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Improve Numerical.integrate() by adding higher precision and supported iterations of up to n = 16.
This commit is contained in:
Jürg Lehni
parent
c1d70c90bb
commit
500942e86d
1 changed files with 45 additions and 29 deletions
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@ -16,46 +16,62 @@
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var Numerical = new function() {
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var abscissa = [
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-0.5773502692, 0.5773502692,
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-0.7745966692, 0.7745966692, 0,
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-0.8611363116, 0.8611363116, -0.3399810436, 0.3399810436,
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-0.9061798459, 0.9061798459, -0.5384693101, 0.5384693101, 0.0000000000,
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-0.9324695142, 0.9324695142, -0.6612093865, 0.6612093865, -0.2386191861, 0.2386191861,
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-0.9491079123, 0.9491079123, -0.7415311856, 0.7415311856, -0.4058451514, 0.4058451514, 0.0000000000,
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-0.9602898565, 0.9602898565, -0.7966664774, 0.7966664774, -0.5255324099, 0.5255324099, -0.1834346425, 0.1834346425
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var abscissas = [
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[0.5773502691896257645091488],
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[0,0.7745966692414833770358531],
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[0.3399810435848562648026658,0.8611363115940525752239465],
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[0,0.5384693101056830910363144,0.9061798459386639927976269],
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[0.2386191860831969086305017,0.6612093864662645136613996,0.9324695142031520278123016],
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[0,0.4058451513773971669066064,0.7415311855993944398638648,0.9491079123427585245261897],
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[0.1834346424956498049394761,0.5255324099163289858177390,0.7966664774136267395915539,0.9602898564975362316835609],
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[0,0.3242534234038089290385380,0.6133714327005903973087020,0.8360311073266357942994298,0.9681602395076260898355762],
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[0.1488743389816312108848260,0.4333953941292471907992659,0.6794095682990244062343274,0.8650633666889845107320967,0.9739065285171717200779640],
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[0,0.2695431559523449723315320,0.5190961292068118159257257,0.7301520055740493240934163,0.8870625997680952990751578,0.9782286581460569928039380],
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[0.1252334085114689154724414,0.3678314989981801937526915,0.5873179542866174472967024,0.7699026741943046870368938,0.9041172563704748566784659,0.9815606342467192506905491],
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[0,0.2304583159551347940655281,0.4484927510364468528779129,0.6423493394403402206439846,0.8015780907333099127942065,0.9175983992229779652065478,0.9841830547185881494728294],
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[0.1080549487073436620662447,0.3191123689278897604356718,0.5152486363581540919652907,0.6872929048116854701480198,0.8272013150697649931897947,0.9284348836635735173363911,0.9862838086968123388415973],
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[0,0.2011940939974345223006283,0.3941513470775633698972074,0.5709721726085388475372267,0.7244177313601700474161861,0.8482065834104272162006483,0.9372733924007059043077589,0.9879925180204854284895657],
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[0.0950125098376374401853193,0.2816035507792589132304605,0.4580167776572273863424194,0.6178762444026437484466718,0.7554044083550030338951012,0.8656312023878317438804679,0.9445750230732325760779884,0.9894009349916499325961542]
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],
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weight = [
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1, 1,
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0.5555555556, 0.5555555556, 0.8888888888,
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0.3478548451, 0.3478548451, 0.6521451549, 0.6521451549,
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0.2369268851, 0.2369268851, 0.4786286705, 0.4786286705, 0.5688888888,
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0.1713244924, 0.1713244924, 0.3607615730, 0.3607615730, 0.4679139346, 0.4679139346,
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0.1294849662, 0.1294849662, 0.2797053915, 0.2797053915, 0.3818300505, 0.3818300505, 0.4179591837,
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0.1012285363, 0.1012285363, 0.2223810345, 0.2223810345, 0.3137066459, 0.3137066459, 0.3626837834, 0.3626837834
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weights = [
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[1],
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[0.8888888888888888888888889,0.5555555555555555555555556],
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[0.6521451548625461426269361,0.3478548451374538573730639],
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[0.5688888888888888888888889,0.4786286704993664680412915,0.2369268850561890875142640],
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[0.4679139345726910473898703,0.3607615730481386075698335,0.1713244923791703450402961],
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[0.4179591836734693877551020,0.3818300505051189449503698,0.2797053914892766679014678,0.1294849661688696932706114],
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[0.3626837833783619829651504,0.3137066458778872873379622,0.2223810344533744705443560,0.1012285362903762591525314],
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[0.3302393550012597631645251,0.3123470770400028400686304,0.2606106964029354623187429,0.1806481606948574040584720,0.0812743883615744119718922],
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[0.2955242247147528701738930,0.2692667193099963550912269,0.2190863625159820439955349,0.1494513491505805931457763,0.0666713443086881375935688],
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[0.2729250867779006307144835,0.2628045445102466621806889,0.2331937645919904799185237,0.1862902109277342514260976,0.1255803694649046246346943,0.0556685671161736664827537],
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[0.2491470458134027850005624,0.2334925365383548087608499,0.2031674267230659217490645,0.1600783285433462263346525,0.1069393259953184309602547,0.0471753363865118271946160],
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[0.2325515532308739101945895,0.2262831802628972384120902,0.2078160475368885023125232,0.1781459807619457382800467,0.1388735102197872384636018,0.0921214998377284479144218,0.0404840047653158795200216],
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[0.2152638534631577901958764,0.2051984637212956039659241,0.1855383974779378137417166,0.1572031671581935345696019,0.1215185706879031846894148,0.0801580871597602098056333,0.0351194603317518630318329],
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[0.2025782419255612728806202,0.1984314853271115764561183,0.1861610000155622110268006,0.1662692058169939335532009,0.1395706779261543144478048,0.1071592204671719350118695,0.0703660474881081247092674,0.0307532419961172683546284],
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[0.1894506104550684962853967,0.1826034150449235888667637,0.1691565193950025381893121,0.1495959888165767320815017,0.1246289712555338720524763,0.0951585116824927848099251,0.0622535239386478928628438,0.0271524594117540948517806]
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];
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return {
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TOLERANCE: 10e-6,
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/**
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* Gauss-Legendre Numerical Integration, ported from Singularity:
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*
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* Copyright (c) 2006-2007, Jim Armstrong (www.algorithmist.net)
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* All Rights Reserved.
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* Gauss-Legendre Numerical Integration
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*/
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integrate: function(f, a, b, n) {
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n = Math.min(Math.max(n, 2), 8);
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var x = abscissas[n - 2],
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w = weights[n - 2],
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A = 0.5 * (b - a),
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B = A + a,
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i = 0,
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m = (n + 1) >> 1,
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sum = n & 1 ? w[i++] * f(B) : 0; // Handle odd n
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while (i < m) {
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var Ax = A * x[i];
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sum += w[i++] * (f(B + Ax) + f(B - Ax));
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}
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var l = n == 2 ? 0 : n * (n - 1) / 2 - 1,
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sum = 0,
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mul = 0.5 * (b - a),
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ab2 = mul + a;
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for(var i = 0; i < n; i++)
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sum += f(ab2 + mul * abscissa[l + i]) * weight[l + i];
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return mul * sum;
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return A * sum;
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},
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findRootNewton: function(f, fd, a, b, n, tol) {
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