001/* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 018package org.apache.commons.math4.legacy.ode.nonstiff; 019 020import org.apache.commons.math4.legacy.exception.DimensionMismatchException; 021import org.apache.commons.math4.legacy.exception.MaxCountExceededException; 022import org.apache.commons.math4.legacy.exception.NoBracketingException; 023import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; 024import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix; 025import org.apache.commons.math4.legacy.linear.RealMatrix; 026import org.apache.commons.math4.legacy.ode.EquationsMapper; 027import org.apache.commons.math4.legacy.ode.ExpandableStatefulODE; 028import org.apache.commons.math4.legacy.ode.sampling.NordsieckStepInterpolator; 029import org.apache.commons.math4.core.jdkmath.JdkMath; 030 031 032/** 033 * This class implements explicit Adams-Bashforth integrators for Ordinary 034 * Differential Equations. 035 * 036 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit 037 * multistep ODE solvers. This implementation is a variation of the classical 038 * one: it uses adaptive stepsize to implement error control, whereas 039 * classical implementations are fixed step size. The value of state vector 040 * at step n+1 is a simple combination of the value at step n and of the 041 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous 042 * steps one wants to use for computing the next value, different formulas 043 * are available:</p> 044 * <ul> 045 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li> 046 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li> 047 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li> 048 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li> 049 * <li>...</li> 050 * </ul> 051 * 052 * <p>A k-steps Adams-Bashforth method is of order k.</p> 053 * 054 * <p><b>Implementation details</b></p> 055 * 056 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 057 * <div style="white-space: pre"><code> 058 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 059 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 060 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 061 * ... 062 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 063 * </code></div> 064 * 065 * <p>The definitions above use the classical representation with several previous first 066 * derivatives. Lets define 067 * <div style="white-space: pre"><code> 068 * q<sub>n</sub> = [ s<sub>1</sub>(n-1) s<sub>1</sub>(n-2) ... s<sub>1</sub>(n-(k-1)) ]<sup>T</sup> 069 * </code></div> 070 * (we omit the k index in the notation for clarity). With these definitions, 071 * Adams-Bashforth methods can be written: 072 * <ul> 073 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li> 074 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ] q<sub>n</sub></li> 075 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [ -16/12 5/12 ] q<sub>n</sub></li> 076 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [ -59/24 37/24 -9/24 ] q<sub>n</sub></li> 077 * <li>...</li> 078 * </ul> 079 * 080 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 081 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with 082 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 083 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 084 * <div style="white-space: pre"><code> 085 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 086 * </code></div> 087 * (here again we omit the k index in the notation for clarity) 088 * 089 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 090 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 091 * for degree k polynomials. 092 * <div style="white-space: pre"><code> 093 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j>0</sub> (j+1) (-i)<sup>j</sup> s<sub>j+1</sub>(n) 094 * </code></div> 095 * The previous formula can be used with several values for i to compute the transform between 096 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 097 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 098 * <div style="white-space: pre"><code> 099 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 100 * </code></div> 101 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 102 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 103 * the column number starting from 1: 104 * <pre> 105 * [ -2 3 -4 5 ... ] 106 * [ -4 12 -32 80 ... ] 107 * P = [ -6 27 -108 405 ... ] 108 * [ -8 48 -256 1280 ... ] 109 * [ ... ] 110 * </pre> 111 * 112 * <p>Using the Nordsieck vector has several advantages: 113 * <ul> 114 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 115 * Taylor series formulas,</li> 116 * <li>it simplifies step changes that occur when discrete events that truncate 117 * the step are triggered,</li> 118 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 119 * </ul> 120 * 121 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: 122 * <ul> 123 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 124 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 125 * <li>r<sub>n+1</sub> = (s<sub>1</sub>(n) - s<sub>1</sub>(n+1)) P<sup>-1</sup> u + P<sup>-1</sup> A P r<sub>n</sub></li> 126 * </ul> 127 * where A is a rows shifting matrix (the lower left part is an identity matrix): 128 * <pre> 129 * [ 0 0 ... 0 0 | 0 ] 130 * [ ---------------+---] 131 * [ 1 0 ... 0 0 | 0 ] 132 * A = [ 0 1 ... 0 0 | 0 ] 133 * [ ... | 0 ] 134 * [ 0 0 ... 1 0 | 0 ] 135 * [ 0 0 ... 0 1 | 0 ] 136 * </pre> 137 * 138 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 139 * they only depend on k and therefore are precomputed once for all.</p> 140 * 141 * @since 2.0 142 */ 143public class AdamsBashforthIntegrator extends AdamsIntegrator { 144 145 /** Integrator method name. */ 146 private static final String METHOD_NAME = "Adams-Bashforth"; 147 148 /** 149 * Build an Adams-Bashforth integrator with the given order and step control parameters. 150 * @param nSteps number of steps of the method excluding the one being computed 151 * @param minStep minimal step (sign is irrelevant, regardless of 152 * integration direction, forward or backward), the last step can 153 * be smaller than this 154 * @param maxStep maximal step (sign is irrelevant, regardless of 155 * integration direction, forward or backward), the last step can 156 * be smaller than this 157 * @param scalAbsoluteTolerance allowed absolute error 158 * @param scalRelativeTolerance allowed relative error 159 * @exception NumberIsTooSmallException if order is 1 or less 160 */ 161 public AdamsBashforthIntegrator(final int nSteps, 162 final double minStep, final double maxStep, 163 final double scalAbsoluteTolerance, 164 final double scalRelativeTolerance) 165 throws NumberIsTooSmallException { 166 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 167 scalAbsoluteTolerance, scalRelativeTolerance); 168 } 169 170 /** 171 * Build an Adams-Bashforth integrator with the given order and step control parameters. 172 * @param nSteps number of steps of the method excluding the one being computed 173 * @param minStep minimal step (sign is irrelevant, regardless of 174 * integration direction, forward or backward), the last step can 175 * be smaller than this 176 * @param maxStep maximal step (sign is irrelevant, regardless of 177 * integration direction, forward or backward), the last step can 178 * be smaller than this 179 * @param vecAbsoluteTolerance allowed absolute error 180 * @param vecRelativeTolerance allowed relative error 181 * @exception IllegalArgumentException if order is 1 or less 182 */ 183 public AdamsBashforthIntegrator(final int nSteps, 184 final double minStep, final double maxStep, 185 final double[] vecAbsoluteTolerance, 186 final double[] vecRelativeTolerance) 187 throws IllegalArgumentException { 188 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 189 vecAbsoluteTolerance, vecRelativeTolerance); 190 } 191 192 /** Estimate error. 193 * <p> 194 * Error is estimated by interpolating back to previous state using 195 * the state Taylor expansion and comparing to real previous state. 196 * </p> 197 * @param previousState state vector at step start 198 * @param predictedState predicted state vector at step end 199 * @param predictedScaled predicted value of the scaled derivatives at step end 200 * @param predictedNordsieck predicted value of the Nordsieck vector at step end 201 * @return estimated normalized local discretization error 202 */ 203 private double errorEstimation(final double[] previousState, 204 final double[] predictedState, 205 final double[] predictedScaled, 206 final RealMatrix predictedNordsieck) { 207 208 double error = 0; 209 for (int i = 0; i < mainSetDimension; ++i) { 210 final double yScale = JdkMath.abs(predictedState[i]); 211 final double tol = (vecAbsoluteTolerance == null) ? 212 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 213 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 214 215 // apply Taylor formula from high order to low order, 216 // for the sake of numerical accuracy 217 double variation = 0; 218 int sign = (predictedNordsieck.getRowDimension() & 1) == 0 ? -1 : 1; 219 for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { 220 variation += sign * predictedNordsieck.getEntry(k, i); 221 sign = -sign; 222 } 223 variation -= predictedScaled[i]; 224 225 final double ratio = (predictedState[i] - previousState[i] + variation) / tol; 226 error += ratio * ratio; 227 } 228 229 return JdkMath.sqrt(error / mainSetDimension); 230 } 231 232 /** {@inheritDoc} */ 233 @Override 234 public void integrate(final ExpandableStatefulODE equations, final double t) 235 throws NumberIsTooSmallException, DimensionMismatchException, 236 MaxCountExceededException, NoBracketingException { 237 238 sanityChecks(equations, t); 239 setEquations(equations); 240 final boolean forward = t > equations.getTime(); 241 242 // initialize working arrays 243 final double[] y = equations.getCompleteState(); 244 final double[] yDot = new double[y.length]; 245 246 // set up an interpolator sharing the integrator arrays 247 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 248 interpolator.reinitialize(y, forward, 249 equations.getPrimaryMapper(), equations.getSecondaryMappers()); 250 251 // set up integration control objects 252 initIntegration(equations.getTime(), y, t); 253 254 // compute the initial Nordsieck vector using the configured starter integrator 255 start(equations.getTime(), y, t); 256 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 257 interpolator.storeTime(stepStart); 258 259 // reuse the step that was chosen by the starter integrator 260 double hNew = stepSize; 261 interpolator.rescale(hNew); 262 263 // main integration loop 264 isLastStep = false; 265 do { 266 267 interpolator.shift(); 268 final double[] predictedY = new double[y.length]; 269 final double[] predictedScaled = new double[y.length]; 270 Array2DRowRealMatrix predictedNordsieck = null; 271 double error = 10; 272 while (error >= 1.0) { 273 274 // predict a first estimate of the state at step end 275 final double stepEnd = stepStart + hNew; 276 interpolator.storeTime(stepEnd); 277 final ExpandableStatefulODE expandable = getExpandable(); 278 final EquationsMapper primary = expandable.getPrimaryMapper(); 279 primary.insertEquationData(interpolator.getInterpolatedState(), predictedY); 280 int index = 0; 281 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { 282 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY); 283 ++index; 284 } 285 286 // evaluate the derivative 287 computeDerivatives(stepEnd, predictedY, yDot); 288 289 // predict Nordsieck vector at step end 290 for (int j = 0; j < predictedScaled.length; ++j) { 291 predictedScaled[j] = hNew * yDot[j]; 292 } 293 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); 294 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); 295 296 // evaluate error 297 error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); 298 299 if (error >= 1.0) { 300 // reject the step and attempt to reduce error by stepsize control 301 final double factor = computeStepGrowShrinkFactor(error); 302 hNew = filterStep(hNew * factor, forward, false); 303 interpolator.rescale(hNew); 304 } 305 } 306 307 stepSize = hNew; 308 final double stepEnd = stepStart + stepSize; 309 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck); 310 311 // discrete events handling 312 interpolator.storeTime(stepEnd); 313 System.arraycopy(predictedY, 0, y, 0, y.length); 314 stepStart = acceptStep(interpolator, y, yDot, t); 315 scaled = predictedScaled; 316 nordsieck = predictedNordsieck; 317 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); 318 319 if (!isLastStep) { 320 321 // prepare next step 322 interpolator.storeTime(stepStart); 323 324 if (resetOccurred) { 325 // some events handler has triggered changes that 326 // invalidate the derivatives, we need to restart from scratch 327 start(stepStart, y, t); 328 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 329 } 330 331 // stepsize control for next step 332 final double factor = computeStepGrowShrinkFactor(error); 333 final double scaledH = stepSize * factor; 334 final double nextT = stepStart + scaledH; 335 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 336 hNew = filterStep(scaledH, forward, nextIsLast); 337 338 final double filteredNextT = stepStart + hNew; 339 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 340 if (filteredNextIsLast) { 341 hNew = t - stepStart; 342 } 343 344 interpolator.rescale(hNew); 345 } 346 } while (!isLastStep); 347 348 // dispatch results 349 equations.setTime(stepStart); 350 equations.setCompleteState(y); 351 352 resetInternalState(); 353 } 354}