl.h +// Made freely available by Stephen L. Moshier without support or guarantee. + // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. - -/* - * Cephes Math Library Release 2.1: December, 1988 - * Copyright 1984, 1987, 1988 by Stephen L. Moshier - * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 - */ +// Copyright ©1984, ©1987, ©1988 by Stephen L. Moshier +// Portions Copyright ©2016 The gonum Authors. All rights reserved. package cephes +import "math" + // polevl evaluates a polynomial of degree N // y = c_0 + c_1 x_1 + c_2 x_2^2 ... // where the coefficients are stored in reverse order, i.e. coef[0] = c_n and @@ -31,3 +32,51 @@ func p1evl(x float64, coef []float64, n int) float64 { } return ans } + +// ratevl evaluates a rational function +func ratevl(x float64, num []float64, m int, denom []float64, n int) float64 { + // Source: Holin et. al., "Polynomial and Rational Function Evaluation", + // http://www.boost.org/doc/libs/1_61_0/libs/math/doc/html/math_toolkit/roots/rational.html + absx := math.Abs(x) + + var dir, idx int + var y float64 + if absx > 1 { + // Evaluate as a polynomial in 1/x + dir = -1 + idx = m + y = 1 / x + } else { + dir = 1 + idx = 0 + y = x + } + + // Evaluate the numerator + numAns := num[idx] + idx += dir + for i := 0; i < m; i++ { + numAns = numAns*y + num[idx] + idx += dir + } + + // Evaluate the denominator + if absx > 1 { + idx = n + } else { + idx = 0 + } + + denomAns := denom[idx] + idx += dir + for i := 0; i < n; i++ { + denomAns = denomAns*y + denom[idx] + idx += dir + } + + if absx > 1 { + pow := float64(n - m) + return math.Pow(x, pow) * numAns / denomAns + } + return numAns / denomAns +} diff --git a/internal/cephes/unity.go b/internal/cephes/unity.go new file mode 100644 index 00000000..1333dd1d --- /dev/null +++ b/internal/cephes/unity.go @@ -0,0 +1,170 @@ +// Derived from SciPy's special/cephes/unity.c +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/unity.c +// Made freely available by Stephen L. Moshier without support or guarantee. + +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Copyright ©1984, ©1996 by Stephen L. Moshier +// Portions Copyright ©2016 The gonum Authors. All rights reserved. + +package cephes + +import "math" + +// Relative error approximations for function arguments near unity. +// log1p(x) = log(1+x) +// expm1(x) = exp(x) - 1 +// cosm1(x) = cos(x) - 1 +// lgam1p(x) = lgam(1+x) + +const ( + invSqrt2 = 1 / math.Sqrt2 + pi4 = math.Pi / 4 + euler = 0.577215664901532860606512090082402431 // Euler constant +) + +// Coefficients for +// log(1+x) = x - \frac{x^2}{2} + \frac{x^3 lP(x)}{lQ(x)} +// for +// \frac{1}{\sqrt{2}} <= x < \sqrt{2} +// Theoretical peak relative error = 2.32e-20 +var lP = [...]float64{ + 4.5270000862445199635215E-5, + 4.9854102823193375972212E-1, + 6.5787325942061044846969E0, + 2.9911919328553073277375E1, + 6.0949667980987787057556E1, + 5.7112963590585538103336E1, + 2.0039553499201281259648E1, +} + +var lQ = [...]float64{ + 1.5062909083469192043167E1, + 8.3047565967967209469434E1, + 2.2176239823732856465394E2, + 3.0909872225312059774938E2, + 2.1642788614495947685003E2, + 6.0118660497603843919306E1, +} + +// log1p computes +// log(1 + x) +func log1p(x float64) float64 { + z := 1 + x + if z < invSqrt2 || z > math.Sqrt2 { + return math.Log(z) + } + z = x * x + z = -0.5*z + x*(z*polevl(x, lP[:], 6)/p1evl(x, lQ[:], 6)) + return x + z +} + +// log1pmx computes +// log(1 + x) - x +func log1pmx(x float64) float64 { + if math.Abs(x) < 0.5 { + xfac := x + res := 0.0 + + var term float64 + for n := 2; n < maxIter; n++ { + xfac *= -x + term = xfac / float64(n) + res += term + if math.Abs(term) < machEp*math.Abs(res) { + break + } + } + return res + } + return log1p(x) - x +} + +// Coefficients for +// e^x = 1 + \frac{2x eP(x^2)}{eQ(x^2) - eP(x^2)} +// for +// -0.5 <= x <= 0.5 +var eP = [...]float64{ + 1.2617719307481059087798E-4, + 3.0299440770744196129956E-2, + 9.9999999999999999991025E-1, +} + +var eQ = [...]float64{ + 3.0019850513866445504159E-6, + 2.5244834034968410419224E-3, + 2.2726554820815502876593E-1, + 2.0000000000000000000897E0, +} + +// expm1 computes +// expm1(x) = e^x - 1 +func expm1(x float64) float64 { + if math.IsInf(x, 0) { + if math.IsNaN(x) || x > 0 { + return x + } + return -1 + } + if x < -0.5 || x > 0.5 { + return math.Exp(x) - 1 + } + xx := x * x + r := x * polevl(xx, eP[:], 2) + r = r / (polevl(xx, eQ[:], 3) - r) + return r + r +} + +var coscof = [...]float64{ + 4.7377507964246204691685E-14, + -1.1470284843425359765671E-11, + 2.0876754287081521758361E-9, + -2.7557319214999787979814E-7, + 2.4801587301570552304991E-5, + -1.3888888888888872993737E-3, + 4.1666666666666666609054E-2, +} + +// cosm1 computes +// cosm1(x) = cos(x) - 1 +func cosm1(x float64) float64 { + if x < -pi4 || x > pi4 { + return math.Cos(x) - 1 + } + xx := x * x + xx = -0.5*xx + xx*xx*polevl(xx, coscof[:], 6) + return xx +} + +// lgam1pTayler computes +// lgam(x + 1) +//around x = 0 using its Taylor series. +func lgam1pTaylor(x float64) float64 { + if x == 0 { + return 0 + } + res := -euler * x + xfac := -x + for n := 2; n < 42; n++ { + nf := float64(n) + xfac *= -x + coeff := Zeta(nf, 1) * xfac / nf + res += coeff + if math.Abs(coeff) < machEp*math.Abs(res) { + break + } + } + + return res +} + +// lgam1p computes +// lgam(x + 1) +func lgam1p(x float64) float64 { + if math.Abs(x) <= 0.5 { + return lgam1pTaylor(x) + } else if math.Abs(x-1) < 0.5 { + return math.Log(x) + lgam1pTaylor(x-1) + } + return lgam(x + 1) +} diff --git a/internal/cephes/zeta.go b/internal/cephes/zeta.go new file mode 100644 index 00000000..14a41b6f --- /dev/null +++ b/internal/cephes/zeta.go @@ -0,0 +1,110 @@ +// Derived from SciPy's special/cephes/zeta.c +// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/zeta.c +// Made freely available by Stephen L. Moshier without support or guarantee. + +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Copyright ©1984, ©1987 by Stephen L. Moshier +// Portions Copyright ©2016 The gonum Authors. All rights reserved. + +package cephes + +import "math" + +// zetaCoegs are the expansion coefficients for Euler-Maclaurin summation +// formula: +// \frac{(2k)!}{B_{2k}} +// where +// B_{2k} +// are Bernoulli numbers. +var zetaCoefs = [...]float64{ + 12.0, + -720.0, + 30240.0, + -1209600.0, + 47900160.0, + -1.307674368e12 / 691, + 7.47242496e10, + -1.067062284288e16 / 3617, + 5.109094217170944e18 / 43867, + -8.028576626982912e20 / 174611, + 1.5511210043330985984e23 / 854513, + -1.6938241367317436694528e27 / 236364091, +} + +// Zeta computes the Riemann zeta function of two arguments. +// Zeta(x,q) = \sum_{k=0}^{\infty} (k+q)^{-x} +// Note that Zeta returns +Inf if x is 1 and will panic if x is less than 1, +// q is either zero or a negative integer, or q is negative and x is not an +// integer. +// +// Note that: +// zeta(x,1) = zetac(x) + 1 +func Zeta(x, q float64) float64 { + // REFERENCE: Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, Series, + // and Products, p. 1073; Academic Press, 1980. + if x == 1 { + return math.Inf(1) + } + + if x < 1 { + panic(badParamOutOfBounds) + } + + if q <= 0 { + if q == math.Floor(q) { + panic(badParamFunctionSingularity) + } + if x != math.Floor(x) { + panic(badParamOutOfBounds) // Because q^-x not defined + } + } + + // Asymptotic expansion: http://dlmf.nist.gov/25.11#E43 + if q > 1e8 { + return (1/(x-1) + 1/(2*q)) * math.Pow(q, 1-x) + } + + // The Euler-Maclaurin summation formula is used to obtain the expansion: + // Zeta(x,q) = \sum_{k=1}^n (k+q)^{-x} + \frac{(n+q)^{1-x}}{x-1} - \frac{1}{2(n+q)^x} + \sum_{j=1}^{\infty} \frac{B_{2j}x(x+1)...(x+2j)}{(2j)! (n+q)^{x+2j+1}} + // where + // B_{2j} + // are Bernoulli numbers. + // Permit negative q but continue sum until n+q > 9. This case should be + // handled by a reflection formula. If q<0 and x is an integer, there is a + // relation to the polyGamma function. + s := math.Pow(q, -x) + a := q + i := 0 + b := 0.0 + for i < 9 || a <= 9 { + i++ + a += 1.0 + b = math.Pow(a, -x) + s += b + if math.Abs(b/s) < machEp { + return s + } + } + + w := a + s += b * w / (x - 1) + s -= 0.5 * b + a = 1.0 + k := 0.0 + for _, coef := range zetaCoefs { + a *= x + k + b /= w + t := a * b / coef + s = s + t + t = math.Abs(t / s) + if t < machEp { + return s + } + k += 1.0 + a *= x + k + b /= w + k += 1.0 + } + return s +} diff --git a/roots.go b/roots.go new file mode 100644 index 00000000..a5cd329d --- /dev/null +++ b/roots.go @@ -0,0 +1,178 @@ +// Derived from SciPy's special/c_misc/fsolve.c and special/c_misc/misc.h +// https://github.com/scipy/scipy/blob/master/scipy/special/c_misc/fsolve.c +// https://github.com/scipy/scipy/blob/master/scipy/special/c_misc/misc.h + +// Copyright ©2017 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import "math" + +type objectiveFunc func(float64, []float64) float64 + +type fSolveResult uint8 + +const ( + // An exact solution was found, in which case the first point on the + // interval is the value + fSolveExact fSolveResult = iota + 1 + // Interval width is less than the tolerance + fSolveConverged + // Root-finding didn't converge in a set number of iterations + fSolveMaxIterations +) + +const ( + machEp = 1.0 / (1 << 53) +) + +// falsePosition uses a combination of bisection and false position to find a +// root of a function within a given interval. This is guaranteed to converge, +// and always keeps a bounding interval, unlike Newton's method. Inputs are: +// x1, x2: initial bounding interval +// f1, f2: value of f() at x1 and x2 +// absErr, relErr: absolute and relative errors on the bounding interval +// bisectTil: if > 0.0, perform bisection until the width of the bounding +// interval is less than this +// f, fExtra: function to find root of is f(x, fExtra) +// Returns: +// result: whether an exact root was found, the process converged to a +// bounding interval small than the required error, or the max number +// of iterations was hit +// bestX: best root approximation +// bestF: function value at bestX +// errEst: error estimation +func falsePosition(x1, x2, f1, f2, absErr, relErr, bisectTil float64, f objectiveFunc, fExtra []float64) (fSolveResult, float64, float64, float64) { + // The false position steps are either unmodified, or modified with the + // Anderson-Bjorck method as appropiate. Theoretically, this has a "speed of + // convergence" of 1.7 (bisection is 1, Newton is 2). + // Note that this routine was designed initially to work with gammaincinv, so + // it may not be tuned right for other problems. Don't use it blindly. + + if f1*f2 >= 0 { + panic("Initial interval is not a bounding interval") + } + + const ( + maxIterations = 100 + bisectIter = 4 + bisectWidth = 4.0 + ) + + const ( + bisect = iota + 1 + falseP + ) + + var state uint8 + if bisectTil > 0 { + state = bisect + } else { + state = falseP + } + + gamma := 1.0 + + w := math.Abs(x2 - x1) + lastBisectWidth := w + + var nFalseP int + var x3, f3, bestX, bestF float64 + for i := 0; i < maxIterations; i++ { + switch state { + case bisect: + x3 = 0.5 * (x1 + x2) + if x3 == x1 || x3 == x2 { + // i.e., x1 and x2 are successive floating-point numbers + bestX = x3 + if x3 == x1 { + bestF = f1 + } else { + bestF = f2 + } + return fSolveConverged, bestX, bestF, w + } + + f3 = f(x3, fExtra) + if f3 == 0 { + return fSolveExact, x3, f3, w + } + + if f3*f2 < 0 { + x1 = x2 + f1 = f2 + } + x2 = x3 + f2 = f3 + w = math.Abs(x2 - x1) + lastBisectWidth = w + if bisectTil > 0 { + if w < bisectTil { + bisectTil = -1.0 + gamma = 1.0 + nFalseP = 0 + state = falseP + } + } else { + gamma = 1.0 + nFalseP = 0 + state = falseP + } + case falseP: + s12 := (f2 - gamma*f1) / (x2 - x1) + x3 = x2 - f2/s12 + f3 = f(x3, fExtra) + if f3 == 0 { + return fSolveExact, x3, f3, w + } + + nFalseP++ + if f3*f2 < 0 { + gamma = 1.0 + x1 = x2 + f1 = f2 + } else { + // Anderson-Bjorck method + g := 1.0 - f3/f2 + if g <= 0 { + g = 0.5 + } + gamma *= g + } + x2 = x3 + f2 = f3 + w = math.Abs(x2 - x1) + + // Sanity check. For every 4 false position checks, see if we really are + // decreasing the interval by comparing to what bisection would have + // achieved (or, rather, a bit more lenient than that -- interval + // decreased by 4 instead of by 16, as the fp could be decreasing gamma + // for a bit). Note that this should guarantee convergence, as it makes + // sure that we always end up decreasing the interval width with a + // bisection. + if nFalseP > bisectIter { + if w*bisectWidth > lastBisectWidth { + state = bisect + } + nFalseP = 0 + lastBisectWidth = w + } + } + + tol := absErr + relErr*math.Max(math.Max(math.Abs(x1), math.Abs(x2)), 1.0) + if w <= tol { + if math.Abs(f1) < math.Abs(f2) { + bestX = x1 + bestF = f1 + } else { + bestX = x2 + bestF = f2 + } + return fSolveConverged, bestX, bestF, w + } + } + + return fSolveMaxIterations, x3, f3, w +} diff --git a/zeta.go b/zeta.go new file mode 100644 index 00000000..e1f2d2dc --- /dev/null +++ b/zeta.go @@ -0,0 +1,20 @@ +// Copyright ©2016 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import "github.com/gonum/mathext/internal/cephes" + +// Zeta computes the Riemann zeta function of two arguments. +// Zeta(x,q) = \sum_{k=0}^{\infty} (k+q)^{-x} +// Note that Zeta returns +Inf if x is 1 and will panic if x is less than 1, +// q is either zero or a negative integer, or q is negative and x is not an +// integer. +// +// See http://mathworld.wolfram.com/HurwitzZetaFunction.html +// or https://en.wikipedia.org/wiki/Multiple_zeta_function#Two_parameters_case +// for more detailed information. +func Zeta(x, q float64) float64 { + return cephes.Zeta(x, q) +} diff --git a/zeta_test.go b/zeta_test.go new file mode 100644 index 00000000..e415818f --- /dev/null +++ b/zeta_test.go @@ -0,0 +1,42 @@ +// Copyright ©2016 The gonum Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package mathext + +import ( + "math" + "testing" +) + +func TestZeta(t *testing.T) { + for i, test := range []struct { + x, q, want float64 + }{ + // Results computed using scipy.special.zeta + {1, 1, math.Inf(1)}, + {1.00001, 0.5, 100001.96352290553}, + {1.0001, 25, 9996.8017690244506}, + {1.001, 1, 1000.5772884760117}, + {1.01, 10, 97.773405639173305}, + {1.5, 2, 1.6123753486854886}, + {1.5, 20, 0.45287361712938717}, + {2, -0.7, 14.28618087263834}, + {2.5, 0.5, 6.2471106345688137}, + {5, 2.5, 0.013073166646113805}, + {7.5, 5, 7.9463377443314306e-06}, + {10, -0.5, 2048.0174503557578}, + {10, 0.5, 1024.0174503557578}, + {10, 7.5, 2.5578265694201971e-9}, + {12, 2.5, 1.7089167198843551e-5}, + {17, 0.5, 131072.00101513157}, + {20, -2.5, 2097152.0006014798}, + {20, 0.75, 315.3368689825316}, + {25, 0.25, 1125899906842624.0}, + {30, 1, 1.0000000009313275}, + } { + if got := Zeta(test.x, test.q); math.Abs(got-test.want) > 1e-10 { + t.Errorf("test %d Zeta(%g, %g) failed: got %g want %g", i, test.x, test.q, got, test.want) + } + } +}