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18 package org.apache.commons.math.ode.nonstiff;
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55 public class DormandPrince853Integrator extends EmbeddedRungeKuttaIntegrator {
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57
58 private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)";
59
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61 private static final double[] staticC = {
62 (12.0 - 2.0 * Math.sqrt(6.0)) / 135.0, (6.0 - Math.sqrt(6.0)) / 45.0, (6.0 - Math.sqrt(6.0)) / 30.0,
63 (6.0 + Math.sqrt(6.0)) / 30.0, 1.0/3.0, 1.0/4.0, 4.0/13.0, 127.0/195.0, 3.0/5.0,
64 6.0/7.0, 1.0, 1.0
65 };
66
67
68 private static final double[][] staticA = {
69
70
71 {(12.0 - 2.0 * Math.sqrt(6.0)) / 135.0},
72
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74 {(6.0 - Math.sqrt(6.0)) / 180.0, (6.0 - Math.sqrt(6.0)) / 60.0},
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77 {(6.0 - Math.sqrt(6.0)) / 120.0, 0.0, (6.0 - Math.sqrt(6.0)) / 40.0},
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80 {(462.0 + 107.0 * Math.sqrt(6.0)) / 3000.0, 0.0,
81 (-402.0 - 197.0 * Math.sqrt(6.0)) / 1000.0, (168.0 + 73.0 * Math.sqrt(6.0)) / 375.0},
82
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84 {1.0 / 27.0, 0.0, 0.0, (16.0 + Math.sqrt(6.0)) / 108.0, (16.0 - Math.sqrt(6.0)) / 108.0},
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87 {19.0 / 512.0, 0.0, 0.0, (118.0 + 23.0 * Math.sqrt(6.0)) / 1024.0,
88 (118.0 - 23.0 * Math.sqrt(6.0)) / 1024.0, -9.0 / 512.0},
89
90
91 {13772.0 / 371293.0, 0.0, 0.0, (51544.0 + 4784.0 * Math.sqrt(6.0)) / 371293.0,
92 (51544.0 - 4784.0 * Math.sqrt(6.0)) / 371293.0, -5688.0 / 371293.0, 3072.0 / 371293.0},
93
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95 {58656157643.0 / 93983540625.0, 0.0, 0.0,
96 (-1324889724104.0 - 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
97 (-1324889724104.0 + 318801444819.0 * Math.sqrt(6.0)) / 626556937500.0,
98 96044563816.0 / 3480871875.0, 5682451879168.0 / 281950621875.0,
99 -165125654.0 / 3796875.0},
100
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102 {8909899.0 / 18653125.0, 0.0, 0.0,
103 (-4521408.0 - 1137963.0 * Math.sqrt(6.0)) / 2937500.0,
104 (-4521408.0 + 1137963.0 * Math.sqrt(6.0)) / 2937500.0,
105 96663078.0 / 4553125.0, 2107245056.0 / 137915625.0,
106 -4913652016.0 / 147609375.0, -78894270.0 / 3880452869.0},
107
108
109 {-20401265806.0 / 21769653311.0, 0.0, 0.0,
110 (354216.0 + 94326.0 * Math.sqrt(6.0)) / 112847.0,
111 (354216.0 - 94326.0 * Math.sqrt(6.0)) / 112847.0,
112 -43306765128.0 / 5313852383.0, -20866708358144.0 / 1126708119789.0,
113 14886003438020.0 / 654632330667.0, 35290686222309375.0 / 14152473387134411.0,
114 -1477884375.0 / 485066827.0},
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117 {39815761.0 / 17514443.0, 0.0, 0.0,
118 (-3457480.0 - 960905.0 * Math.sqrt(6.0)) / 551636.0,
119 (-3457480.0 + 960905.0 * Math.sqrt(6.0)) / 551636.0,
120 -844554132.0 / 47026969.0, 8444996352.0 / 302158619.0,
121 -2509602342.0 / 877790785.0, -28388795297996250.0 / 3199510091356783.0,
122 226716250.0 / 18341897.0, 1371316744.0 / 2131383595.0},
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127 {104257.0/1920240.0, 0.0, 0.0, 0.0, 0.0, 3399327.0/763840.0,
128 66578432.0/35198415.0, -1674902723.0/288716400.0,
129 54980371265625.0/176692375811392.0, -734375.0/4826304.0,
130 171414593.0/851261400.0, 137909.0/3084480.0}
131
132 };
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135 private static final double[] staticB = {
136 104257.0/1920240.0,
137 0.0,
138 0.0,
139 0.0,
140 0.0,
141 3399327.0/763840.0,
142 66578432.0/35198415.0,
143 -1674902723.0/288716400.0,
144 54980371265625.0/176692375811392.0,
145 -734375.0/4826304.0,
146 171414593.0/851261400.0,
147 137909.0/3084480.0,
148 0.0
149 };
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152 private static final double e1_01 = 116092271.0 / 8848465920.0;
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157 private static final double e1_06 = -1871647.0 / 1527680.0;
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160 private static final double e1_07 = -69799717.0 / 140793660.0;
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163 private static final double e1_08 = 1230164450203.0 / 739113984000.0;
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166 private static final double e1_09 = -1980813971228885.0 / 5654156025964544.0;
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169 private static final double e1_10 = 464500805.0 / 1389975552.0;
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172 private static final double e1_11 = 1606764981773.0 / 19613062656000.0;
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175 private static final double e1_12 = -137909.0 / 6168960.0;
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179 private static final double e2_01 = -364463.0 / 1920240.0;
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184 private static final double e2_06 = 3399327.0 / 763840.0;
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187 private static final double e2_07 = 66578432.0 / 35198415.0;
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190 private static final double e2_08 = -1674902723.0 / 288716400.0;
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193 private static final double e2_09 = -74684743568175.0 / 176692375811392.0;
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196 private static final double e2_10 = -734375.0 / 4826304.0;
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199 private static final double e2_11 = 171414593.0 / 851261400.0;
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202 private static final double e2_12 = 69869.0 / 3084480.0;
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213 public DormandPrince853Integrator(final double minStep, final double maxStep,
214 final double scalAbsoluteTolerance,
215 final double scalRelativeTolerance) {
216 super(METHOD_NAME, true, staticC, staticA, staticB,
217 new DormandPrince853StepInterpolator(),
218 minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance);
219 }
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230 public DormandPrince853Integrator(final double minStep, final double maxStep,
231 final double[] vecAbsoluteTolerance,
232 final double[] vecRelativeTolerance) {
233 super(METHOD_NAME, true, staticC, staticA, staticB,
234 new DormandPrince853StepInterpolator(),
235 minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance);
236 }
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239 @Override
240 public int getOrder() {
241 return 8;
242 }
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245 @Override
246 protected double estimateError(final double[][] yDotK,
247 final double[] y0, final double[] y1,
248 final double h) {
249 double error1 = 0;
250 double error2 = 0;
251
252 for (int j = 0; j < y0.length; ++j) {
253 final double errSum1 = e1_01 * yDotK[0][j] + e1_06 * yDotK[5][j] +
254 e1_07 * yDotK[6][j] + e1_08 * yDotK[7][j] +
255 e1_09 * yDotK[8][j] + e1_10 * yDotK[9][j] +
256 e1_11 * yDotK[10][j] + e1_12 * yDotK[11][j];
257 final double errSum2 = e2_01 * yDotK[0][j] + e2_06 * yDotK[5][j] +
258 e2_07 * yDotK[6][j] + e2_08 * yDotK[7][j] +
259 e2_09 * yDotK[8][j] + e2_10 * yDotK[9][j] +
260 e2_11 * yDotK[10][j] + e2_12 * yDotK[11][j];
261
262 final double yScale = Math.max(Math.abs(y0[j]), Math.abs(y1[j]));
263 final double tol = (vecAbsoluteTolerance == null) ?
264 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) :
265 (vecAbsoluteTolerance[j] + vecRelativeTolerance[j] * yScale);
266 final double ratio1 = errSum1 / tol;
267 error1 += ratio1 * ratio1;
268 final double ratio2 = errSum2 / tol;
269 error2 += ratio2 * ratio2;
270 }
271
272 double den = error1 + 0.01 * error2;
273 if (den <= 0.0) {
274 den = 1.0;
275 }
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277 return Math.abs(h) * error1 / Math.sqrt(y0.length * den);
278
279 }
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281 }