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.math3.ode.nonstiff; 019 020import org.apache.commons.math3.exception.DimensionMismatchException; 021import org.apache.commons.math3.exception.MaxCountExceededException; 022import org.apache.commons.math3.exception.NoBracketingException; 023import org.apache.commons.math3.exception.NumberIsTooSmallException; 024import org.apache.commons.math3.linear.Array2DRowRealMatrix; 025import org.apache.commons.math3.linear.RealMatrix; 026import org.apache.commons.math3.ode.EquationsMapper; 027import org.apache.commons.math3.ode.ExpandableStatefulODE; 028import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator; 029import org.apache.commons.math3.util.FastMath; 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 * <h3>Implementation details</h3> 055 * 056 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 057 * <pre> 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 * </pre></p> 064 * 065 * <p>The definitions above use the classical representation with several previous first 066 * derivatives. Lets define 067 * <pre> 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 * </pre> 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></p> 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 * <pre> 085 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 086 * </pre> 087 * (here again we omit the k index in the notation for clarity) 088 * </p> 089 * 090 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 091 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 092 * for degree k polynomials. 093 * <pre> 094 * 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) 095 * </pre> 096 * The previous formula can be used with several values for i to compute the transform between 097 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 098 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 099 * <pre> 100 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 101 * </pre> 102 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 103 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 104 * the column number starting from 1: 105 * <pre> 106 * [ -2 3 -4 5 ... ] 107 * [ -4 12 -32 80 ... ] 108 * P = [ -6 27 -108 405 ... ] 109 * [ -8 48 -256 1280 ... ] 110 * [ ... ] 111 * </pre></p> 112 * 113 * <p>Using the Nordsieck vector has several advantages: 114 * <ul> 115 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 116 * Taylor series formulas,</li> 117 * <li>it simplifies step changes that occur when discrete events that truncate 118 * the step are triggered,</li> 119 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 120 * </ul></p> 121 * 122 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: 123 * <ul> 124 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 125 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 126 * <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> 127 * </ul> 128 * where A is a rows shifting matrix (the lower left part is an identity matrix): 129 * <pre> 130 * [ 0 0 ... 0 0 | 0 ] 131 * [ ---------------+---] 132 * [ 1 0 ... 0 0 | 0 ] 133 * A = [ 0 1 ... 0 0 | 0 ] 134 * [ ... | 0 ] 135 * [ 0 0 ... 1 0 | 0 ] 136 * [ 0 0 ... 0 1 | 0 ] 137 * </pre></p> 138 * 139 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 140 * they only depend on k and therefore are precomputed once for all.</p> 141 * 142 * @since 2.0 143 */ 144public class AdamsBashforthIntegrator extends AdamsIntegrator { 145 146 /** Integrator method name. */ 147 private static final String METHOD_NAME = "Adams-Bashforth"; 148 149 /** 150 * Build an Adams-Bashforth integrator with the given order and step control parameters. 151 * @param nSteps number of steps of the method excluding the one being computed 152 * @param minStep minimal step (sign is irrelevant, regardless of 153 * integration direction, forward or backward), the last step can 154 * be smaller than this 155 * @param maxStep maximal step (sign is irrelevant, regardless of 156 * integration direction, forward or backward), the last step can 157 * be smaller than this 158 * @param scalAbsoluteTolerance allowed absolute error 159 * @param scalRelativeTolerance allowed relative error 160 * @exception NumberIsTooSmallException if order is 1 or less 161 */ 162 public AdamsBashforthIntegrator(final int nSteps, 163 final double minStep, final double maxStep, 164 final double scalAbsoluteTolerance, 165 final double scalRelativeTolerance) 166 throws NumberIsTooSmallException { 167 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 168 scalAbsoluteTolerance, scalRelativeTolerance); 169 } 170 171 /** 172 * Build an Adams-Bashforth integrator with the given order and step control parameters. 173 * @param nSteps number of steps of the method excluding the one being computed 174 * @param minStep minimal step (sign is irrelevant, regardless of 175 * integration direction, forward or backward), the last step can 176 * be smaller than this 177 * @param maxStep maximal step (sign is irrelevant, regardless of 178 * integration direction, forward or backward), the last step can 179 * be smaller than this 180 * @param vecAbsoluteTolerance allowed absolute error 181 * @param vecRelativeTolerance allowed relative error 182 * @exception IllegalArgumentException if order is 1 or less 183 */ 184 public AdamsBashforthIntegrator(final int nSteps, 185 final double minStep, final double maxStep, 186 final double[] vecAbsoluteTolerance, 187 final double[] vecRelativeTolerance) 188 throws IllegalArgumentException { 189 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 190 vecAbsoluteTolerance, vecRelativeTolerance); 191 } 192 193 /** Estimate error. 194 * <p> 195 * Error is estimated by interpolating back to previous state using 196 * the state Taylor expansion and comparing to real previous state. 197 * </p> 198 * @param previousState state vector at step start 199 * @param predictedState predicted state vector at step end 200 * @param predictedScaled predicted value of the scaled derivatives at step end 201 * @param predictedNordsieck predicted value of the Nordsieck vector at step end 202 * @return estimated normalized local discretization error 203 */ 204 private double errorEstimation(final double[] previousState, 205 final double[] predictedState, 206 final double[] predictedScaled, 207 final RealMatrix predictedNordsieck) { 208 209 double error = 0; 210 for (int i = 0; i < mainSetDimension; ++i) { 211 final double yScale = FastMath.abs(predictedState[i]); 212 final double tol = (vecAbsoluteTolerance == null) ? 213 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 214 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 215 216 // apply Taylor formula from high order to low order, 217 // for the sake of numerical accuracy 218 double variation = 0; 219 int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1; 220 for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { 221 variation += sign * predictedNordsieck.getEntry(k, i); 222 sign = -sign; 223 } 224 variation -= predictedScaled[i]; 225 226 final double ratio = (predictedState[i] - previousState[i] + variation) / tol; 227 error += ratio * ratio; 228 229 } 230 231 return FastMath.sqrt(error / mainSetDimension); 232 233 } 234 235 /** {@inheritDoc} */ 236 @Override 237 public void integrate(final ExpandableStatefulODE equations, final double t) 238 throws NumberIsTooSmallException, DimensionMismatchException, 239 MaxCountExceededException, NoBracketingException { 240 241 sanityChecks(equations, t); 242 setEquations(equations); 243 final boolean forward = t > equations.getTime(); 244 245 // initialize working arrays 246 final double[] y = equations.getCompleteState(); 247 final double[] yDot = new double[y.length]; 248 249 // set up an interpolator sharing the integrator arrays 250 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 251 interpolator.reinitialize(y, forward, 252 equations.getPrimaryMapper(), equations.getSecondaryMappers()); 253 254 // set up integration control objects 255 initIntegration(equations.getTime(), y, t); 256 257 // compute the initial Nordsieck vector using the configured starter integrator 258 start(equations.getTime(), y, t); 259 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 260 interpolator.storeTime(stepStart); 261 262 // reuse the step that was chosen by the starter integrator 263 double hNew = stepSize; 264 interpolator.rescale(hNew); 265 266 // main integration loop 267 isLastStep = false; 268 do { 269 270 interpolator.shift(); 271 final double[] predictedY = new double[y.length]; 272 final double[] predictedScaled = new double[y.length]; 273 Array2DRowRealMatrix predictedNordsieck = null; 274 double error = 10; 275 while (error >= 1.0) { 276 277 // predict a first estimate of the state at step end 278 final double stepEnd = stepStart + hNew; 279 interpolator.storeTime(stepEnd); 280 final ExpandableStatefulODE expandable = getExpandable(); 281 final EquationsMapper primary = expandable.getPrimaryMapper(); 282 primary.insertEquationData(interpolator.getInterpolatedState(), predictedY); 283 int index = 0; 284 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { 285 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY); 286 ++index; 287 } 288 289 // evaluate the derivative 290 computeDerivatives(stepEnd, predictedY, yDot); 291 292 // predict Nordsieck vector at step end 293 for (int j = 0; j < predictedScaled.length; ++j) { 294 predictedScaled[j] = hNew * yDot[j]; 295 } 296 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); 297 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); 298 299 // evaluate error 300 error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); 301 302 if (error >= 1.0) { 303 // reject the step and attempt to reduce error by stepsize control 304 final double factor = computeStepGrowShrinkFactor(error); 305 hNew = filterStep(hNew * factor, forward, false); 306 interpolator.rescale(hNew); 307 308 } 309 } 310 311 stepSize = hNew; 312 final double stepEnd = stepStart + stepSize; 313 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck); 314 315 // discrete events handling 316 interpolator.storeTime(stepEnd); 317 System.arraycopy(predictedY, 0, y, 0, y.length); 318 stepStart = acceptStep(interpolator, y, yDot, t); 319 scaled = predictedScaled; 320 nordsieck = predictedNordsieck; 321 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); 322 323 if (!isLastStep) { 324 325 // prepare next step 326 interpolator.storeTime(stepStart); 327 328 if (resetOccurred) { 329 // some events handler has triggered changes that 330 // invalidate the derivatives, we need to restart from scratch 331 start(stepStart, y, t); 332 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 333 } 334 335 // stepsize control for next step 336 final double factor = computeStepGrowShrinkFactor(error); 337 final double scaledH = stepSize * factor; 338 final double nextT = stepStart + scaledH; 339 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 340 hNew = filterStep(scaledH, forward, nextIsLast); 341 342 final double filteredNextT = stepStart + hNew; 343 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 344 if (filteredNextIsLast) { 345 hNew = t - stepStart; 346 } 347 348 interpolator.rescale(hNew); 349 350 } 351 352 } while (!isLastStep); 353 354 // dispatch results 355 equations.setTime(stepStart); 356 equations.setCompleteState(y); 357 358 resetInternalState(); 359 360 } 361 362}