1 /* 2 * Licensed to the Apache Software Foundation (ASF) under one or more 3 * contributor license agreements. See the NOTICE file distributed with 4 * this work for additional information regarding copyright ownership. 5 * The ASF licenses this file to You under the Apache License, Version 2.0 6 * (the "License"); you may not use this file except in compliance with 7 * the License. You may obtain a copy of the License at 8 * 9 * http://www.apache.org/licenses/LICENSE-2.0 10 * 11 * Unless required by applicable law or agreed to in writing, software 12 * distributed under the License is distributed on an "AS IS" BASIS, 13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14 * See the License for the specific language governing permissions and 15 * limitations under the License. 16 */ 17 18 package org.apache.commons.math.ode.nonstiff; 19 20 import java.util.Arrays; 21 22 import org.apache.commons.math.linear.Array2DRowRealMatrix; 23 import org.apache.commons.math.linear.MatrixVisitorException; 24 import org.apache.commons.math.linear.RealMatrixPreservingVisitor; 25 import org.apache.commons.math.ode.DerivativeException; 26 import org.apache.commons.math.ode.FirstOrderDifferentialEquations; 27 import org.apache.commons.math.ode.IntegratorException; 28 import org.apache.commons.math.ode.events.CombinedEventsManager; 29 import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator; 30 import org.apache.commons.math.ode.sampling.StepHandler; 31 32 33 /** 34 * This class implements implicit Adams-Moulton integrators for Ordinary 35 * Differential Equations. 36 * 37 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit 38 * multistep ODE solvers. This implementation is a variation of the classical 39 * one: it uses adaptive stepsize to implement error control, whereas 40 * classical implementations are fixed step size. The value of state vector 41 * at step n+1 is a simple combination of the value at step n and of the 42 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to 43 * compute y<sub>n+1</sub>,another method must be used to compute a first 44 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute 45 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending 46 * on the number k of previous steps one wants to use for computing the next 47 * value, different formulas are available for the final estimate:</p> 48 * <ul> 49 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li> 50 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li> 51 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li> 52 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li> 53 * <li>...</li> 54 * </ul> 55 * 56 * <p>A k-steps Adams-Moulton method is of order k+1.</p> 57 * 58 * <h3>Implementation details</h3> 59 * 60 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 61 * <pre> 62 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 63 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 64 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 65 * ... 66 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative 67 * </pre></p> 68 * 69 * <p>The definitions above use the classical representation with several previous first 70 * derivatives. Lets define 71 * <pre> 72 * 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> 73 * </pre> 74 * (we omit the k index in the notation for clarity). With these definitions, 75 * Adams-Moulton methods can be written: 76 * <ul> 77 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li> 78 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li> 79 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li> 80 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</sub></li> 81 * <li>...</li> 82 * </ul></p> 83 * 84 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 85 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with 86 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 87 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 88 * <pre> 89 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 90 * </pre> 91 * (here again we omit the k index in the notation for clarity) 92 * </p> 93 * 94 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 95 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 96 * for degree k polynomials. 97 * <pre> 98 * s<sub>1</sub>(n-i) = s<sub>1</sub>(n) + ∑<sub>j</sub> j (-i)<sup>j-1</sup> s<sub>j</sub>(n) 99 * </pre> 100 * The previous formula can be used with several values for i to compute the transform between 101 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 102 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 103 * <pre> 104 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 105 * </pre> 106 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 107 * with the j (-i)<sup>j-1</sup> terms: 108 * <pre> 109 * [ -2 3 -4 5 ... ] 110 * [ -4 12 -32 80 ... ] 111 * P = [ -6 27 -108 405 ... ] 112 * [ -8 48 -256 1280 ... ] 113 * [ ... ] 114 * </pre></p> 115 * 116 * <p>Using the Nordsieck vector has several advantages: 117 * <ul> 118 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 119 * Taylor series formulas,</li> 120 * <li>it simplifies step changes that occur when discrete events that truncate 121 * the step are triggered,</li> 122 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 123 * </ul></p> 124 * 125 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step 126 * n as follows: 127 * <ul> 128 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 129 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 130 * <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> 131 * </ul> 132 * where A is a rows shifting matrix (the lower left part is an identity matrix): 133 * <pre> 134 * [ 0 0 ... 0 0 | 0 ] 135 * [ ---------------+---] 136 * [ 1 0 ... 0 0 | 0 ] 137 * A = [ 0 1 ... 0 0 | 0 ] 138 * [ ... | 0 ] 139 * [ 0 0 ... 1 0 | 0 ] 140 * [ 0 0 ... 0 1 | 0 ] 141 * </pre> 142 * From this predicted vector, the corrected vector is computed as follows: 143 * <ul> 144 * <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub></li> 145 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 146 * <li>r<sub>n+1</sub> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li> 147 * </ul> 148 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 149 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 150 * represent the corrected states.</p> 151 * 152 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 153 * they only depend on k and therefore are precomputed once for all.</p> 154 * 155 * @version $Revision: 789159 $ $Date: 2009-06-28 17:56:20 -0400 (Sun, 28 Jun 2009) $ 156 * @since 2.0 157 */ 158 public class AdamsMoultonIntegrator extends AdamsIntegrator { 159 160 /** 161 * Build an Adams-Moulton integrator with the given order and error control parameters. 162 * @param nSteps number of steps of the method excluding the one being computed 163 * @param minStep minimal step (must be positive even for backward 164 * integration), the last step can be smaller than this 165 * @param maxStep maximal step (must be positive even for backward 166 * integration) 167 * @param scalAbsoluteTolerance allowed absolute error 168 * @param scalRelativeTolerance allowed relative error 169 * @exception IllegalArgumentException if order is 1 or less 170 */ 171 public AdamsMoultonIntegrator(final int nSteps, 172 final double minStep, final double maxStep, 173 final double scalAbsoluteTolerance, 174 final double scalRelativeTolerance) 175 throws IllegalArgumentException { 176 super("Adams-Moulton", nSteps, nSteps + 1, minStep, maxStep, 177 scalAbsoluteTolerance, scalRelativeTolerance); 178 } 179 180 /** 181 * Build an Adams-Moulton integrator with the given order and error control parameters. 182 * @param nSteps number of steps of the method excluding the one being computed 183 * @param minStep minimal step (must be positive even for backward 184 * integration), the last step can be smaller than this 185 * @param maxStep maximal step (must be positive even for backward 186 * integration) 187 * @param vecAbsoluteTolerance allowed absolute error 188 * @param vecRelativeTolerance allowed relative error 189 * @exception IllegalArgumentException if order is 1 or less 190 */ 191 public AdamsMoultonIntegrator(final int nSteps, 192 final double minStep, final double maxStep, 193 final double[] vecAbsoluteTolerance, 194 final double[] vecRelativeTolerance) 195 throws IllegalArgumentException { 196 super("Adams-Moulton", nSteps, nSteps + 1, minStep, maxStep, 197 vecAbsoluteTolerance, vecRelativeTolerance); 198 } 199 200 201 /** {@inheritDoc} */ 202 @Override 203 public double integrate(final FirstOrderDifferentialEquations equations, 204 final double t0, final double[] y0, 205 final double t, final double[] y) 206 throws DerivativeException, IntegratorException { 207 208 final int n = y0.length; 209 sanityChecks(equations, t0, y0, t, y); 210 setEquations(equations); 211 resetEvaluations(); 212 final boolean forward = (t > t0); 213 214 // initialize working arrays 215 if (y != y0) { 216 System.arraycopy(y0, 0, y, 0, n); 217 } 218 final double[] yDot = new double[y0.length]; 219 final double[] yTmp = new double[y0.length]; 220 221 // set up two interpolators sharing the integrator arrays 222 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 223 interpolator.reinitialize(y, forward); 224 final NordsieckStepInterpolator interpolatorTmp = new NordsieckStepInterpolator(); 225 interpolatorTmp.reinitialize(yTmp, forward); 226 227 // set up integration control objects 228 for (StepHandler handler : stepHandlers) { 229 handler.reset(); 230 } 231 CombinedEventsManager manager = addEndTimeChecker(t0, t, eventsHandlersManager); 232 233 234 // compute the initial Nordsieck vector using the configured starter integrator 235 start(t0, y, t); 236 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 237 interpolator.storeTime(stepStart); 238 239 double hNew = stepSize; 240 interpolator.rescale(hNew); 241 242 boolean lastStep = false; 243 while (!lastStep) { 244 245 // shift all data 246 interpolator.shift(); 247 248 double error = 0; 249 for (boolean loop = true; loop;) { 250 251 stepSize = hNew; 252 253 // predict a first estimate of the state at step end (P in the PECE sequence) 254 final double stepEnd = stepStart + stepSize; 255 interpolator.setInterpolatedTime(stepEnd); 256 System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length); 257 258 // evaluate a first estimate of the derivative (first E in the PECE sequence) 259 computeDerivatives(stepEnd, yTmp, yDot); 260 261 // update Nordsieck vector 262 final double[] predictedScaled = new double[y0.length]; 263 for (int j = 0; j < y0.length; ++j) { 264 predictedScaled[j] = stepSize * yDot[j]; 265 } 266 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck); 267 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp); 268 269 // apply correction (C in the PECE sequence) 270 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp)); 271 272 if (error <= 1.0) { 273 274 // evaluate a final estimate of the derivative (second E in the PECE sequence) 275 computeDerivatives(stepEnd, yTmp, yDot); 276 277 // update Nordsieck vector 278 final double[] correctedScaled = new double[y0.length]; 279 for (int j = 0; j < y0.length; ++j) { 280 correctedScaled[j] = stepSize * yDot[j]; 281 } 282 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp); 283 284 // discrete events handling 285 interpolatorTmp.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp); 286 interpolatorTmp.storeTime(stepStart); 287 interpolatorTmp.shift(); 288 interpolatorTmp.storeTime(stepEnd); 289 if (manager.evaluateStep(interpolatorTmp)) { 290 final double dt = manager.getEventTime() - stepStart; 291 if (Math.abs(dt) <= Math.ulp(stepStart)) { 292 // rejecting the step would lead to a too small next step, we accept it 293 loop = false; 294 } else { 295 // reject the step to match exactly the next switch time 296 hNew = dt; 297 interpolator.rescale(hNew); 298 } 299 } else { 300 // accept the step 301 scaled = correctedScaled; 302 nordsieck = nordsieckTmp; 303 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); 304 loop = false; 305 } 306 307 } else { 308 // reject the step and attempt to reduce error by stepsize control 309 final double factor = computeStepGrowShrinkFactor(error); 310 hNew = filterStep(stepSize * factor, forward, false); 311 interpolator.rescale(hNew); 312 } 313 314 } 315 316 // the step has been accepted (may have been truncated) 317 final double nextStep = stepStart + stepSize; 318 System.arraycopy(yTmp, 0, y, 0, n); 319 interpolator.storeTime(nextStep); 320 manager.stepAccepted(nextStep, y); 321 lastStep = manager.stop(); 322 323 // provide the step data to the step handler 324 for (StepHandler handler : stepHandlers) { 325 interpolator.setInterpolatedTime(nextStep); 326 handler.handleStep(interpolator, lastStep); 327 } 328 stepStart = nextStep; 329 330 if (!lastStep && manager.reset(stepStart, y)) { 331 332 // some events handler has triggered changes that 333 // invalidate the derivatives, we need to restart from scratch 334 start(stepStart, y, t); 335 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 336 337 } 338 339 if (! lastStep) { 340 // in some rare cases we may get here with stepSize = 0, for example 341 // when an event occurs at integration start, reducing the first step 342 // to zero; we have to reset the step to some safe non zero value 343 stepSize = filterStep(stepSize, forward, true); 344 345 // stepsize control for next step 346 final double factor = computeStepGrowShrinkFactor(error); 347 final double scaledH = stepSize * factor; 348 final double nextT = stepStart + scaledH; 349 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 350 hNew = filterStep(scaledH, forward, nextIsLast); 351 interpolator.rescale(hNew); 352 } 353 354 } 355 356 final double stopTime = stepStart; 357 stepStart = Double.NaN; 358 stepSize = Double.NaN; 359 return stopTime; 360 361 } 362 363 /** Corrector for current state in Adams-Moulton method. 364 * <p> 365 * This visitor implements the Taylor series formula: 366 * <pre> 367 * Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub> 368 * </pre> 369 * </p> 370 */ 371 private class Corrector implements RealMatrixPreservingVisitor { 372 373 /** Previous state. */ 374 private final double[] previous; 375 376 /** Current scaled first derivative. */ 377 private final double[] scaled; 378 379 /** Current state before correction. */ 380 private final double[] before; 381 382 /** Current state after correction. */ 383 private final double[] after; 384 385 /** Simple constructor. 386 * @param previous previous state 387 * @param scaled current scaled first derivative 388 * @param state state to correct (will be overwritten after visit) 389 */ 390 public Corrector(final double[] previous, final double[] scaled, final double[] state) { 391 this.previous = previous; 392 this.scaled = scaled; 393 this.after = state; 394 this.before = state.clone(); 395 } 396 397 /** {@inheritDoc} */ 398 public void start(int rows, int columns, 399 int startRow, int endRow, int startColumn, int endColumn) { 400 Arrays.fill(after, 0.0); 401 } 402 403 /** {@inheritDoc} */ 404 public void visit(int row, int column, double value) 405 throws MatrixVisitorException { 406 if ((row & 0x1) == 0) { 407 after[column] -= value; 408 } else { 409 after[column] += value; 410 } 411 } 412 413 /** 414 * End visiting te Nordsieck vector. 415 * <p>The correction is used to control stepsize. So its amplitude is 416 * considered to be an error, which must be normalized according to 417 * error control settings. If the normalized value is greater than 1, 418 * the correction was too large and the step must be rejected.</p> 419 * @return the normalized correction, if greater than 1, the step 420 * must be rejected 421 */ 422 public double end() { 423 424 double error = 0; 425 for (int i = 0; i < after.length; ++i) { 426 after[i] += previous[i] + scaled[i]; 427 final double yScale = Math.max(Math.abs(previous[i]), Math.abs(after[i])); 428 final double tol = (vecAbsoluteTolerance == null) ? 429 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 430 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 431 final double ratio = (after[i] - before[i]) / tol; 432 error += ratio * ratio; 433 } 434 435 return Math.sqrt(error / after.length); 436 437 } 438 } 439 440 }