| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF licenses this file to You under the Apache License, Version 2.0 |
| * (the "License"); you may not use this file except in compliance with |
| * the License. You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| |
| package org.apache.commons.math4.legacy.ode.nonstiff; |
| |
| import org.apache.commons.math4.legacy.core.Field; |
| import org.apache.commons.math4.legacy.core.RealFieldElement; |
| import org.apache.commons.math4.legacy.ode.FieldEquationsMapper; |
| import org.apache.commons.math4.legacy.ode.FieldODEStateAndDerivative; |
| import org.apache.commons.math4.legacy.core.MathArrays; |
| |
| |
| /** |
| * This class implements the 8(5,3) Dormand-Prince integrator for Ordinary |
| * Differential Equations. |
| * |
| * <p>This integrator is an embedded Runge-Kutta integrator |
| * of order 8(5,3) used in local extrapolation mode (i.e. the solution |
| * is computed using the high order formula) with stepsize control |
| * (and automatic step initialization) and continuous output. This |
| * method uses 12 functions evaluations per step for integration and 4 |
| * evaluations for interpolation. However, since the first |
| * interpolation evaluation is the same as the first integration |
| * evaluation of the next step, we have included it in the integrator |
| * rather than in the interpolator and specified the method was an |
| * <i>fsal</i>. Hence, despite we have 13 stages here, the cost is |
| * really 12 evaluations per step even if no interpolation is done, |
| * and the overcost of interpolation is only 3 evaluations.</p> |
| * |
| * <p>This method is based on an 8(6) method by Dormand and Prince |
| * (i.e. order 8 for the integration and order 6 for error estimation) |
| * modified by Hairer and Wanner to use a 5th order error estimator |
| * with 3rd order correction. This modification was introduced because |
| * the original method failed in some cases (wrong steps can be |
| * accepted when step size is too large, for example in the |
| * Brusselator problem) and also had <i>severe difficulties when |
| * applied to problems with discontinuities</i>. This modification is |
| * explained in the second edition of the first volume (Nonstiff |
| * Problems) of the reference book by Hairer, Norsett and Wanner: |
| * <i>Solving Ordinary Differential Equations</i> (Springer-Verlag, |
| * ISBN 3-540-56670-8).</p> |
| * |
| * @param <T> the type of the field elements |
| * @since 3.6 |
| */ |
| |
| public class DormandPrince853FieldIntegrator<T extends RealFieldElement<T>> |
| extends EmbeddedRungeKuttaFieldIntegrator<T> { |
| |
| /** Integrator method name. */ |
| private static final String METHOD_NAME = "Dormand-Prince 8 (5, 3)"; |
| |
| /** First error weights array, element 1. */ |
| private final T e1_01; |
| |
| // elements 2 to 5 are zero, so they are neither stored nor used |
| |
| /** First error weights array, element 6. */ |
| private final T e1_06; |
| |
| /** First error weights array, element 7. */ |
| private final T e1_07; |
| |
| /** First error weights array, element 8. */ |
| private final T e1_08; |
| |
| /** First error weights array, element 9. */ |
| private final T e1_09; |
| |
| /** First error weights array, element 10. */ |
| private final T e1_10; |
| |
| /** First error weights array, element 11. */ |
| private final T e1_11; |
| |
| /** First error weights array, element 12. */ |
| private final T e1_12; |
| |
| |
| /** Second error weights array, element 1. */ |
| private final T e2_01; |
| |
| // elements 2 to 5 are zero, so they are neither stored nor used |
| |
| /** Second error weights array, element 6. */ |
| private final T e2_06; |
| |
| /** Second error weights array, element 7. */ |
| private final T e2_07; |
| |
| /** Second error weights array, element 8. */ |
| private final T e2_08; |
| |
| /** Second error weights array, element 9. */ |
| private final T e2_09; |
| |
| /** Second error weights array, element 10. */ |
| private final T e2_10; |
| |
| /** Second error weights array, element 11. */ |
| private final T e2_11; |
| |
| /** Second error weights array, element 12. */ |
| private final T e2_12; |
| |
| /** Simple constructor. |
| * Build an eighth order Dormand-Prince integrator with the given step bounds |
| * @param field field to which the time and state vector elements belong |
| * @param minStep minimal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param maxStep maximal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param scalAbsoluteTolerance allowed absolute error |
| * @param scalRelativeTolerance allowed relative error |
| */ |
| public DormandPrince853FieldIntegrator(final Field<T> field, |
| final double minStep, final double maxStep, |
| final double scalAbsoluteTolerance, |
| final double scalRelativeTolerance) { |
| super(field, METHOD_NAME, 12, |
| minStep, maxStep, scalAbsoluteTolerance, scalRelativeTolerance); |
| e1_01 = fraction( 116092271.0, 8848465920.0); |
| e1_06 = fraction( -1871647.0, 1527680.0); |
| e1_07 = fraction( -69799717.0, 140793660.0); |
| e1_08 = fraction( 1230164450203.0, 739113984000.0); |
| e1_09 = fraction(-1980813971228885.0, 5654156025964544.0); |
| e1_10 = fraction( 464500805.0, 1389975552.0); |
| e1_11 = fraction( 1606764981773.0, 19613062656000.0); |
| e1_12 = fraction( -137909.0, 6168960.0); |
| e2_01 = fraction( -364463.0, 1920240.0); |
| e2_06 = fraction( 3399327.0, 763840.0); |
| e2_07 = fraction( 66578432.0, 35198415.0); |
| e2_08 = fraction( -1674902723.0, 288716400.0); |
| e2_09 = fraction( -74684743568175.0, 176692375811392.0); |
| e2_10 = fraction( -734375.0, 4826304.0); |
| e2_11 = fraction( 171414593.0, 851261400.0); |
| e2_12 = fraction( 69869.0, 3084480.0); |
| } |
| |
| /** Simple constructor. |
| * Build an eighth order Dormand-Prince integrator with the given step bounds |
| * @param field field to which the time and state vector elements belong |
| * @param minStep minimal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param maxStep maximal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param vecAbsoluteTolerance allowed absolute error |
| * @param vecRelativeTolerance allowed relative error |
| */ |
| public DormandPrince853FieldIntegrator(final Field<T> field, |
| final double minStep, final double maxStep, |
| final double[] vecAbsoluteTolerance, |
| final double[] vecRelativeTolerance) { |
| super(field, METHOD_NAME, 12, |
| minStep, maxStep, vecAbsoluteTolerance, vecRelativeTolerance); |
| e1_01 = fraction( 116092271.0, 8848465920.0); |
| e1_06 = fraction( -1871647.0, 1527680.0); |
| e1_07 = fraction( -69799717.0, 140793660.0); |
| e1_08 = fraction( 1230164450203.0, 739113984000.0); |
| e1_09 = fraction(-1980813971228885.0, 5654156025964544.0); |
| e1_10 = fraction( 464500805.0, 1389975552.0); |
| e1_11 = fraction( 1606764981773.0, 19613062656000.0); |
| e1_12 = fraction( -137909.0, 6168960.0); |
| e2_01 = fraction( -364463.0, 1920240.0); |
| e2_06 = fraction( 3399327.0, 763840.0); |
| e2_07 = fraction( 66578432.0, 35198415.0); |
| e2_08 = fraction( -1674902723.0, 288716400.0); |
| e2_09 = fraction( -74684743568175.0, 176692375811392.0); |
| e2_10 = fraction( -734375.0, 4826304.0); |
| e2_11 = fraction( 171414593.0, 851261400.0); |
| e2_12 = fraction( 69869.0, 3084480.0); |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public T[] getC() { |
| |
| final T sqrt6 = getField().getOne().multiply(6).sqrt(); |
| |
| final T[] c = MathArrays.buildArray(getField(), 15); |
| c[ 0] = sqrt6.add(-6).divide(-67.5); |
| c[ 1] = sqrt6.add(-6).divide(-45.0); |
| c[ 2] = sqrt6.add(-6).divide(-30.0); |
| c[ 3] = sqrt6.add( 6).divide( 30.0); |
| c[ 4] = fraction(1, 3); |
| c[ 5] = fraction(1, 4); |
| c[ 6] = fraction(4, 13); |
| c[ 7] = fraction(127, 195); |
| c[ 8] = fraction(3, 5); |
| c[ 9] = fraction(6, 7); |
| c[10] = getField().getOne(); |
| c[11] = getField().getOne(); |
| c[12] = fraction(1.0, 10.0); |
| c[13] = fraction(1.0, 5.0); |
| c[14] = fraction(7.0, 9.0); |
| |
| return c; |
| |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public T[][] getA() { |
| |
| final T sqrt6 = getField().getOne().multiply(6).sqrt(); |
| |
| final T[][] a = MathArrays.buildArray(getField(), 15, -1); |
| for (int i = 0; i < a.length; ++i) { |
| a[i] = MathArrays.buildArray(getField(), i + 1); |
| } |
| |
| a[ 0][ 0] = sqrt6.add(-6).divide(-67.5); |
| |
| a[ 1][ 0] = sqrt6.add(-6).divide(-180); |
| a[ 1][ 1] = sqrt6.add(-6).divide( -60); |
| |
| a[ 2][ 0] = sqrt6.add(-6).divide(-120); |
| a[ 2][ 1] = getField().getZero(); |
| a[ 2][ 2] = sqrt6.add(-6).divide( -40); |
| |
| a[ 3][ 0] = sqrt6.multiply(107).add(462).divide( 3000); |
| a[ 3][ 1] = getField().getZero(); |
| a[ 3][ 2] = sqrt6.multiply(197).add(402).divide(-1000); |
| a[ 3][ 3] = sqrt6.multiply( 73).add(168).divide( 375); |
| |
| a[ 4][ 0] = fraction(1, 27); |
| a[ 4][ 1] = getField().getZero(); |
| a[ 4][ 2] = getField().getZero(); |
| a[ 4][ 3] = sqrt6.add( 16).divide( 108); |
| a[ 4][ 4] = sqrt6.add(-16).divide(-108); |
| |
| a[ 5][ 0] = fraction(19, 512); |
| a[ 5][ 1] = getField().getZero(); |
| a[ 5][ 2] = getField().getZero(); |
| a[ 5][ 3] = sqrt6.multiply( 23).add(118).divide(1024); |
| a[ 5][ 4] = sqrt6.multiply(-23).add(118).divide(1024); |
| a[ 5][ 5] = fraction(-9, 512); |
| |
| a[ 6][ 0] = fraction(13772, 371293); |
| a[ 6][ 1] = getField().getZero(); |
| a[ 6][ 2] = getField().getZero(); |
| a[ 6][ 3] = sqrt6.multiply( 4784).add(51544).divide(371293); |
| a[ 6][ 4] = sqrt6.multiply(-4784).add(51544).divide(371293); |
| a[ 6][ 5] = fraction(-5688, 371293); |
| a[ 6][ 6] = fraction( 3072, 371293); |
| |
| a[ 7][ 0] = fraction(58656157643.0, 93983540625.0); |
| a[ 7][ 1] = getField().getZero(); |
| a[ 7][ 2] = getField().getZero(); |
| a[ 7][ 3] = sqrt6.multiply(-318801444819.0).add(-1324889724104.0).divide(626556937500.0); |
| a[ 7][ 4] = sqrt6.multiply( 318801444819.0).add(-1324889724104.0).divide(626556937500.0); |
| a[ 7][ 5] = fraction(96044563816.0, 3480871875.0); |
| a[ 7][ 6] = fraction(5682451879168.0, 281950621875.0); |
| a[ 7][ 7] = fraction(-165125654.0, 3796875.0); |
| |
| a[ 8][ 0] = fraction(8909899.0, 18653125.0); |
| a[ 8][ 1] = getField().getZero(); |
| a[ 8][ 2] = getField().getZero(); |
| a[ 8][ 3] = sqrt6.multiply(-1137963.0).add(-4521408.0).divide(2937500.0); |
| a[ 8][ 4] = sqrt6.multiply( 1137963.0).add(-4521408.0).divide(2937500.0); |
| a[ 8][ 5] = fraction(96663078.0, 4553125.0); |
| a[ 8][ 6] = fraction(2107245056.0, 137915625.0); |
| a[ 8][ 7] = fraction(-4913652016.0, 147609375.0); |
| a[ 8][ 8] = fraction(-78894270.0, 3880452869.0); |
| |
| a[ 9][ 0] = fraction(-20401265806.0, 21769653311.0); |
| a[ 9][ 1] = getField().getZero(); |
| a[ 9][ 2] = getField().getZero(); |
| a[ 9][ 3] = sqrt6.multiply( 94326.0).add(354216.0).divide(112847.0); |
| a[ 9][ 4] = sqrt6.multiply(-94326.0).add(354216.0).divide(112847.0); |
| a[ 9][ 5] = fraction(-43306765128.0, 5313852383.0); |
| a[ 9][ 6] = fraction(-20866708358144.0, 1126708119789.0); |
| a[ 9][ 7] = fraction(14886003438020.0, 654632330667.0); |
| a[ 9][ 8] = fraction(35290686222309375.0, 14152473387134411.0); |
| a[ 9][ 9] = fraction(-1477884375.0, 485066827.0); |
| |
| a[10][ 0] = fraction(39815761.0, 17514443.0); |
| a[10][ 1] = getField().getZero(); |
| a[10][ 2] = getField().getZero(); |
| a[10][ 3] = sqrt6.multiply(-960905.0).add(-3457480.0).divide(551636.0); |
| a[10][ 4] = sqrt6.multiply( 960905.0).add(-3457480.0).divide(551636.0); |
| a[10][ 5] = fraction(-844554132.0, 47026969.0); |
| a[10][ 6] = fraction(8444996352.0, 302158619.0); |
| a[10][ 7] = fraction(-2509602342.0, 877790785.0); |
| a[10][ 8] = fraction(-28388795297996250.0, 3199510091356783.0); |
| a[10][ 9] = fraction(226716250.0, 18341897.0); |
| a[10][10] = fraction(1371316744.0, 2131383595.0); |
| |
| // the following stage is both for interpolation and the first stage in next step |
| // (the coefficients are identical to the B array) |
| a[11][ 0] = fraction(104257.0, 1920240.0); |
| a[11][ 1] = getField().getZero(); |
| a[11][ 2] = getField().getZero(); |
| a[11][ 3] = getField().getZero(); |
| a[11][ 4] = getField().getZero(); |
| a[11][ 5] = fraction(3399327.0, 763840.0); |
| a[11][ 6] = fraction(66578432.0, 35198415.0); |
| a[11][ 7] = fraction(-1674902723.0, 288716400.0); |
| a[11][ 8] = fraction(54980371265625.0, 176692375811392.0); |
| a[11][ 9] = fraction(-734375.0, 4826304.0); |
| a[11][10] = fraction(171414593.0, 851261400.0); |
| a[11][11] = fraction(137909.0, 3084480.0); |
| |
| // the following stages are for interpolation only |
| a[12][ 0] = fraction( 13481885573.0, 240030000000.0); |
| a[12][ 1] = getField().getZero(); |
| a[12][ 2] = getField().getZero(); |
| a[12][ 3] = getField().getZero(); |
| a[12][ 4] = getField().getZero(); |
| a[12][ 5] = getField().getZero(); |
| a[12][ 6] = fraction( 139418837528.0, 549975234375.0); |
| a[12][ 7] = fraction( -11108320068443.0, 45111937500000.0); |
| a[12][ 8] = fraction(-1769651421925959.0, 14249385146080000.0); |
| a[12][ 9] = fraction( 57799439.0, 377055000.0); |
| a[12][10] = fraction( 793322643029.0, 96734250000000.0); |
| a[12][11] = fraction( 1458939311.0, 192780000000.0); |
| a[12][12] = fraction( -4149.0, 500000.0); |
| |
| a[13][ 0] = fraction( 1595561272731.0, 50120273500000.0); |
| a[13][ 1] = getField().getZero(); |
| a[13][ 2] = getField().getZero(); |
| a[13][ 3] = getField().getZero(); |
| a[13][ 4] = getField().getZero(); |
| a[13][ 5] = fraction( 975183916491.0, 34457688031250.0); |
| a[13][ 6] = fraction( 38492013932672.0, 718912673015625.0); |
| a[13][ 7] = fraction(-1114881286517557.0, 20298710767500000.0); |
| a[13][ 8] = getField().getZero(); |
| a[13][ 9] = getField().getZero(); |
| a[13][10] = fraction( -2538710946863.0, 23431227861250000.0); |
| a[13][11] = fraction( 8824659001.0, 23066716781250.0); |
| a[13][12] = fraction( -11518334563.0, 33831184612500.0); |
| a[13][13] = fraction( 1912306948.0, 13532473845.0); |
| |
| a[14][ 0] = fraction( -13613986967.0, 31741908048.0); |
| a[14][ 1] = getField().getZero(); |
| a[14][ 2] = getField().getZero(); |
| a[14][ 3] = getField().getZero(); |
| a[14][ 4] = getField().getZero(); |
| a[14][ 5] = fraction( -4755612631.0, 1012344804.0); |
| a[14][ 6] = fraction( 42939257944576.0, 5588559685701.0); |
| a[14][ 7] = fraction( 77881972900277.0, 19140370552944.0); |
| a[14][ 8] = fraction( 22719829234375.0, 63689648654052.0); |
| a[14][ 9] = getField().getZero(); |
| a[14][10] = getField().getZero(); |
| a[14][11] = getField().getZero(); |
| a[14][12] = fraction( -1199007803.0, 857031517296.0); |
| a[14][13] = fraction( 157882067000.0, 53564469831.0); |
| a[14][14] = fraction( -290468882375.0, 31741908048.0); |
| |
| return a; |
| |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public T[] getB() { |
| final T[] b = MathArrays.buildArray(getField(), 16); |
| b[ 0] = fraction(104257, 1920240); |
| b[ 1] = getField().getZero(); |
| b[ 2] = getField().getZero(); |
| b[ 3] = getField().getZero(); |
| b[ 4] = getField().getZero(); |
| b[ 5] = fraction( 3399327.0, 763840.0); |
| b[ 6] = fraction( 66578432.0, 35198415.0); |
| b[ 7] = fraction( -1674902723.0, 288716400.0); |
| b[ 8] = fraction( 54980371265625.0, 176692375811392.0); |
| b[ 9] = fraction( -734375.0, 4826304.0); |
| b[10] = fraction( 171414593.0, 851261400.0); |
| b[11] = fraction( 137909.0, 3084480.0); |
| b[12] = getField().getZero(); |
| b[13] = getField().getZero(); |
| b[14] = getField().getZero(); |
| b[15] = getField().getZero(); |
| return b; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| protected DormandPrince853FieldStepInterpolator<T> |
| createInterpolator(final boolean forward, T[][] yDotK, |
| final FieldODEStateAndDerivative<T> globalPreviousState, |
| final FieldODEStateAndDerivative<T> globalCurrentState, final FieldEquationsMapper<T> mapper) { |
| return new DormandPrince853FieldStepInterpolator<>(getField(), forward, yDotK, |
| globalPreviousState, globalCurrentState, |
| globalPreviousState, globalCurrentState, |
| mapper); |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public int getOrder() { |
| return 8; |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| protected T estimateError(final T[][] yDotK, final T[] y0, final T[] y1, final T h) { |
| T error1 = h.getField().getZero(); |
| T error2 = h.getField().getZero(); |
| |
| for (int j = 0; j < mainSetDimension; ++j) { |
| final T errSum1 = yDotK[ 0][j].multiply(e1_01). |
| add(yDotK[ 5][j].multiply(e1_06)). |
| add(yDotK[ 6][j].multiply(e1_07)). |
| add(yDotK[ 7][j].multiply(e1_08)). |
| add(yDotK[ 8][j].multiply(e1_09)). |
| add(yDotK[ 9][j].multiply(e1_10)). |
| add(yDotK[10][j].multiply(e1_11)). |
| add(yDotK[11][j].multiply(e1_12)); |
| final T errSum2 = yDotK[ 0][j].multiply(e2_01). |
| add(yDotK[ 5][j].multiply(e2_06)). |
| add(yDotK[ 6][j].multiply(e2_07)). |
| add(yDotK[ 7][j].multiply(e2_08)). |
| add(yDotK[ 8][j].multiply(e2_09)). |
| add(yDotK[ 9][j].multiply(e2_10)). |
| add(yDotK[10][j].multiply(e2_11)). |
| add(yDotK[11][j].multiply(e2_12)); |
| |
| final T yScale = RealFieldElement.max(y0[j].abs(), y1[j].abs()); |
| final T tol = vecAbsoluteTolerance == null ? |
| yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : |
| yScale.multiply(vecRelativeTolerance[j]).add(vecAbsoluteTolerance[j]); |
| final T ratio1 = errSum1.divide(tol); |
| error1 = error1.add(ratio1.multiply(ratio1)); |
| final T ratio2 = errSum2.divide(tol); |
| error2 = error2.add(ratio2.multiply(ratio2)); |
| } |
| |
| T den = error1.add(error2.multiply(0.01)); |
| if (den.getReal() <= 0.0) { |
| den = h.getField().getOne(); |
| } |
| |
| return h.abs().multiply(error1).divide(den.multiply(mainSetDimension).sqrt()); |
| |
| } |
| |
| } |
