Integrate BACKBEAT SDK and resolve KACHING license validation
Major integrations and fixes: - Added BACKBEAT SDK integration for P2P operation timing - Implemented beat-aware status tracking for distributed operations - Added Docker secrets support for secure license management - Resolved KACHING license validation via HTTPS/TLS - Updated docker-compose configuration for clean stack deployment - Disabled rollback policies to prevent deployment failures - Added license credential storage (CHORUS-DEV-MULTI-001) Technical improvements: - BACKBEAT P2P operation tracking with phase management - Enhanced configuration system with file-based secrets - Improved error handling for license validation - Clean separation of KACHING and CHORUS deployment stacks 🤖 Generated with [Claude Code](https://claude.ai/code) Co-Authored-By: Claude <noreply@anthropic.com>
This commit is contained in:
anthonyrawlins
28
vendor/gonum.org/v1/gonum/mathext/internal/cephes/cephes.go
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vendor/gonum.org/v1/gonum/mathext/internal/cephes/cephes.go
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// Copyright ©2016 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package cephes
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import "math"
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/*
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Additional copyright information:
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Code in this package is adapted from the Cephes library (http://www.netlib.org/cephes/).
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There is no explicit licence on Netlib, but the author has agreed to a BSD release.
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See https://github.com/deepmind/torch-cephes/blob/master/LICENSE.txt and
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https://lists.debian.org/debian-legal/2004/12/msg00295.html
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*/
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const (
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paramOutOfBounds = "cephes: parameter out of bounds"
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errParamFunctionSingularity = "cephes: function singularity"
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)
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const (
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machEp = 1.0 / (1 << 53)
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maxLog = 1024 * math.Ln2
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minLog = -1075 * math.Ln2
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maxIter = 2000
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)
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6
vendor/gonum.org/v1/gonum/mathext/internal/cephes/doc.go
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vendor/gonum.org/v1/gonum/mathext/internal/cephes/doc.go
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// Copyright ©2017 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Package cephes implements functions originally in the Netlib code by Stephen Mosher.
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package cephes // import "gonum.org/v1/gonum/mathext/internal/cephes"
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320
vendor/gonum.org/v1/gonum/mathext/internal/cephes/igam.go
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vendor/gonum.org/v1/gonum/mathext/internal/cephes/igam.go
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// Derived from SciPy's special/cephes/igam.c and special/cephes/igam.h
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// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.c
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// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igam.h
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// Made freely available by Stephen L. Moshier without support or guarantee.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Copyright ©1985, ©1987 by Stephen L. Moshier
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// Portions Copyright ©2016 The Gonum Authors. All rights reserved.
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package cephes
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import "math"
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const (
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igamDimK = 25
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igamDimN = 25
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igam = 1
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igamC = 0
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igamSmall = 20
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igamLarge = 200
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igamSmallRatio = 0.3
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igamLargeRatio = 4.5
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)
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var igamCoefs = [igamDimK][igamDimN]float64{
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{-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},
|
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{-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},
|
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{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},
|
||||
{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},
|
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{-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},
|
||||
{-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},
|
||||
{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},
|
||||
{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},
|
||||
{-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},
|
||||
{-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},
|
||||
{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},
|
||||
{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},
|
||||
{-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},
|
||||
{-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},
|
||||
{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},
|
||||
{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},
|
||||
{-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},
|
||||
{-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},
|
||||
{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},
|
||||
{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},
|
||||
{-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},
|
||||
{-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},
|
||||
{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},
|
||||
{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},
|
||||
{-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(paramOutOfBounds)
|
||||
}
|
||||
|
||||
// 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(paramOutOfBounds)
|
||||
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
|
||||
}
|
||||
155
vendor/gonum.org/v1/gonum/mathext/internal/cephes/igami.go
generated
vendored
Normal file
155
vendor/gonum.org/v1/gonum/mathext/internal/cephes/igami.go
generated
vendored
Normal file
@@ -0,0 +1,155 @@
|
||||
// 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(paramOutOfBounds)
|
||||
}
|
||||
|
||||
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
|
||||
}
|
||||
312
vendor/gonum.org/v1/gonum/mathext/internal/cephes/incbeta.go
generated
vendored
Normal file
312
vendor/gonum.org/v1/gonum/mathext/internal/cephes/incbeta.go
generated
vendored
Normal file
@@ -0,0 +1,312 @@
|
||||
// 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.
|
||||
|
||||
/*
|
||||
* Cephes Math Library, Release 2.3: March, 1995
|
||||
* Copyright 1984, 1995 by Stephen L. Moshier
|
||||
*/
|
||||
|
||||
package cephes
|
||||
|
||||
import (
|
||||
"math"
|
||||
|
||||
"gonum.org/v1/gonum/mathext/internal/gonum"
|
||||
)
|
||||
|
||||
const (
|
||||
maxGam = 171.624376956302725
|
||||
big = 4.503599627370496e15
|
||||
biginv = 2.22044604925031308085e-16
|
||||
)
|
||||
|
||||
// Incbet computes the regularized incomplete beta function.
|
||||
func Incbet(aa, bb, xx float64) float64 {
|
||||
if aa <= 0 || bb <= 0 {
|
||||
panic(paramOutOfBounds)
|
||||
}
|
||||
if xx <= 0 || xx >= 1 {
|
||||
if xx == 0 {
|
||||
return 0
|
||||
}
|
||||
if xx == 1 {
|
||||
return 1
|
||||
}
|
||||
panic(paramOutOfBounds)
|
||||
}
|
||||
|
||||
var flag int
|
||||
if bb*xx <= 1 && xx <= 0.95 {
|
||||
t := pseries(aa, bb, xx)
|
||||
return transformT(t, flag)
|
||||
}
|
||||
|
||||
w := 1 - xx
|
||||
|
||||
// Reverse a and b if x is greater than the mean.
|
||||
var a, b, xc, x float64
|
||||
if xx > aa/(aa+bb) {
|
||||
flag = 1
|
||||
a = bb
|
||||
b = aa
|
||||
xc = xx
|
||||
x = w
|
||||
} else {
|
||||
a = aa
|
||||
b = bb
|
||||
xc = w
|
||||
x = xx
|
||||
}
|
||||
|
||||
if flag == 1 && (b*x) <= 1.0 && x <= 0.95 {
|
||||
t := pseries(a, b, x)
|
||||
return transformT(t, flag)
|
||||
}
|
||||
|
||||
// Choose expansion for better convergence.
|
||||
y := x*(a+b-2.0) - (a - 1.0)
|
||||
if y < 0.0 {
|
||||
w = incbcf(a, b, x)
|
||||
} else {
|
||||
w = incbd(a, b, x) / xc
|
||||
}
|
||||
|
||||
// Multiply w by the factor
|
||||
// x^a * (1-x)^b * Γ(a+b) / (a*Γ(a)*Γ(b))
|
||||
var t float64
|
||||
y = a * math.Log(x)
|
||||
t = b * math.Log(xc)
|
||||
if (a+b) < maxGam && math.Abs(y) < maxLog && math.Abs(t) < maxLog {
|
||||
t = math.Pow(xc, b)
|
||||
t *= math.Pow(x, a)
|
||||
t /= a
|
||||
t *= w
|
||||
t *= 1.0 / gonum.Beta(a, b)
|
||||
return transformT(t, flag)
|
||||
}
|
||||
|
||||
// Resort to logarithms.
|
||||
y += t - gonum.Lbeta(a, b)
|
||||
y += math.Log(w / a)
|
||||
if y < minLog {
|
||||
t = 0.0
|
||||
} else {
|
||||
t = math.Exp(y)
|
||||
}
|
||||
|
||||
return transformT(t, flag)
|
||||
}
|
||||
|
||||
func transformT(t float64, flag int) float64 {
|
||||
if flag == 1 {
|
||||
if t <= machEp {
|
||||
t = 1.0 - machEp
|
||||
} else {
|
||||
t = 1.0 - t
|
||||
}
|
||||
}
|
||||
return t
|
||||
}
|
||||
|
||||
// incbcf returns the incomplete beta integral evaluated by a continued fraction
|
||||
// expansion.
|
||||
func incbcf(a, b, x float64) float64 {
|
||||
var xk, pk, pkm1, pkm2, qk, qkm1, qkm2 float64
|
||||
var k1, k2, k3, k4, k5, k6, k7, k8 float64
|
||||
var r, t, ans, thresh float64
|
||||
var n int
|
||||
|
||||
k1 = a
|
||||
k2 = a + b
|
||||
k3 = a
|
||||
k4 = a + 1.0
|
||||
k5 = 1.0
|
||||
k6 = b - 1.0
|
||||
k7 = k4
|
||||
k8 = a + 2.0
|
||||
|
||||
pkm2 = 0.0
|
||||
qkm2 = 1.0
|
||||
pkm1 = 1.0
|
||||
qkm1 = 1.0
|
||||
ans = 1.0
|
||||
r = 1.0
|
||||
thresh = 3.0 * machEp
|
||||
|
||||
for n = 0; n <= 300; n++ {
|
||||
|
||||
xk = -(x * k1 * k2) / (k3 * k4)
|
||||
pk = pkm1 + pkm2*xk
|
||||
qk = qkm1 + qkm2*xk
|
||||
pkm2 = pkm1
|
||||
pkm1 = pk
|
||||
qkm2 = qkm1
|
||||
qkm1 = qk
|
||||
|
||||
xk = (x * k5 * k6) / (k7 * k8)
|
||||
pk = pkm1 + pkm2*xk
|
||||
qk = qkm1 + qkm2*xk
|
||||
pkm2 = pkm1
|
||||
pkm1 = pk
|
||||
qkm2 = qkm1
|
||||
qkm1 = qk
|
||||
|
||||
if qk != 0 {
|
||||
r = pk / qk
|
||||
}
|
||||
if r != 0 {
|
||||
t = math.Abs((ans - r) / r)
|
||||
ans = r
|
||||
} else {
|
||||
t = 1.0
|
||||
}
|
||||
|
||||
if t < thresh {
|
||||
return ans
|
||||
}
|
||||
|
||||
k1 += 1.0
|
||||
k2 += 1.0
|
||||
k3 += 2.0
|
||||
k4 += 2.0
|
||||
k5 += 1.0
|
||||
k6 -= 1.0
|
||||
k7 += 2.0
|
||||
k8 += 2.0
|
||||
|
||||
if (math.Abs(qk) + math.Abs(pk)) > big {
|
||||
pkm2 *= biginv
|
||||
pkm1 *= biginv
|
||||
qkm2 *= biginv
|
||||
qkm1 *= biginv
|
||||
}
|
||||
if (math.Abs(qk) < biginv) || (math.Abs(pk) < biginv) {
|
||||
pkm2 *= big
|
||||
pkm1 *= big
|
||||
qkm2 *= big
|
||||
qkm1 *= big
|
||||
}
|
||||
}
|
||||
|
||||
return ans
|
||||
}
|
||||
|
||||
// incbd returns the incomplete beta integral evaluated by a continued fraction
|
||||
// expansion.
|
||||
func incbd(a, b, x float64) float64 {
|
||||
var xk, pk, pkm1, pkm2, qk, qkm1, qkm2 float64
|
||||
var k1, k2, k3, k4, k5, k6, k7, k8 float64
|
||||
var r, t, ans, z, thresh float64
|
||||
var n int
|
||||
|
||||
k1 = a
|
||||
k2 = b - 1.0
|
||||
k3 = a
|
||||
k4 = a + 1.0
|
||||
k5 = 1.0
|
||||
k6 = a + b
|
||||
k7 = a + 1.0
|
||||
k8 = a + 2.0
|
||||
|
||||
pkm2 = 0.0
|
||||
qkm2 = 1.0
|
||||
pkm1 = 1.0
|
||||
qkm1 = 1.0
|
||||
z = x / (1.0 - x)
|
||||
ans = 1.0
|
||||
r = 1.0
|
||||
thresh = 3.0 * machEp
|
||||
for n = 0; n <= 300; n++ {
|
||||
|
||||
xk = -(z * k1 * k2) / (k3 * k4)
|
||||
pk = pkm1 + pkm2*xk
|
||||
qk = qkm1 + qkm2*xk
|
||||
pkm2 = pkm1
|
||||
pkm1 = pk
|
||||
qkm2 = qkm1
|
||||
qkm1 = qk
|
||||
|
||||
xk = (z * k5 * k6) / (k7 * k8)
|
||||
pk = pkm1 + pkm2*xk
|
||||
qk = qkm1 + qkm2*xk
|
||||
pkm2 = pkm1
|
||||
pkm1 = pk
|
||||
qkm2 = qkm1
|
||||
qkm1 = qk
|
||||
|
||||
if qk != 0 {
|
||||
r = pk / qk
|
||||
}
|
||||
if r != 0 {
|
||||
t = math.Abs((ans - r) / r)
|
||||
ans = r
|
||||
} else {
|
||||
t = 1.0
|
||||
}
|
||||
|
||||
if t < thresh {
|
||||
return ans
|
||||
}
|
||||
|
||||
k1 += 1.0
|
||||
k2 -= 1.0
|
||||
k3 += 2.0
|
||||
k4 += 2.0
|
||||
k5 += 1.0
|
||||
k6 += 1.0
|
||||
k7 += 2.0
|
||||
k8 += 2.0
|
||||
|
||||
if (math.Abs(qk) + math.Abs(pk)) > big {
|
||||
pkm2 *= biginv
|
||||
pkm1 *= biginv
|
||||
qkm2 *= biginv
|
||||
qkm1 *= biginv
|
||||
}
|
||||
if (math.Abs(qk) < biginv) || (math.Abs(pk) < biginv) {
|
||||
pkm2 *= big
|
||||
pkm1 *= big
|
||||
qkm2 *= big
|
||||
qkm1 *= big
|
||||
}
|
||||
}
|
||||
return ans
|
||||
}
|
||||
|
||||
// pseries returns the incomplete beta integral evaluated by a power series. Use
|
||||
// when b*x is small and x not too close to 1.
|
||||
func pseries(a, b, x float64) float64 {
|
||||
var s, t, u, v, n, t1, z, ai float64
|
||||
ai = 1.0 / a
|
||||
u = (1.0 - b) * x
|
||||
v = u / (a + 1.0)
|
||||
t1 = v
|
||||
t = u
|
||||
n = 2.0
|
||||
s = 0.0
|
||||
z = machEp * ai
|
||||
for math.Abs(v) > z {
|
||||
u = (n - b) * x / n
|
||||
t *= u
|
||||
v = t / (a + n)
|
||||
s += v
|
||||
n += 1.0
|
||||
}
|
||||
s += t1
|
||||
s += ai
|
||||
|
||||
u = a * math.Log(x)
|
||||
if (a+b) < maxGam && math.Abs(u) < maxLog {
|
||||
t = 1.0 / gonum.Beta(a, b)
|
||||
s = s * t * math.Pow(x, a)
|
||||
} else {
|
||||
t = -gonum.Lbeta(a, b) + u + math.Log(s)
|
||||
if t < minLog {
|
||||
s = 0.0
|
||||
} else {
|
||||
s = math.Exp(t)
|
||||
}
|
||||
}
|
||||
return (s)
|
||||
}
|
||||
247
vendor/gonum.org/v1/gonum/mathext/internal/cephes/incbi.go
generated
vendored
Normal file
247
vendor/gonum.org/v1/gonum/mathext/internal/cephes/incbi.go
generated
vendored
Normal file
@@ -0,0 +1,247 @@
|
||||
// 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.
|
||||
|
||||
/*
|
||||
* Cephes Math Library Release 2.4: March,1996
|
||||
* Copyright 1984, 1996 by Stephen L. Moshier
|
||||
*/
|
||||
|
||||
package cephes
|
||||
|
||||
import "math"
|
||||
|
||||
// Incbi computes the inverse of the regularized incomplete beta integral.
|
||||
func Incbi(aa, bb, yy0 float64) float64 {
|
||||
var a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt float64
|
||||
var i, rflg, dir, nflg int
|
||||
|
||||
if yy0 <= 0 {
|
||||
return (0.0)
|
||||
}
|
||||
if yy0 >= 1.0 {
|
||||
return (1.0)
|
||||
}
|
||||
x0 = 0.0
|
||||
yl = 0.0
|
||||
x1 = 1.0
|
||||
yh = 1.0
|
||||
nflg = 0
|
||||
|
||||
if aa <= 1.0 || bb <= 1.0 {
|
||||
dithresh = 1.0e-6
|
||||
rflg = 0
|
||||
a = aa
|
||||
b = bb
|
||||
y0 = yy0
|
||||
x = a / (a + b)
|
||||
y = Incbet(a, b, x)
|
||||
goto ihalve
|
||||
} else {
|
||||
dithresh = 1.0e-4
|
||||
}
|
||||
// Approximation to inverse function
|
||||
yp = -Ndtri(yy0)
|
||||
|
||||
if yy0 > 0.5 {
|
||||
rflg = 1
|
||||
a = bb
|
||||
b = aa
|
||||
y0 = 1.0 - yy0
|
||||
yp = -yp
|
||||
} else {
|
||||
rflg = 0
|
||||
a = aa
|
||||
b = bb
|
||||
y0 = yy0
|
||||
}
|
||||
|
||||
lgm = (yp*yp - 3.0) / 6.0
|
||||
x = 2.0 / (1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0))
|
||||
d = yp*math.Sqrt(x+lgm)/x - (1.0/(2.0*b-1.0)-1.0/(2.0*a-1.0))*(lgm+5.0/6.0-2.0/(3.0*x))
|
||||
d = 2.0 * d
|
||||
if d < minLog {
|
||||
// mtherr("incbi", UNDERFLOW)
|
||||
x = 0
|
||||
goto done
|
||||
}
|
||||
x = a / (a + b*math.Exp(d))
|
||||
y = Incbet(a, b, x)
|
||||
yp = (y - y0) / y0
|
||||
if math.Abs(yp) < 0.2 {
|
||||
goto newt
|
||||
}
|
||||
|
||||
/* Resort to interval halving if not close enough. */
|
||||
ihalve:
|
||||
|
||||
dir = 0
|
||||
di = 0.5
|
||||
for i = 0; i < 100; i++ {
|
||||
if i != 0 {
|
||||
x = x0 + di*(x1-x0)
|
||||
if x == 1.0 {
|
||||
x = 1.0 - machEp
|
||||
}
|
||||
if x == 0.0 {
|
||||
di = 0.5
|
||||
x = x0 + di*(x1-x0)
|
||||
if x == 0.0 {
|
||||
// mtherr("incbi", UNDERFLOW)
|
||||
goto done
|
||||
}
|
||||
}
|
||||
y = Incbet(a, b, x)
|
||||
yp = (x1 - x0) / (x1 + x0)
|
||||
if math.Abs(yp) < dithresh {
|
||||
goto newt
|
||||
}
|
||||
yp = (y - y0) / y0
|
||||
if math.Abs(yp) < dithresh {
|
||||
goto newt
|
||||
}
|
||||
}
|
||||
if y < y0 {
|
||||
x0 = x
|
||||
yl = y
|
||||
if dir < 0 {
|
||||
dir = 0
|
||||
di = 0.5
|
||||
} else if dir > 3 {
|
||||
di = 1.0 - (1.0-di)*(1.0-di)
|
||||
} else if dir > 1 {
|
||||
di = 0.5*di + 0.5
|
||||
} else {
|
||||
di = (y0 - y) / (yh - yl)
|
||||
}
|
||||
dir += 1
|
||||
if x0 > 0.75 {
|
||||
if rflg == 1 {
|
||||
rflg = 0
|
||||
a = aa
|
||||
b = bb
|
||||
y0 = yy0
|
||||
} else {
|
||||
rflg = 1
|
||||
a = bb
|
||||
b = aa
|
||||
y0 = 1.0 - yy0
|
||||
}
|
||||
x = 1.0 - x
|
||||
y = Incbet(a, b, x)
|
||||
x0 = 0.0
|
||||
yl = 0.0
|
||||
x1 = 1.0
|
||||
yh = 1.0
|
||||
goto ihalve
|
||||
}
|
||||
} else {
|
||||
x1 = x
|
||||
if rflg == 1 && x1 < machEp {
|
||||
x = 0.0
|
||||
goto done
|
||||
}
|
||||
yh = y
|
||||
if dir > 0 {
|
||||
dir = 0
|
||||
di = 0.5
|
||||
} else if dir < -3 {
|
||||
di = di * di
|
||||
} else if dir < -1 {
|
||||
di = 0.5 * di
|
||||
} else {
|
||||
di = (y - y0) / (yh - yl)
|
||||
}
|
||||
dir -= 1
|
||||
}
|
||||
}
|
||||
// mtherr("incbi", PLOSS)
|
||||
if x0 >= 1.0 {
|
||||
x = 1.0 - machEp
|
||||
goto done
|
||||
}
|
||||
if x <= 0.0 {
|
||||
// mtherr("incbi", UNDERFLOW)
|
||||
x = 0.0
|
||||
goto done
|
||||
}
|
||||
|
||||
newt:
|
||||
if nflg > 0 {
|
||||
goto done
|
||||
}
|
||||
nflg = 1
|
||||
lgm = lgam(a+b) - lgam(a) - lgam(b)
|
||||
|
||||
for i = 0; i < 8; i++ {
|
||||
/* Compute the function at this point. */
|
||||
if i != 0 {
|
||||
y = Incbet(a, b, x)
|
||||
}
|
||||
if y < yl {
|
||||
x = x0
|
||||
y = yl
|
||||
} else if y > yh {
|
||||
x = x1
|
||||
y = yh
|
||||
} else if y < y0 {
|
||||
x0 = x
|
||||
yl = y
|
||||
} else {
|
||||
x1 = x
|
||||
yh = y
|
||||
}
|
||||
if x == 1.0 || x == 0.0 {
|
||||
break
|
||||
}
|
||||
/* Compute the derivative of the function at this point. */
|
||||
d = (a-1.0)*math.Log(x) + (b-1.0)*math.Log(1.0-x) + lgm
|
||||
if d < minLog {
|
||||
goto done
|
||||
}
|
||||
if d > maxLog {
|
||||
break
|
||||
}
|
||||
d = math.Exp(d)
|
||||
/* Compute the step to the next approximation of x. */
|
||||
d = (y - y0) / d
|
||||
xt = x - d
|
||||
if xt <= x0 {
|
||||
y = (x - x0) / (x1 - x0)
|
||||
xt = x0 + 0.5*y*(x-x0)
|
||||
if xt <= 0.0 {
|
||||
break
|
||||
}
|
||||
}
|
||||
if xt >= x1 {
|
||||
y = (x1 - x) / (x1 - x0)
|
||||
xt = x1 - 0.5*y*(x1-x)
|
||||
if xt >= 1.0 {
|
||||
break
|
||||
}
|
||||
}
|
||||
x = xt
|
||||
if math.Abs(d/x) < 128.0*machEp {
|
||||
goto done
|
||||
}
|
||||
}
|
||||
/* Did not converge. */
|
||||
dithresh = 256.0 * machEp
|
||||
goto ihalve
|
||||
|
||||
done:
|
||||
|
||||
if rflg > 0 {
|
||||
if x <= machEp {
|
||||
x = 1.0 - machEp
|
||||
} else {
|
||||
x = 1.0 - x
|
||||
}
|
||||
}
|
||||
return (x)
|
||||
}
|
||||
|
||||
func lgam(a float64) float64 {
|
||||
lg, _ := math.Lgamma(a)
|
||||
return lg
|
||||
}
|
||||
153
vendor/gonum.org/v1/gonum/mathext/internal/cephes/lanczos.go
generated
vendored
Normal file
153
vendor/gonum.org/v1/gonum/mathext/internal/cephes/lanczos.go
generated
vendored
Normal file
@@ -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
|
||||
}
|
||||
150
vendor/gonum.org/v1/gonum/mathext/internal/cephes/ndtri.go
generated
vendored
Normal file
150
vendor/gonum.org/v1/gonum/mathext/internal/cephes/ndtri.go
generated
vendored
Normal file
@@ -0,0 +1,150 @@
|
||||
// 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.
|
||||
|
||||
/*
|
||||
* Cephes Math Library Release 2.1: January, 1989
|
||||
* Copyright 1984, 1987, 1989 by Stephen L. Moshier
|
||||
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
|
||||
*/
|
||||
|
||||
package cephes
|
||||
|
||||
import "math"
|
||||
|
||||
// TODO(btracey): There is currently an implementation of this functionality
|
||||
// in gonum/stat/distuv. Find out which implementation is better, and rectify
|
||||
// by having distuv call this, or moving this implementation into
|
||||
// gonum/mathext/internal/gonum.
|
||||
|
||||
// math.Sqrt(2*pi)
|
||||
const s2pi = 2.50662827463100050242e0
|
||||
|
||||
// approximation for 0 <= |y - 0.5| <= 3/8
|
||||
var P0 = [5]float64{
|
||||
-5.99633501014107895267e1,
|
||||
9.80010754185999661536e1,
|
||||
-5.66762857469070293439e1,
|
||||
1.39312609387279679503e1,
|
||||
-1.23916583867381258016e0,
|
||||
}
|
||||
|
||||
var Q0 = [8]float64{
|
||||
/* 1.00000000000000000000E0, */
|
||||
1.95448858338141759834e0,
|
||||
4.67627912898881538453e0,
|
||||
8.63602421390890590575e1,
|
||||
-2.25462687854119370527e2,
|
||||
2.00260212380060660359e2,
|
||||
-8.20372256168333339912e1,
|
||||
1.59056225126211695515e1,
|
||||
-1.18331621121330003142e0,
|
||||
}
|
||||
|
||||
// Approximation for interval z = math.Sqrt(-2 log y ) between 2 and 8
|
||||
// i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
|
||||
var P1 = [9]float64{
|
||||
4.05544892305962419923e0,
|
||||
3.15251094599893866154e1,
|
||||
5.71628192246421288162e1,
|
||||
4.40805073893200834700e1,
|
||||
1.46849561928858024014e1,
|
||||
2.18663306850790267539e0,
|
||||
-1.40256079171354495875e-1,
|
||||
-3.50424626827848203418e-2,
|
||||
-8.57456785154685413611e-4,
|
||||
}
|
||||
|
||||
var Q1 = [8]float64{
|
||||
/* 1.00000000000000000000E0, */
|
||||
1.57799883256466749731e1,
|
||||
4.53907635128879210584e1,
|
||||
4.13172038254672030440e1,
|
||||
1.50425385692907503408e1,
|
||||
2.50464946208309415979e0,
|
||||
-1.42182922854787788574e-1,
|
||||
-3.80806407691578277194e-2,
|
||||
-9.33259480895457427372e-4,
|
||||
}
|
||||
|
||||
// Approximation for interval z = math.Sqrt(-2 log y ) between 8 and 64
|
||||
// i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
|
||||
var P2 = [9]float64{
|
||||
3.23774891776946035970e0,
|
||||
6.91522889068984211695e0,
|
||||
3.93881025292474443415e0,
|
||||
1.33303460815807542389e0,
|
||||
2.01485389549179081538e-1,
|
||||
1.23716634817820021358e-2,
|
||||
3.01581553508235416007e-4,
|
||||
2.65806974686737550832e-6,
|
||||
6.23974539184983293730e-9,
|
||||
}
|
||||
|
||||
var Q2 = [8]float64{
|
||||
/* 1.00000000000000000000E0, */
|
||||
6.02427039364742014255e0,
|
||||
3.67983563856160859403e0,
|
||||
1.37702099489081330271e0,
|
||||
2.16236993594496635890e-1,
|
||||
1.34204006088543189037e-2,
|
||||
3.28014464682127739104e-4,
|
||||
2.89247864745380683936e-6,
|
||||
6.79019408009981274425e-9,
|
||||
}
|
||||
|
||||
// Ndtri returns the argument, x, for which the area under the
|
||||
// Gaussian probability density function (integrated from
|
||||
// minus infinity to x) is equal to y.
|
||||
func Ndtri(y0 float64) float64 {
|
||||
// For small arguments 0 < y < exp(-2), the program computes
|
||||
// z = math.Sqrt( -2.0 * math.Log(y) ); then the approximation is
|
||||
// x = z - math.Log(z)/z - (1/z) P(1/z) / Q(1/z).
|
||||
// There are two rational functions P/Q, one for 0 < y < exp(-32)
|
||||
// and the other for y up to exp(-2). For larger arguments,
|
||||
// w = y - 0.5, and x/math.Sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
|
||||
var x, y, z, y2, x0, x1 float64
|
||||
var code int
|
||||
|
||||
if y0 <= 0.0 {
|
||||
if y0 < 0 {
|
||||
panic(paramOutOfBounds)
|
||||
}
|
||||
return math.Inf(-1)
|
||||
}
|
||||
if y0 >= 1.0 {
|
||||
if y0 > 1 {
|
||||
panic(paramOutOfBounds)
|
||||
}
|
||||
return math.Inf(1)
|
||||
}
|
||||
code = 1
|
||||
y = y0
|
||||
if y > (1.0 - 0.13533528323661269189) { /* 0.135... = exp(-2) */
|
||||
y = 1.0 - y
|
||||
code = 0
|
||||
}
|
||||
|
||||
if y > 0.13533528323661269189 {
|
||||
y = y - 0.5
|
||||
y2 = y * y
|
||||
x = y + y*(y2*polevl(y2, P0[:], 4)/p1evl(y2, Q0[:], 8))
|
||||
x = x * s2pi
|
||||
return (x)
|
||||
}
|
||||
|
||||
x = math.Sqrt(-2.0 * math.Log(y))
|
||||
x0 = x - math.Log(x)/x
|
||||
|
||||
z = 1.0 / x
|
||||
if x < 8.0 { /* y > exp(-32) = 1.2664165549e-14 */
|
||||
x1 = z * polevl(z, P1[:], 8) / p1evl(z, Q1[:], 8)
|
||||
} else {
|
||||
x1 = z * polevl(z, P2[:], 8) / p1evl(z, Q2[:], 8)
|
||||
}
|
||||
x = x0 - x1
|
||||
if code != 0 {
|
||||
x = -x
|
||||
}
|
||||
return (x)
|
||||
}
|
||||
84
vendor/gonum.org/v1/gonum/mathext/internal/cephes/polevl.go
generated
vendored
Normal file
84
vendor/gonum.org/v1/gonum/mathext/internal/cephes/polevl.go
generated
vendored
Normal file
@@ -0,0 +1,84 @@
|
||||
// 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.
|
||||
// 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
|
||||
// coef[n] = c_0.
|
||||
func polevl(x float64, coef []float64, n int) float64 {
|
||||
ans := coef[0]
|
||||
for i := 1; i <= n; i++ {
|
||||
ans = ans*x + coef[i]
|
||||
}
|
||||
return ans
|
||||
}
|
||||
|
||||
// p1evl is the same as polevl, except c_n is assumed to be 1 and is not included
|
||||
// in the slice.
|
||||
func p1evl(x float64, coef []float64, n int) float64 {
|
||||
ans := x + coef[0]
|
||||
for i := 1; i <= n-1; i++ {
|
||||
ans = ans*x + coef[i]
|
||||
}
|
||||
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
|
||||
}
|
||||
1
vendor/gonum.org/v1/gonum/mathext/internal/cephes/staticcheck.conf
generated
vendored
Normal file
1
vendor/gonum.org/v1/gonum/mathext/internal/cephes/staticcheck.conf
generated
vendored
Normal file
@@ -0,0 +1 @@
|
||||
checks = []
|
||||
184
vendor/gonum.org/v1/gonum/mathext/internal/cephes/unity.go
generated
vendored
Normal file
184
vendor/gonum.org/v1/gonum/mathext/internal/cephes/unity.go
generated
vendored
Normal file
@@ -0,0 +1,184 @@
|
||||
// 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)
|
||||
}
|
||||
117
vendor/gonum.org/v1/gonum/mathext/internal/cephes/zeta.go
generated
vendored
Normal file
117
vendor/gonum.org/v1/gonum/mathext/internal/cephes/zeta.go
generated
vendored
Normal file
@@ -0,0 +1,117 @@
|
||||
// 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(paramOutOfBounds)
|
||||
}
|
||||
|
||||
if q <= 0 {
|
||||
if q == math.Floor(q) {
|
||||
panic(errParamFunctionSingularity)
|
||||
}
|
||||
if x != math.Floor(x) {
|
||||
panic(paramOutOfBounds) // 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
|
||||
}
|
||||
Reference in New Issue
Block a user