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.RealMatrixPreservingVisitor; 24 import org.apache.commons.math.ode.DerivativeException; 25 import org.apache.commons.math.ode.FirstOrderDifferentialEquations; 26 import org.apache.commons.math.ode.IntegratorException; 27 import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator; 28 import org.apache.commons.math.ode.sampling.StepHandler; 29 import org.apache.commons.math.util.FastMath; 30 31 32 /** 33 * This class implements implicit Adams-Moulton integrators for Ordinary 34 * Differential Equations. 35 * 36 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit 37 * multistep ODE solvers. This implementation is a variation of the classical 38 * one: it uses adaptive stepsize to implement error control, whereas 39 * classical implementations are fixed step size. The value of state vector 40 * at step n+1 is a simple combination of the value at step n and of the 41 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to 42 * compute y<sub>n+1</sub>,another method must be used to compute a first 43 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute 44 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending 45 * on the number k of previous steps one wants to use for computing the next 46 * value, different formulas are available for the final estimate:</p> 47 * <ul> 48 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li> 49 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li> 50 * <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> 51 * <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> 52 * <li>...</li> 53 * </ul> 54 * 55 * <p>A k-steps Adams-Moulton method is of order k+1.</p> 56 * 57 * <h3>Implementation details</h3> 58 * 59 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 60 * <pre> 61 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 62 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 63 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 64 * ... 65 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative 66 * </pre></p> 67 * 68 * <p>The definitions above use the classical representation with several previous first 69 * derivatives. Lets define 70 * <pre> 71 * 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> 72 * </pre> 73 * (we omit the k index in the notation for clarity). With these definitions, 74 * Adams-Moulton methods can be written: 75 * <ul> 76 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li> 77 * <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> 78 * <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> 79 * <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> 80 * <li>...</li> 81 * </ul></p> 82 * 83 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 84 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with 85 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 86 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 87 * <pre> 88 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 89 * </pre> 90 * (here again we omit the k index in the notation for clarity) 91 * </p> 92 * 93 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 94 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 95 * for degree k polynomials. 96 * <pre> 97 * 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) 98 * </pre> 99 * The previous formula can be used with several values for i to compute the transform between 100 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 101 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 102 * <pre> 103 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 104 * </pre> 105 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 106 * with the j (-i)<sup>j-1</sup> terms: 107 * <pre> 108 * [ -2 3 -4 5 ... ] 109 * [ -4 12 -32 80 ... ] 110 * P = [ -6 27 -108 405 ... ] 111 * [ -8 48 -256 1280 ... ] 112 * [ ... ] 113 * </pre></p> 114 * 115 * <p>Using the Nordsieck vector has several advantages: 116 * <ul> 117 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 118 * Taylor series formulas,</li> 119 * <li>it simplifies step changes that occur when discrete events that truncate 120 * the step are triggered,</li> 121 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 122 * </ul></p> 123 * 124 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step 125 * n as follows: 126 * <ul> 127 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 128 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 129 * <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> 130 * </ul> 131 * where A is a rows shifting matrix (the lower left part is an identity matrix): 132 * <pre> 133 * [ 0 0 ... 0 0 | 0 ] 134 * [ ---------------+---] 135 * [ 1 0 ... 0 0 | 0 ] 136 * A = [ 0 1 ... 0 0 | 0 ] 137 * [ ... | 0 ] 138 * [ 0 0 ... 1 0 | 0 ] 139 * [ 0 0 ... 0 1 | 0 ] 140 * </pre> 141 * From this predicted vector, the corrected vector is computed as follows: 142 * <ul> 143 * <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> 144 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 145 * <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> 146 * </ul> 147 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 148 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 149 * represent the corrected states.</p> 150 * 151 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 152 * they only depend on k and therefore are precomputed once for all.</p> 153 * 154 * @version $Revision: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 fvr. 2011) $ 155 * @since 2.0 156 */ 157 public class AdamsMoultonIntegrator extends AdamsIntegrator { 158 159 /** Integrator method name. */ 160 private static final String METHOD_NAME = "Adams-Moulton"; 161 162 /** 163 * Build an Adams-Moulton integrator with the given order and error control parameters. 164 * @param nSteps number of steps of the method excluding the one being computed 165 * @param minStep minimal step (must be positive even for backward 166 * integration), the last step can be smaller than this 167 * @param maxStep maximal step (must be positive even for backward 168 * integration) 169 * @param scalAbsoluteTolerance allowed absolute error 170 * @param scalRelativeTolerance allowed relative error 171 * @exception IllegalArgumentException if order is 1 or less 172 */ 173 public AdamsMoultonIntegrator(final int nSteps, 174 final double minStep, final double maxStep, 175 final double scalAbsoluteTolerance, 176 final double scalRelativeTolerance) 177 throws IllegalArgumentException { 178 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 179 scalAbsoluteTolerance, scalRelativeTolerance); 180 } 181 182 /** 183 * Build an Adams-Moulton integrator with the given order and error control parameters. 184 * @param nSteps number of steps of the method excluding the one being computed 185 * @param minStep minimal step (must be positive even for backward 186 * integration), the last step can be smaller than this 187 * @param maxStep maximal step (must be positive even for backward 188 * integration) 189 * @param vecAbsoluteTolerance allowed absolute error 190 * @param vecRelativeTolerance allowed relative error 191 * @exception IllegalArgumentException if order is 1 or less 192 */ 193 public AdamsMoultonIntegrator(final int nSteps, 194 final double minStep, final double maxStep, 195 final double[] vecAbsoluteTolerance, 196 final double[] vecRelativeTolerance) 197 throws IllegalArgumentException { 198 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 199 vecAbsoluteTolerance, vecRelativeTolerance); 200 } 201 202 203 /** {@inheritDoc} */ 204 @Override 205 public double integrate(final FirstOrderDifferentialEquations equations, 206 final double t0, final double[] y0, 207 final double t, final double[] y) 208 throws DerivativeException, IntegratorException { 209 210 final int n = y0.length; 211 sanityChecks(equations, t0, y0, t, y); 212 setEquations(equations); 213 resetEvaluations(); 214 final boolean forward = t > t0; 215 216 // initialize working arrays 217 if (y != y0) { 218 System.arraycopy(y0, 0, y, 0, n); 219 } 220 final double[] yDot = new double[y0.length]; 221 final double[] yTmp = new double[y0.length]; 222 final double[] predictedScaled = new double[y0.length]; 223 Array2DRowRealMatrix nordsieckTmp = null; 224 225 // set up two interpolators sharing the integrator arrays 226 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 227 interpolator.reinitialize(y, forward); 228 229 // set up integration control objects 230 for (StepHandler handler : stepHandlers) { 231 handler.reset(); 232 } 233 setStateInitialized(false); 234 235 // compute the initial Nordsieck vector using the configured starter integrator 236 start(t0, y, t); 237 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 238 interpolator.storeTime(stepStart); 239 240 double hNew = stepSize; 241 interpolator.rescale(hNew); 242 243 isLastStep = false; 244 do { 245 246 double error = 10; 247 while (error >= 1.0) { 248 249 stepSize = hNew; 250 251 // predict a first estimate of the state at step end (P in the PECE sequence) 252 final double stepEnd = stepStart + stepSize; 253 interpolator.setInterpolatedTime(stepEnd); 254 System.arraycopy(interpolator.getInterpolatedState(), 0, yTmp, 0, y0.length); 255 256 // evaluate a first estimate of the derivative (first E in the PECE sequence) 257 computeDerivatives(stepEnd, yTmp, yDot); 258 259 // update Nordsieck vector 260 for (int j = 0; j < y0.length; ++j) { 261 predictedScaled[j] = stepSize * yDot[j]; 262 } 263 nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck); 264 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp); 265 266 // apply correction (C in the PECE sequence) 267 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp)); 268 269 if (error >= 1.0) { 270 // reject the step and attempt to reduce error by stepsize control 271 final double factor = computeStepGrowShrinkFactor(error); 272 hNew = filterStep(stepSize * factor, forward, false); 273 interpolator.rescale(hNew); 274 } 275 } 276 277 // evaluate a final estimate of the derivative (second E in the PECE sequence) 278 final double stepEnd = stepStart + stepSize; 279 computeDerivatives(stepEnd, yTmp, yDot); 280 281 // update Nordsieck vector 282 final double[] correctedScaled = new double[y0.length]; 283 for (int j = 0; j < y0.length; ++j) { 284 correctedScaled[j] = stepSize * yDot[j]; 285 } 286 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp); 287 288 // discrete events handling 289 System.arraycopy(yTmp, 0, y, 0, n); 290 interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp); 291 interpolator.storeTime(stepStart); 292 interpolator.shift(); 293 interpolator.storeTime(stepEnd); 294 stepStart = acceptStep(interpolator, y, yDot, t); 295 scaled = correctedScaled; 296 nordsieck = nordsieckTmp; 297 298 if (!isLastStep) { 299 300 // prepare next step 301 interpolator.storeTime(stepStart); 302 303 if (resetOccurred) { 304 // some events handler has triggered changes that 305 // invalidate the derivatives, we need to restart from scratch 306 start(stepStart, y, t); 307 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 308 309 } 310 311 // stepsize control for next step 312 final double factor = computeStepGrowShrinkFactor(error); 313 final double scaledH = stepSize * factor; 314 final double nextT = stepStart + scaledH; 315 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 316 hNew = filterStep(scaledH, forward, nextIsLast); 317 318 final double filteredNextT = stepStart + hNew; 319 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 320 if (filteredNextIsLast) { 321 hNew = t - stepStart; 322 } 323 324 interpolator.rescale(hNew); 325 } 326 327 } while (!isLastStep); 328 329 final double stopTime = stepStart; 330 stepStart = Double.NaN; 331 stepSize = Double.NaN; 332 return stopTime; 333 334 } 335 336 /** Corrector for current state in Adams-Moulton method. 337 * <p> 338 * This visitor implements the Taylor series formula: 339 * <pre> 340 * 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> 341 * </pre> 342 * </p> 343 */ 344 private class Corrector implements RealMatrixPreservingVisitor { 345 346 /** Previous state. */ 347 private final double[] previous; 348 349 /** Current scaled first derivative. */ 350 private final double[] scaled; 351 352 /** Current state before correction. */ 353 private final double[] before; 354 355 /** Current state after correction. */ 356 private final double[] after; 357 358 /** Simple constructor. 359 * @param previous previous state 360 * @param scaled current scaled first derivative 361 * @param state state to correct (will be overwritten after visit) 362 */ 363 public Corrector(final double[] previous, final double[] scaled, final double[] state) { 364 this.previous = previous; 365 this.scaled = scaled; 366 this.after = state; 367 this.before = state.clone(); 368 } 369 370 /** {@inheritDoc} */ 371 public void start(int rows, int columns, 372 int startRow, int endRow, int startColumn, int endColumn) { 373 Arrays.fill(after, 0.0); 374 } 375 376 /** {@inheritDoc} */ 377 public void visit(int row, int column, double value) { 378 if ((row & 0x1) == 0) { 379 after[column] -= value; 380 } else { 381 after[column] += value; 382 } 383 } 384 385 /** 386 * End visiting the Nordsieck vector. 387 * <p>The correction is used to control stepsize. So its amplitude is 388 * considered to be an error, which must be normalized according to 389 * error control settings. If the normalized value is greater than 1, 390 * the correction was too large and the step must be rejected.</p> 391 * @return the normalized correction, if greater than 1, the step 392 * must be rejected 393 */ 394 public double end() { 395 396 double error = 0; 397 for (int i = 0; i < after.length; ++i) { 398 after[i] += previous[i] + scaled[i]; 399 if (i < mainSetDimension) { 400 final double yScale = FastMath.max(FastMath.abs(previous[i]), FastMath.abs(after[i])); 401 final double tol = (vecAbsoluteTolerance == null) ? 402 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 403 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 404 final double ratio = (after[i] - before[i]) / tol; 405 error += ratio * ratio; 406 } 407 } 408 409 return FastMath.sqrt(error / mainSetDimension); 410 411 } 412 } 413 414 } 415
