From 3ec482c2be7b961eed46e9518496517be7fe64fd Mon Sep 17 00:00:00 2001 From: Larz White Date: Mon, 18 Jan 2016 23:22:17 -0600 Subject: [PATCH 1/7] Gauss-Legendre Rule Feature - Support for obtaining the Gauss-Legendre abscissas/weights - Integration in 1D and 2D. - Added Unit tests --- src/Numerics/Integration/GaussLegendreRule.cs | 263 ++++++++++++++++++ .../GaussRule/GaussLegendrePointFactory.cs | 200 +++++++++++++ .../Integration/GaussRule/GaussPoint.cs | 61 ++++ src/Numerics/Numerics.csproj | 3 + .../IntegrationTests/IntegrationTest.cs | 102 ++++++- src/UnitTests/UnitTests.csproj | 11 +- 6 files changed, 632 insertions(+), 8 deletions(-) create mode 100644 src/Numerics/Integration/GaussLegendreRule.cs create mode 100644 src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs create mode 100644 src/Numerics/Integration/GaussRule/GaussPoint.cs diff --git a/src/Numerics/Integration/GaussLegendreRule.cs b/src/Numerics/Integration/GaussLegendreRule.cs new file mode 100644 index 00000000..dce9784a --- /dev/null +++ b/src/Numerics/Integration/GaussLegendreRule.cs @@ -0,0 +1,263 @@ +// +// Math.NET Numerics, part of the Math.NET Project +// http://numerics.mathdotnet.com +// http://github.com/mathnet/mathnet-numerics +// http://mathnetnumerics.codeplex.com +// +// Copyright (c) 2009-2016 Math.NET +// +// Permission is hereby granted, free of charge, to any person +// obtaining a copy of this software and associated documentation +// files (the "Software"), to deal in the Software without +// restriction, including without limitation the rights to use, +// copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the +// Software is furnished to do so, subject to the following +// conditions: +// +// The above copyright notice and this permission notice shall be +// included in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, +// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES +// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND +// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT +// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, +// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR +// OTHER DEALINGS IN THE SOFTWARE. +// + +using System; +using MathNet.Numerics.Integration.GaussRule; + +namespace MathNet.Numerics.Integration +{ + /// + /// Approximates a definite integral using an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// + public class GaussLegendreRule + { + private readonly GaussPoint gaussLegendrePoint; + private static GaussPoint nonNegativeGaussLegendrePoint; + + /// + /// Initializes a new instance of the class. + /// + /// Where the interval starts, inclusive and finite. + /// Where the interval stops, inclusive and finite. + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + public GaussLegendreRule(double intervalBegin, double intervalEnd, int order) + { + gaussLegendrePoint = Map(GaussLegendrePointFactory.GetGaussPoint(order), intervalBegin, intervalEnd); + } + + /// + /// Gettter for the ith abscissa. + /// + /// Index of the ith abscissa. + /// The ith abscissa. + public double GetAbscissa(int index) + { + return gaussLegendrePoint.Abscissas[index]; + } + + /// + /// Getter for the ith weight. + /// + /// Index of the ith weight. + /// The ith weight. + public double GetWeight(int index) + { + return gaussLegendrePoint.Weights[index]; + } + + /// + /// Getter for the order. + /// + /// The order, which defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + public int GetOrder() + { + return gaussLegendrePoint.Order; + } + + /// + /// Getter for the InvervalBegin. + /// + /// Where the interval starts, inclusive and finite. + public double GetIntervalBegin() + { + return gaussLegendrePoint.IntervalBegin; + } + + /// + /// Getter for the InvervalEnd. + /// + /// Where the interval stops, inclusive and finite. + public double GetIntervalEnd() + { + return gaussLegendrePoint.IntervalEnd; + } + + /// + /// Getter for the GaussPoint. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. + private static GaussPoint GetGaussPoint(int order) + { + // Try to get the point from the static field first. If it's not there, or it's the wrong order, then generate it on the fly and store it in the static field. + if (nonNegativeGaussLegendrePoint == null || nonNegativeGaussLegendrePoint.Order != order) + { + return GaussLegendrePointFactory.GetGaussPoint(order); + } + + return nonNegativeGaussLegendrePoint; + } + + /// + /// Maps the non-negative abscissas/weights from the interval [-1, 1] to the interval [intervalBegin, intervalEnd]. + /// + /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. + /// Where the interval starts, inclusive and finite. + /// Where the interval stops, inclusive and finite. + /// Object containing the abscissas/weights, order, and intervalBegin/intervalEnd. + private static GaussPoint Map(GaussPoint gaussPoint, double intervalBegin, double intervalEnd) + { + double[] abscissas = new double[gaussPoint.Order]; + double[] weights = new double[gaussPoint.Order]; + + double a = 0.5*(intervalEnd - intervalBegin); + double b = 0.5*(intervalEnd + intervalBegin); + + int m = (gaussPoint.Order + 1) >> 1; + + for (int i = 1; i <= m; i++) + { + int index1 = gaussPoint.Order - i; + int index2 = i - 1; + int index3 = m - i; + + abscissas[index1] = gaussPoint.Abscissas[index3]*a + b; + abscissas[index2] = -gaussPoint.Abscissas[index3]*a + b; + + weights[index1] = gaussPoint.Weights[index3]*a; + weights[index2] = gaussPoint.Weights[index3]*a; + } + + return new GaussPoint(intervalBegin, intervalEnd, gaussPoint.Order, abscissas, weights); + } + + /// + /// Approximates a definite integral using an Nth order Gauss-Legendre rule. + /// + /// The analytic smooth function to integrate. + /// Where the interval starts, exclusive and finite. + /// Where the interval ends, exclusive and finite. + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Approximation of the finite integral in the given interval. + public static double Integrate(Func f, double invervalBegin, double invervalEnd, int order) + { + nonNegativeGaussLegendrePoint = GetGaussPoint(order); + + double sum, ax; + int i; + int m = (order + 1) >> 1; + + double a = 0.5*(invervalEnd - invervalBegin); + double b = 0.5*(invervalEnd + invervalBegin); + + if (order.IsOdd()) + { + sum = nonNegativeGaussLegendrePoint.Weights[0]*f(b); + for (i = 1; i < m; i++) + { + ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + sum += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); + } + } + else + { + sum = 0.0; + for (i = 0; i < m; i++) + { + ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + sum += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); + } + } + + return a*sum; + } + + /// + /// Approximates a 2-dimensional definite integral using an Nth order Gauss-Legendre rule over the rectangle [a,b] x [c,d]. + /// + /// The 2-dimensional analytic smooth function to integrate. + /// Where the interval starts for the first (inside) integral, exclusive and finite. + /// Where the interval ends for the first (inside) integral, exclusive and finite. + /// Where the interval starts for the second (outside) integral, exclusive and finite. + /// /// Where the interval ends for the second (outside) integral, exclusive and finite. + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Approximation of the finite integral in the given interval. + public static double Integrate(Func f, double invervalBeginA, double invervalEndA, double invervalBeginB, double invervalEndB, int order) + { + nonNegativeGaussLegendrePoint = GetGaussPoint(order); + + double ax, cy, sum; + int i, j; + int m = (order + 1) >> 1; + + double a = 0.5*(invervalEndA - invervalBeginA); + double b = 0.5*(invervalEndA + invervalBeginA); + double c = 0.5*(invervalEndB - invervalBeginB); + double d = 0.5*(invervalEndB + invervalBeginB); + + if (order.IsOdd()) + { + sum = nonNegativeGaussLegendrePoint.Weights[0]*nonNegativeGaussLegendrePoint.Weights[0]*f(b, d); + + double t; + for (j = 1, t = 0.0; j < m; j++) + { + cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; + t += nonNegativeGaussLegendrePoint.Weights[j]*(f(b, d + cy) + f(b, d - cy)); + } + + sum += nonNegativeGaussLegendrePoint.Weights[0]*t; + + for (i = 1, t = 0.0; i < m; i++) + { + ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + t += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax, d) + f(b - ax, d)); + } + + sum += nonNegativeGaussLegendrePoint.Weights[0]*t; + + for (i = 1; i < m; i++) + { + ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + for (j = 1; j < m; j++) + { + cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; + sum += nonNegativeGaussLegendrePoint.Weights[i]*nonNegativeGaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); + } + } + } + else + { + sum = 0.0; + for (i = 0; i < m; i++) + { + ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + for (j = 0; j < m; j++) + { + cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; + sum += nonNegativeGaussLegendrePoint.Weights[i]*nonNegativeGaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); + } + } + } + + return c*a*sum; + } + } +} \ No newline at end of file diff --git a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs new file mode 100644 index 00000000..d221915d --- /dev/null +++ b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs @@ -0,0 +1,200 @@ +// +// Math.NET Numerics, part of the Math.NET Project +// http://numerics.mathdotnet.com +// http://github.com/mathnet/mathnet-numerics +// http://mathnetnumerics.codeplex.com +// +// Copyright (c) 2009-2016 Math.NET +// +// Permission is hereby granted, free of charge, to any person +// obtaining a copy of this software and associated documentation +// files (the "Software"), to deal in the Software without +// restriction, including without limitation the rights to use, +// copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the +// Software is furnished to do so, subject to the following +// conditions: +// +// The above copyright notice and this permission notice shall be +// included in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, +// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES +// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND +// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT +// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, +// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR +// OTHER DEALINGS IN THE SOFTWARE. +// + +using System; +using System.Collections.Generic; + +namespace MathNet.Numerics.Integration.GaussRule +{ + /// + /// Contains precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 and a method to calculate them on the fly. + /// + static class GaussLegendrePointFactory + { + /// + /// Getter for the GaussPoint. + /// + /// Defines an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. + public static GaussPoint GetGaussPoint(int order) + { + GaussPoint gaussLegendrePoint; + + // Try to find it in the precomputed dictionary first. + if (!GaussLegendrePoints.TryGetValue(order, out gaussLegendrePoint)) + { + gaussLegendrePoint = GenerateGaussLegendrePoint(order, 1e-10); // Generate the abscissas/weights on the fly. + } + + return gaussLegendrePoint; + } + + /// + /// Precomputed abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 + /// + private static readonly Dictionary GaussLegendrePoints = new Dictionary + { + // Even + { 2, new GaussPoint(2, new[] { 0.5773502691896257645091488 }, new[] { 1.0000000000000000000000000 }) }, + { 4, new GaussPoint(4, new[] { 0.3399810435848562648026658, 0.8611363115940525752239465 }, new[] { 0.6521451548625461426269361, 0.3478548451374538573730639 }) }, + { 6, new GaussPoint(6, new[] { 0.2386191860831969086305017, 0.6612093864662645136613996, 0.9324695142031520278123016 }, new[] { 0.4679139345726910473898703, 0.3607615730481386075698335, 0.1713244923791703450402961 }) }, + { 8, new GaussPoint(8, new[] { 0.1834346424956498049394761, 0.5255324099163289858177390, 0.7966664774136267395915539, 0.9602898564975362316835609 }, new[] { 0.3626837833783619829651504, 0.3137066458778872873379622, 0.2223810344533744705443560, 0.1012285362903762591525314 }) }, + { 10, new GaussPoint(10, new[] { 0.1488743389816312108848260, 0.4333953941292471907992659, 0.6794095682990244062343274, 0.8650633666889845107320967, 0.9739065285171717200779640 }, new[] { 0.2955242247147528701738930, 0.2692667193099963550912269, 0.2190863625159820439955349, 0.1494513491505805931457763, 0.0666713443086881375935688 }) }, + { 12, new GaussPoint(12, new[] { 0.1252334085114689154724414, 0.3678314989981801937526915, 0.5873179542866174472967024, 0.7699026741943046870368938, 0.9041172563704748566784659, 0.9815606342467192506905491 }, new[] { 0.2491470458134027850005624, 0.2334925365383548087608499, 0.2031674267230659217490645, 0.1600783285433462263346525, 0.1069393259953184309602547, 0.0471753363865118271946160 }) }, + { 14, new GaussPoint(14, new[] { 0.1080549487073436620662447, 0.3191123689278897604356718, 0.5152486363581540919652907, 0.6872929048116854701480198, 0.8272013150697649931897947, 0.9284348836635735173363911, 0.9862838086968123388415973 }, new[] { 0.2152638534631577901958764, 0.2051984637212956039659241, 0.1855383974779378137417166, 0.1572031671581935345696019, 0.1215185706879031846894148, 0.0801580871597602098056333, 0.0351194603317518630318329 }) }, + { 16, new GaussPoint(16, new[] { 0.0950125098376374401853193, 0.2816035507792589132304605, 0.4580167776572273863424194, 0.6178762444026437484466718, 0.7554044083550030338951012, 0.8656312023878317438804679, 0.9445750230732325760779884, 0.9894009349916499325961542 }, new[] { 0.1894506104550684962853967, 0.1826034150449235888667637, 0.1691565193950025381893121, 0.1495959888165767320815017, 0.1246289712555338720524763, 0.0951585116824927848099251, 0.0622535239386478928628438, 0.0271524594117540948517806 }) }, + { 18, new GaussPoint(18, new[] { 0.0847750130417353012422619, 0.2518862256915055095889729, 0.4117511614628426460359318, 0.5597708310739475346078715, 0.6916870430603532078748911, 0.8037049589725231156824175, 0.8926024664975557392060606, 0.9558239495713977551811959, 0.9915651684209309467300160 }, new[] { 0.1691423829631435918406565, 0.1642764837458327229860538, 0.1546846751262652449254180, 0.1406429146706506512047313, 0.1225552067114784601845191, 0.1009420441062871655628140, 0.0764257302548890565291297, 0.0497145488949697964533349, 0.0216160135264833103133427 }) }, + { 20, new GaussPoint(20, new[] { 0.0765265211334973337546404, 0.2277858511416450780804962, 0.3737060887154195606725482, 0.5108670019508270980043641, 0.6360536807265150254528367, 0.7463319064601507926143051, 0.8391169718222188233945291, 0.9122344282513259058677524, 0.9639719272779137912676661, 0.9931285991850949247861224 }, new[] { 0.1527533871307258506980843, 0.1491729864726037467878287, 0.1420961093183820513292983, 0.1316886384491766268984945, 0.1181945319615184173123774, 0.1019301198172404350367501, 0.0832767415767047487247581, 0.0626720483341090635695065, 0.0406014298003869413310400, 0.0176140071391521183118620 }) }, + { 32, new GaussPoint(32, new[] { 0.0483076656877383162348126, 0.1444719615827964934851864, 0.2392873622521370745446032, 0.3318686022821276497799168, 0.4213512761306353453641194, 0.5068999089322293900237475, 0.5877157572407623290407455, 0.6630442669302152009751152, 0.7321821187402896803874267, 0.7944837959679424069630973, 0.8493676137325699701336930, 0.8963211557660521239653072, 0.9349060759377396891709191, 0.9647622555875064307738119, 0.9856115115452683354001750, 0.9972638618494815635449811 }, new[] { 0.0965400885147278005667648, 0.0956387200792748594190820, 0.0938443990808045656391802, 0.0911738786957638847128686, 0.0876520930044038111427715, 0.0833119242269467552221991, 0.0781938957870703064717409, 0.0723457941088485062253994, 0.0658222227763618468376501, 0.0586840934785355471452836, 0.0509980592623761761961632, 0.0428358980222266806568786, 0.0342738629130214331026877, 0.0253920653092620594557526, 0.0162743947309056706051706, 0.0070186100094700966004071 }) }, + { 64, new GaussPoint(64, new[] { 0.0243502926634244325089558, 0.0729931217877990394495429, 0.1214628192961205544703765, 0.1696444204239928180373136, 0.2174236437400070841496487, 0.2646871622087674163739642, 0.3113228719902109561575127, 0.3572201583376681159504426, 0.4022701579639916036957668, 0.4463660172534640879849477, 0.4894031457070529574785263, 0.5312794640198945456580139, 0.5718956462026340342838781, 0.6111553551723932502488530, 0.6489654712546573398577612, 0.6852363130542332425635584, 0.7198818501716108268489402, 0.7528199072605318966118638, 0.7839723589433414076102205, 0.8132653151227975597419233, 0.8406292962525803627516915, 0.8659993981540928197607834, 0.8893154459951141058534040, 0.9105221370785028057563807, 0.9295691721319395758214902, 0.9464113748584028160624815, 0.9610087996520537189186141, 0.9733268277899109637418535, 0.9833362538846259569312993, 0.9910133714767443207393824, 0.9963401167719552793469245, 0.9993050417357721394569056 }, new[] { 0.0486909570091397203833654, 0.0485754674415034269347991, 0.0483447622348029571697695, 0.0479993885964583077281262, 0.0475401657148303086622822, 0.0469681828162100173253263, 0.0462847965813144172959532, 0.0454916279274181444797710, 0.0445905581637565630601347, 0.0435837245293234533768279, 0.0424735151236535890073398, 0.0412625632426235286101563, 0.0399537411327203413866569, 0.0385501531786156291289625, 0.0370551285402400460404151, 0.0354722132568823838106931, 0.0338051618371416093915655, 0.0320579283548515535854675, 0.0302346570724024788679741, 0.0283396726142594832275113, 0.0263774697150546586716918, 0.0243527025687108733381776, 0.0222701738083832541592983, 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0.5688888888888888888888889, 0.4786286704993664680412915, 0.2369268850561890875142640 }) }, + { 7, new GaussPoint(7, new[] { 0.0000000000000000000000000, 0.4058451513773971669066064, 0.7415311855993944398638648, 0.9491079123427585245261897 }, new[] { 0.4179591836734693877551020, 0.3818300505051189449503698, 0.2797053914892766679014678, 0.1294849661688696932706114 }) }, + { 9, new GaussPoint(9, new[] { 0.0000000000000000000000000, 0.3242534234038089290385380, 0.6133714327005903973087020, 0.8360311073266357942994298, 0.9681602395076260898355762 }, new[] { 0.3302393550012597631645251, 0.3123470770400028400686304, 0.2606106964029354623187429, 0.1806481606948574040584720, 0.0812743883615744119718922 }) }, + { 11, new GaussPoint(11, new[] { 0.0000000000000000000000000, 0.2695431559523449723315320, 0.5190961292068118159257257, 0.7301520055740493240934163, 0.8870625997680952990751578, 0.9782286581460569928039380 }, new[] { 0.2729250867779006307144835, 0.2628045445102466621806889, 0.2331937645919904799185237, 0.1862902109277342514260976, 0.1255803694649046246346943, 0.0556685671161736664827537 }) }, + { 13, new GaussPoint(13, new[] { 0.0000000000000000000000000, 0.2304583159551347940655281, 0.4484927510364468528779129, 0.6423493394403402206439846, 0.8015780907333099127942065, 0.9175983992229779652065478, 0.9841830547185881494728294 }, new[] { 0.2325515532308739101945895, 0.2262831802628972384120902, 0.2078160475368885023125232, 0.1781459807619457382800467, 0.1388735102197872384636018, 0.0921214998377284479144218, 0.0404840047653158795200216 }) }, + { 15, new GaussPoint(15, new[] { 0.0000000000000000000000000, 0.2011940939974345223006283, 0.3941513470775633698972074, 0.5709721726085388475372267, 0.7244177313601700474161861, 0.8482065834104272162006483, 0.9372733924007059043077589, 0.9879925180204854284895657 }, new[] { 0.2025782419255612728806202, 0.1984314853271115764561183, 0.1861610000155622110268006, 0.1662692058169939335532009, 0.1395706779261543144478048, 0.1071592204671719350118695, 0.0703660474881081247092674, 0.0307532419961172683546284 }) }, + { 17, new GaussPoint(17, new[] { 0.0000000000000000000000000, 0.1784841814958478558506775, 0.3512317634538763152971855, 0.5126905370864769678862466, 0.6576711592166907658503022, 0.7815140038968014069252301, 0.8802391537269859021229557, 0.9506755217687677612227170, 0.9905754753144173356754340 }, new[] { 0.1794464703562065254582656, 0.1765627053669926463252710, 0.1680041021564500445099707, 0.1540457610768102880814316, 0.1351363684685254732863200, 0.1118838471934039710947884, 0.0850361483171791808835354, 0.0554595293739872011294402, 0.0241483028685479319601100 }) }, + { 19, new GaussPoint(19, new[] { 0.0000000000000000000000000, 0.1603586456402253758680961, 0.3165640999636298319901173, 0.4645707413759609457172671, 0.6005453046616810234696382, 0.7209661773352293786170959, 0.8227146565371428249789225, 0.9031559036148179016426609, 0.9602081521348300308527788, 0.9924068438435844031890177 }, new[] { 0.1610544498487836959791636, 0.1589688433939543476499564, 0.1527660420658596667788554, 0.1426067021736066117757461, 0.1287539625393362276755158, 0.1115666455473339947160239, 0.0914900216224499994644621, 0.0690445427376412265807083, 0.0448142267656996003328382, 0.0194617882297264770363120 }) } + }; + + /// + /// Computes the Gauss-Legendre abscissas/weights. + /// See Pavel Holoborodko for a description of the algorithm. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + /// Required precision to compute the abscissas/weights. 1e-10 is usually fine. + /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. + private static GaussPoint GenerateGaussLegendrePoint(int order, double eps) + { + // Look up table for fast calculation of Legendre polynomial for n < 1024 + double[] ltbl = { 0.00000000000000000000, 0.00000000000000000000, 0.50000000000000000000, 0.66666666666666674000, 0.75000000000000000000, 0.80000000000000004000, 0.83333333333333337000, 0.85714285714285721000, 0.87500000000000000000, 0.88888888888888884000, 0.90000000000000002000, 0.90909090909090906000, 0.91666666666666663000, 0.92307692307692313000, 0.92857142857142860000, 0.93333333333333335000, 0.93750000000000000000, 0.94117647058823528000, 0.94444444444444442000, 0.94736842105263164000, 0.94999999999999996000, 0.95238095238095233000, 0.95454545454545459000, 0.95652173913043481000, 0.95833333333333337000, 0.95999999999999996000, 0.96153846153846156000, 0.96296296296296302000, 0.96428571428571430000, 0.96551724137931039000, 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0.99901380670611439000, 0.99901477832512320000, 0.99901574803149606000, 0.99901671583087515000, 0.99901768172888017000, 0.99901864573110888000, 0.99901960784313726000, 0.99902056807051909000, 0.99902152641878672000, 0.99902248289345064000 }; + const double pi = 3.1415926535897932384626433832795028841971693993751; + + double w0 = 0; // Weights + int m = (order + 1) >> 1; + double[] abscissas = new double[m]; + double[] weights = new double[m]; + double t0 = 1.0 - (1.0 - 1.0/order)/(8.0*order*order); + double t1 = 1.0/(4.0*order + 2.0); + + // Find ith root of Legendre polynomial + for (int i = 1; i <= m; i++) + { + int j = 0; + double x0 = Math.Cos(pi*((i << 2) - 1)*t1)*t0; // Initial guess + double x1, dx; // Abscissas + double w1, dw; // Weights + + // Newton iterations, at least one + do + { + // Legendre polynomial values + double p1 = 1.0; + double p0 = x0; + double p2; + int k; + double t2; + + // Optimized version using lookup tables + if (order < 1024) + { + // Use fast algorithm for small orders + for (k = 2; k <= order; k++) + { + p2 = p1; + p1 = p0; + t2 = x0*p1; + + p0 = t2 + ltbl[k]*(t2 - p2); + } + } + else + { + // Use general algorithm for other orders + for (k = 2; k < 1024; k++) + { + p2 = p1; + p1 = p0; + t2 = x0*p1; + + p0 = t2 + ltbl[k]*(t2 - p2); + } + + for (k = 1024; k <= order; k++) + { + p2 = p1; + p1 = p0; + t2 = x0*p1; + double t3 = (k - 1.0)/k; + + p0 = t2 + t3*(t2 - p2); + } + } + + double dpdx = ((x0*p0 - p1)*order)/(x0*x0 - 1.0); // Compute Legendre polynomial derivative at x0 + + x1 = x0 - p0/dpdx; // Newton step + + w1 = 2.0/((1.0 - x1*x1)*dpdx*dpdx); // Weight computing + + // Compute weight w0 on first iteration, needed for dw + if (j == 0) + { + w0 = 2.0/((1.0 - x0*x0)*dpdx*dpdx); + } + + dx = x0 - x1; + dw = w0 - w1; + + x0 = x1; + w0 = w1; + j++; + } + while ((Math.Abs(dx) > eps || Math.Abs(dw) > eps) && j < 100); + + int index = (m - 1) - (i - 1); + + abscissas[index] = x1; + weights[index] = w1; + } + + return new GaussPoint(order, abscissas, weights); + } + } +} \ No newline at end of file diff --git a/src/Numerics/Integration/GaussRule/GaussPoint.cs b/src/Numerics/Integration/GaussRule/GaussPoint.cs new file mode 100644 index 00000000..b4b7ee63 --- /dev/null +++ b/src/Numerics/Integration/GaussRule/GaussPoint.cs @@ -0,0 +1,61 @@ +// +// Math.NET Numerics, part of the Math.NET Project +// http://numerics.mathdotnet.com +// http://github.com/mathnet/mathnet-numerics +// http://mathnetnumerics.codeplex.com +// +// Copyright (c) 2009-2016 Math.NET +// +// Permission is hereby granted, free of charge, to any person +// obtaining a copy of this software and associated documentation +// files (the "Software"), to deal in the Software without +// restriction, including without limitation the rights to use, +// copy, modify, merge, publish, distribute, sublicense, and/or sell +// copies of the Software, and to permit persons to whom the +// Software is furnished to do so, subject to the following +// conditions: +// +// The above copyright notice and this permission notice shall be +// included in all copies or substantial portions of the Software. +// +// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, +// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES +// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND +// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT +// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, +// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING +// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR +// OTHER DEALINGS IN THE SOFTWARE. +// + +namespace MathNet.Numerics.Integration.GaussRule +{ + /// + /// Contains the abscissas/weights, order, and intervalBegin/intervalEnd. + /// + class GaussPoint + { + public double[] Abscissas { get; private set; } + + public double[] Weights { get; private set; } + + public double IntervalBegin { get; private set; } + + public double IntervalEnd { get; private set; } + + public int Order { get; private set; } + + public GaussPoint(double intervalBegin, double intervalEnd, int order, double[] abscissas, double[] weights) + { + Abscissas = abscissas; + Weights = weights; + IntervalBegin = intervalBegin; + IntervalEnd = intervalEnd; + Order = order; + } + + public GaussPoint(int order, double[] abscissas, double[] weights) : this(-1, 1, order, abscissas, weights) + { + } + } +} \ No newline at end of file diff --git a/src/Numerics/Numerics.csproj b/src/Numerics/Numerics.csproj index c724bab7..b6004e90 100644 --- a/src/Numerics/Numerics.csproj +++ b/src/Numerics/Numerics.csproj @@ -96,6 +96,9 @@ + + + diff --git a/src/UnitTests/IntegrationTests/IntegrationTest.cs b/src/UnitTests/IntegrationTests/IntegrationTest.cs index d65b646d..42e7e1af 100644 --- a/src/UnitTests/IntegrationTests/IntegrationTest.cs +++ b/src/UnitTests/IntegrationTests/IntegrationTest.cs @@ -46,6 +46,17 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests return Math.Exp(-x / 5) * (2 + Math.Sin(2 * x)); } + /// + /// Test Function: f(x,y) = exp(-x/5) (2 + sin(x * y)) + /// + /// First input value. + /// Second input value. + /// Function result. + private static double TargeFunctionB(double x, double y) + { + return Math.Exp(-x / 5) * (2 + Math.Sin(2 * y)); + } + /// /// Test Function Start point. /// @@ -56,11 +67,26 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests /// private const double StopA = 10; + /// + /// Test Function Start point. + /// + private const double StartB = 0; + + /// + /// Test Function Stop point. + /// + private const double StopB = 1; + /// /// Target area square. /// private const double TargetAreaA = 9.1082396073229965070; + /// + /// Target area. + /// + private const double TargetAreaB = 11.7078776759298776163; + /// /// Test integrate portal. /// @@ -178,5 +204,79 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests "Composite {0} Partitions", partitions); } + + /// + /// Gauss-Legendre rule supports integration. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + [TestCase(19)] + [TestCase(20)] + [TestCase(21)] + [TestCase(22)] + public void TestGaussLegendreRuleIntegration(int order) + { + double appoximateArea = GaussLegendreRule.Integrate(TargetFunctionA, StartA, StopA, order); + double relativeError = Math.Abs(TargetAreaA - appoximateArea) / TargetAreaA; + Assert.Less(relativeError, 5e-16); + } + + /// + /// Gauss-Legendre rule supports 2-dimensional integration over the rectangle. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + [TestCase(19)] + [TestCase(20)] + [TestCase(21)] + [TestCase(22)] + public void TestGaussLegendreRuleIntegrate2D(int order) + { + double appoximateArea = GaussLegendreRule.Integrate(TargeFunctionB, StartA, StopA, StartB, StopB, order); + double relativeError = Math.Abs(TargetAreaB - appoximateArea) / TargetAreaB; + Assert.Less(relativeError, 1e-15); + } + + /// + /// Gauss-Legendre rule supports obtaining the abscissas/weights. In this case, they're used for integration. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + [TestCase(19)] + [TestCase(20)] + [TestCase(21)] + [TestCase(22)] + public void TestGaussLegendreRuleGetAbscissasGetWeightsGetOrderViaIntegration(int order) + { + GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); + + double appoximateArea = 0; + for (int i = 0; i < gaussLegendre.GetOrder(); i++) + { + appoximateArea += gaussLegendre.GetWeight(i) * TargetFunctionA(gaussLegendre.GetAbscissa(i)); + } + + double relativeError = Math.Abs(TargetAreaA - appoximateArea) / TargetAreaA; + Assert.Less(relativeError, 5e-16); + } + + /// + /// Gauss-Legendre rule supports obtaining IntervalBegin. + /// + [TestCase] + public void TestGetGaussLegendreRuleIntervalBegin() + { + const int order = 19; + GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); + Assert.AreEqual(gaussLegendre.GetIntervalBegin(), StartA); + } + + /// + /// Gauss-Legendre rule supports obtaining IntervalEnd. + /// + [TestCase] + public void TestGaussLegendreRuleGetIntervalEnd() + { + const int order = 19; + GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); + Assert.AreEqual(gaussLegendre.GetIntervalEnd(), StopA); + } } -} +} \ No newline at end of file diff --git a/src/UnitTests/UnitTests.csproj b/src/UnitTests/UnitTests.csproj index 114a4b63..63fddfdb 100644 --- a/src/UnitTests/UnitTests.csproj +++ b/src/UnitTests/UnitTests.csproj @@ -65,6 +65,10 @@ 5 + + False + ..\..\packages\NUnit\lib\nunit.framework.dll + @@ -416,11 +420,4 @@ - - - ..\..\packages\test\NUnit\lib\nunit.framework.dll - True - True - - \ No newline at end of file From 2d33bea77b33ec2128ee9c46db927d2b036d5090 Mon Sep 17 00:00:00 2001 From: Larz White Date: Tue, 9 Feb 2016 22:44:38 -0600 Subject: [PATCH 2/7] Ran build.cmd to fix missing NUnit dependency --- src/UnitTests/UnitTests.csproj | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/src/UnitTests/UnitTests.csproj b/src/UnitTests/UnitTests.csproj index 63fddfdb..114a4b63 100644 --- a/src/UnitTests/UnitTests.csproj +++ b/src/UnitTests/UnitTests.csproj @@ -65,10 +65,6 @@ 5 - - False - ..\..\packages\NUnit\lib\nunit.framework.dll - @@ -420,4 +416,11 @@ + + + ..\..\packages\test\NUnit\lib\nunit.framework.dll + True + True + + \ No newline at end of file From 4f64aa52ab68b206c0ec8a84d2129507a9931b49 Mon Sep 17 00:00:00 2001 From: Larz White Date: Wed, 10 Feb 2016 19:12:24 -0600 Subject: [PATCH 3/7] Fixed typo in TargetFunctionB --- src/UnitTests/IntegrationTests/IntegrationTest.cs | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/UnitTests/IntegrationTests/IntegrationTest.cs b/src/UnitTests/IntegrationTests/IntegrationTest.cs index 42e7e1af..26c6851c 100644 --- a/src/UnitTests/IntegrationTests/IntegrationTest.cs +++ b/src/UnitTests/IntegrationTests/IntegrationTest.cs @@ -52,7 +52,7 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests /// First input value. /// Second input value. /// Function result. - private static double TargeFunctionB(double x, double y) + private static double TargetFunctionB(double x, double y) { return Math.Exp(-x / 5) * (2 + Math.Sin(2 * y)); } @@ -230,7 +230,7 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests [TestCase(22)] public void TestGaussLegendreRuleIntegrate2D(int order) { - double appoximateArea = GaussLegendreRule.Integrate(TargeFunctionB, StartA, StopA, StartB, StopB, order); + double appoximateArea = GaussLegendreRule.Integrate(TargetFunctionB, StartA, StopA, StartB, StopB, order); double relativeError = Math.Abs(TargetAreaB - appoximateArea) / TargetAreaB; Assert.Less(relativeError, 1e-15); } From e60cb1af1b1978833d2c210b2c0aeceeb0cfddc5 Mon Sep 17 00:00:00 2001 From: Larz White Date: Wed, 10 Feb 2016 20:19:18 -0600 Subject: [PATCH 4/7] Moved the caching to the factory, thereby making the rule stateless --- src/Numerics/Integration/GaussLegendreRule.cs | 59 +++++++------------ .../GaussRule/GaussLegendrePointFactory.cs | 14 +++-- 2 files changed, 31 insertions(+), 42 deletions(-) diff --git a/src/Numerics/Integration/GaussLegendreRule.cs b/src/Numerics/Integration/GaussLegendreRule.cs index dce9784a..8ecf8f3c 100644 --- a/src/Numerics/Integration/GaussLegendreRule.cs +++ b/src/Numerics/Integration/GaussLegendreRule.cs @@ -39,8 +39,7 @@ namespace MathNet.Numerics.Integration public class GaussLegendreRule { private readonly GaussPoint gaussLegendrePoint; - private static GaussPoint nonNegativeGaussLegendrePoint; - + /// /// Initializes a new instance of the class. /// @@ -99,22 +98,6 @@ namespace MathNet.Numerics.Integration return gaussLegendrePoint.IntervalEnd; } - /// - /// Getter for the GaussPoint. - /// - /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. - /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. - private static GaussPoint GetGaussPoint(int order) - { - // Try to get the point from the static field first. If it's not there, or it's the wrong order, then generate it on the fly and store it in the static field. - if (nonNegativeGaussLegendrePoint == null || nonNegativeGaussLegendrePoint.Order != order) - { - return GaussLegendrePointFactory.GetGaussPoint(order); - } - - return nonNegativeGaussLegendrePoint; - } - /// /// Maps the non-negative abscissas/weights from the interval [-1, 1] to the interval [intervalBegin, intervalEnd]. /// @@ -158,7 +141,7 @@ namespace MathNet.Numerics.Integration /// Approximation of the finite integral in the given interval. public static double Integrate(Func f, double invervalBegin, double invervalEnd, int order) { - nonNegativeGaussLegendrePoint = GetGaussPoint(order); + GaussPoint gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(order); double sum, ax; int i; @@ -169,11 +152,11 @@ namespace MathNet.Numerics.Integration if (order.IsOdd()) { - sum = nonNegativeGaussLegendrePoint.Weights[0]*f(b); + sum = gaussLegendrePoint.Weights[0]*f(b); for (i = 1; i < m; i++) { - ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; - sum += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); + ax = a*gaussLegendrePoint.Abscissas[i]; + sum += gaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); } } else @@ -181,8 +164,8 @@ namespace MathNet.Numerics.Integration sum = 0.0; for (i = 0; i < m; i++) { - ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; - sum += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); + ax = a*gaussLegendrePoint.Abscissas[i]; + sum += gaussLegendrePoint.Weights[i]*(f(b + ax) + f(b - ax)); } } @@ -201,7 +184,7 @@ namespace MathNet.Numerics.Integration /// Approximation of the finite integral in the given interval. public static double Integrate(Func f, double invervalBeginA, double invervalEndA, double invervalBeginB, double invervalEndB, int order) { - nonNegativeGaussLegendrePoint = GetGaussPoint(order); + GaussPoint gaussLegendrePoint = GaussLegendrePointFactory.GetGaussPoint(order); double ax, cy, sum; int i, j; @@ -214,32 +197,32 @@ namespace MathNet.Numerics.Integration if (order.IsOdd()) { - sum = nonNegativeGaussLegendrePoint.Weights[0]*nonNegativeGaussLegendrePoint.Weights[0]*f(b, d); + sum = gaussLegendrePoint.Weights[0]*gaussLegendrePoint.Weights[0]*f(b, d); double t; for (j = 1, t = 0.0; j < m; j++) { - cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; - t += nonNegativeGaussLegendrePoint.Weights[j]*(f(b, d + cy) + f(b, d - cy)); + cy = c*gaussLegendrePoint.Abscissas[j]; + t += gaussLegendrePoint.Weights[j]*(f(b, d + cy) + f(b, d - cy)); } - sum += nonNegativeGaussLegendrePoint.Weights[0]*t; + sum += gaussLegendrePoint.Weights[0]*t; for (i = 1, t = 0.0; i < m; i++) { - ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; - t += nonNegativeGaussLegendrePoint.Weights[i]*(f(b + ax, d) + f(b - ax, d)); + ax = a*gaussLegendrePoint.Abscissas[i]; + t += gaussLegendrePoint.Weights[i]*(f(b + ax, d) + f(b - ax, d)); } - sum += nonNegativeGaussLegendrePoint.Weights[0]*t; + sum += gaussLegendrePoint.Weights[0]*t; for (i = 1; i < m; i++) { - ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + ax = a*gaussLegendrePoint.Abscissas[i]; for (j = 1; j < m; j++) { - cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; - sum += nonNegativeGaussLegendrePoint.Weights[i]*nonNegativeGaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); + cy = c*gaussLegendrePoint.Abscissas[j]; + sum += gaussLegendrePoint.Weights[i]*gaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); } } } @@ -248,11 +231,11 @@ namespace MathNet.Numerics.Integration sum = 0.0; for (i = 0; i < m; i++) { - ax = a*nonNegativeGaussLegendrePoint.Abscissas[i]; + ax = a*gaussLegendrePoint.Abscissas[i]; for (j = 0; j < m; j++) { - cy = c*nonNegativeGaussLegendrePoint.Abscissas[j]; - sum += nonNegativeGaussLegendrePoint.Weights[i]*nonNegativeGaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); + cy = c*gaussLegendrePoint.Abscissas[j]; + sum += gaussLegendrePoint.Weights[i]*gaussLegendrePoint.Weights[j]*(f(b + ax, d + cy) + f(ax + b, d - cy) + f(b - ax, d + cy) + f(b - ax, d - cy)); } } } diff --git a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs index d221915d..06c40395 100644 --- a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs +++ b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs @@ -38,19 +38,25 @@ namespace MathNet.Numerics.Integration.GaussRule /// static class GaussLegendrePointFactory { + private static GaussPoint gaussLegendrePoint; + /// /// Getter for the GaussPoint. /// - /// Defines an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Defines an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. Furthermore, caching in a private static field is used to speed up the algorithm. /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. public static GaussPoint GetGaussPoint(int order) { - GaussPoint gaussLegendrePoint; + // Try to get the GaussPoint from the cached static field. + if (gaussLegendrePoint != null && gaussLegendrePoint.Order == order) + { + return gaussLegendrePoint; + } - // Try to find it in the precomputed dictionary first. + // Try to find the GaussPoint in the precomputed dictionary. if (!GaussLegendrePoints.TryGetValue(order, out gaussLegendrePoint)) { - gaussLegendrePoint = GenerateGaussLegendrePoint(order, 1e-10); // Generate the abscissas/weights on the fly. + gaussLegendrePoint = GenerateGaussLegendrePoint(order, 1e-10); // Generate the GaussPoint on the fly. } return gaussLegendrePoint; From 3401cac8d79b4ede5672e43851fd2159500cca93 Mon Sep 17 00:00:00 2001 From: Larz White Date: Wed, 10 Feb 2016 22:19:12 -0600 Subject: [PATCH 5/7] Made the cached field thread safe --- src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs | 1 + 1 file changed, 1 insertion(+) diff --git a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs index 06c40395..78967b4b 100644 --- a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs +++ b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs @@ -38,6 +38,7 @@ namespace MathNet.Numerics.Integration.GaussRule /// static class GaussLegendrePointFactory { + [ThreadStatic] private static GaussPoint gaussLegendrePoint; /// From 630c0d48796a287915564ee040f8fddba057d8a4 Mon Sep 17 00:00:00 2001 From: Larz White Date: Wed, 10 Feb 2016 23:00:50 -0600 Subject: [PATCH 6/7] Exposed Gauss-Legendre 1D and 2D integration methods to MathNet.Numerics.Integrate and added NUnit tests for them. --- src/Numerics/Integrate.cs | 28 +++++++++++++++++ .../GaussRule/GaussLegendrePointFactory.cs | 2 +- .../IntegrationTests/IntegrationTest.cs | 30 +++++++++++++++++++ 3 files changed, 59 insertions(+), 1 deletion(-) diff --git a/src/Numerics/Integrate.cs b/src/Numerics/Integrate.cs index 84398dbe..01400e98 100644 --- a/src/Numerics/Integrate.cs +++ b/src/Numerics/Integrate.cs @@ -62,5 +62,33 @@ namespace MathNet.Numerics { return DoubleExponentialTransformation.Integrate(f, intervalBegin, intervalEnd, 1e-8); } + + /// + /// Approximates a definite integral using an Nth order Gauss-Legendre rule. + /// + /// The analytic smooth function to integrate. + /// Where the interval starts, exclusive and finite. + /// Where the interval ends, exclusive and finite. + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Approximation of the finite integral in the given interval. + public static double GaussLegendre(Func f, double invervalBegin, double invervalEnd, int order) + { + return GaussLegendreRule.Integrate(f, invervalBegin, invervalEnd, order); + } + + /// + /// Approximates a 2-dimensional definite integral using an Nth order Gauss-Legendre rule over the rectangle [a,b] x [c,d]. + /// + /// The 2-dimensional analytic smooth function to integrate. + /// Where the interval starts for the first (inside) integral, exclusive and finite. + /// Where the interval ends for the first (inside) integral, exclusive and finite. + /// Where the interval starts for the second (outside) integral, exclusive and finite. + /// /// Where the interval ends for the second (outside) integral, exclusive and finite. + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. + /// Approximation of the finite integral in the given interval. + public static double GaussLegendre(Func f, double invervalBeginA, double invervalEndA, double invervalBeginB, double invervalEndB, int order) + { + return GaussLegendreRule.Integrate(f, invervalBeginA, invervalEndA, invervalBeginB, invervalEndB, order); + } } } diff --git a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs index 78967b4b..3dbd961f 100644 --- a/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs +++ b/src/Numerics/Integration/GaussRule/GaussLegendrePointFactory.cs @@ -44,7 +44,7 @@ namespace MathNet.Numerics.Integration.GaussRule /// /// Getter for the GaussPoint. /// - /// Defines an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. Furthermore, caching in a private static field is used to speed up the algorithm. + /// Defines an Nth order Gauss-Legendre rule. Precomputed Gauss-Legendre abscissas/weights for orders 2,. . ., 20, 32, 64, 96, 100, 128, 256, 512, 1024 are used, otherwise they're calulcated on the fly. /// Object containing the non-negative abscissas/weights, order, and intervalBegin/intervalEnd. The non-negative abscissas/weights are generated over the interval [-1,1] for the given order. public static GaussPoint GetGaussPoint(int order) { diff --git a/src/UnitTests/IntegrationTests/IntegrationTest.cs b/src/UnitTests/IntegrationTests/IntegrationTest.cs index 26c6851c..2a6f3fe0 100644 --- a/src/UnitTests/IntegrationTests/IntegrationTest.cs +++ b/src/UnitTests/IntegrationTests/IntegrationTest.cs @@ -220,6 +220,21 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests Assert.Less(relativeError, 5e-16); } + /// + /// Gauss-Legendre rule supports integration. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + [TestCase(19)] + [TestCase(20)] + [TestCase(21)] + [TestCase(22)] + public void TestIntegrateGaussLegendre(int order) + { + double appoximateArea = Integrate.GaussLegendre(TargetFunctionA, StartA, StopA, order); + double relativeError = Math.Abs(TargetAreaA - appoximateArea) / TargetAreaA; + Assert.Less(relativeError, 5e-16); + } + /// /// Gauss-Legendre rule supports 2-dimensional integration over the rectangle. /// @@ -235,6 +250,21 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests Assert.Less(relativeError, 1e-15); } + /// + /// Gauss-Legendre rule supports 2-dimensional integration over the rectangle. + /// + /// Defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. + [TestCase(19)] + [TestCase(20)] + [TestCase(21)] + [TestCase(22)] + public void TestIntegrateGaussLegendre2D(int order) + { + double appoximateArea = Integrate.GaussLegendre(TargetFunctionB, StartA, StopA, StartB, StopB, order); + double relativeError = Math.Abs(TargetAreaB - appoximateArea) / TargetAreaB; + Assert.Less(relativeError, 1e-15); + } + /// /// Gauss-Legendre rule supports obtaining the abscissas/weights. In this case, they're used for integration. /// From f3632a1cac52d5c7a820d26a71a975f19b8e1d07 Mon Sep 17 00:00:00 2001 From: Larz White Date: Fri, 12 Feb 2016 09:42:11 -0600 Subject: [PATCH 7/7] Switched to using properties for the order and InvervalBegin/IntervalEnd. --- src/Numerics/Integration/GaussLegendreRule.cs | 24 ++++++++++++------- .../IntegrationTests/IntegrationTest.cs | 10 ++++---- 2 files changed, 20 insertions(+), 14 deletions(-) diff --git a/src/Numerics/Integration/GaussLegendreRule.cs b/src/Numerics/Integration/GaussLegendreRule.cs index 8ecf8f3c..238a945e 100644 --- a/src/Numerics/Integration/GaussLegendreRule.cs +++ b/src/Numerics/Integration/GaussLegendreRule.cs @@ -74,28 +74,34 @@ namespace MathNet.Numerics.Integration /// /// Getter for the order. /// - /// The order, which defines an Nth order Gauss-Legendre rule. The order also defines the number of abscissas and weights for the rule. - public int GetOrder() + public int Order { - return gaussLegendrePoint.Order; + get + { + return gaussLegendrePoint.Order; + } } /// /// Getter for the InvervalBegin. /// - /// Where the interval starts, inclusive and finite. - public double GetIntervalBegin() + public double IntervalBegin { - return gaussLegendrePoint.IntervalBegin; + get + { + return gaussLegendrePoint.IntervalBegin; + } } /// /// Getter for the InvervalEnd. /// - /// Where the interval stops, inclusive and finite. - public double GetIntervalEnd() + public double IntervalEnd { - return gaussLegendrePoint.IntervalEnd; + get + { + return gaussLegendrePoint.IntervalEnd; + } } /// diff --git a/src/UnitTests/IntegrationTests/IntegrationTest.cs b/src/UnitTests/IntegrationTests/IntegrationTest.cs index 2a6f3fe0..a0bc3dfb 100644 --- a/src/UnitTests/IntegrationTests/IntegrationTest.cs +++ b/src/UnitTests/IntegrationTests/IntegrationTest.cs @@ -273,12 +273,12 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests [TestCase(20)] [TestCase(21)] [TestCase(22)] - public void TestGaussLegendreRuleGetAbscissasGetWeightsGetOrderViaIntegration(int order) + public void TestGaussLegendreRuleGetAbscissasGetWeightsOrderViaIntegration(int order) { GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); double appoximateArea = 0; - for (int i = 0; i < gaussLegendre.GetOrder(); i++) + for (int i = 0; i < gaussLegendre.Order; i++) { appoximateArea += gaussLegendre.GetWeight(i) * TargetFunctionA(gaussLegendre.GetAbscissa(i)); } @@ -295,18 +295,18 @@ namespace MathNet.Numerics.UnitTests.IntegrationTests { const int order = 19; GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); - Assert.AreEqual(gaussLegendre.GetIntervalBegin(), StartA); + Assert.AreEqual(gaussLegendre.IntervalBegin, StartA); } /// /// Gauss-Legendre rule supports obtaining IntervalEnd. /// [TestCase] - public void TestGaussLegendreRuleGetIntervalEnd() + public void TestGaussLegendreRuleIntervalEnd() { const int order = 19; GaussLegendreRule gaussLegendre = new GaussLegendreRule(StartA, StopA, order); - Assert.AreEqual(gaussLegendre.GetIntervalEnd(), StopA); + Assert.AreEqual(gaussLegendre.IntervalEnd, StopA); } } } \ No newline at end of file