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 java.util.Arrays; 021 022import org.apache.commons.math3.exception.DimensionMismatchException; 023import org.apache.commons.math3.exception.MaxCountExceededException; 024import org.apache.commons.math3.exception.NoBracketingException; 025import org.apache.commons.math3.exception.NumberIsTooSmallException; 026import org.apache.commons.math3.linear.Array2DRowRealMatrix; 027import org.apache.commons.math3.linear.RealMatrixPreservingVisitor; 028import org.apache.commons.math3.ode.EquationsMapper; 029import org.apache.commons.math3.ode.ExpandableStatefulODE; 030import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator; 031import org.apache.commons.math3.util.FastMath; 032 033 034/** 035 * This class implements implicit Adams-Moulton integrators for Ordinary 036 * Differential Equations. 037 * 038 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit 039 * multistep ODE solvers. This implementation is a variation of the classical 040 * one: it uses adaptive stepsize to implement error control, whereas 041 * classical implementations are fixed step size. The value of state vector 042 * at step n+1 is a simple combination of the value at step n and of the 043 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to 044 * compute y<sub>n+1</sub>, another method must be used to compute a first 045 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute 046 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending 047 * on the number k of previous steps one wants to use for computing the next 048 * value, different formulas are available for the final estimate:</p> 049 * <ul> 050 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li> 051 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li> 052 * <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> 053 * <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> 054 * <li>...</li> 055 * </ul> 056 * 057 * <p>A k-steps Adams-Moulton method is of order k+1.</p> 058 * 059 * <h3>Implementation details</h3> 060 * 061 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 062 * <pre> 063 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 064 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 065 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 066 * ... 067 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 068 * </pre></p> 069 * 070 * <p>The definitions above use the classical representation with several previous first 071 * derivatives. Lets define 072 * <pre> 073 * 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> 074 * </pre> 075 * (we omit the k index in the notation for clarity). With these definitions, 076 * Adams-Moulton methods can be written: 077 * <ul> 078 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li> 079 * <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> 080 * <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> 081 * <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> 082 * <li>...</li> 083 * </ul></p> 084 * 085 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 086 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with 087 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 088 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 089 * <pre> 090 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 091 * </pre> 092 * (here again we omit the k index in the notation for clarity) 093 * </p> 094 * 095 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 096 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 097 * for degree k polynomials. 098 * <pre> 099 * 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) 100 * </pre> 101 * The previous formula can be used with several values for i to compute the transform between 102 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 103 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 104 * <pre> 105 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 106 * </pre> 107 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 108 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 109 * the column number starting from 1: 110 * <pre> 111 * [ -2 3 -4 5 ... ] 112 * [ -4 12 -32 80 ... ] 113 * P = [ -6 27 -108 405 ... ] 114 * [ -8 48 -256 1280 ... ] 115 * [ ... ] 116 * </pre></p> 117 * 118 * <p>Using the Nordsieck vector has several advantages: 119 * <ul> 120 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 121 * Taylor series formulas,</li> 122 * <li>it simplifies step changes that occur when discrete events that truncate 123 * the step are triggered,</li> 124 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 125 * </ul></p> 126 * 127 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step 128 * n as follows: 129 * <ul> 130 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 131 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 132 * <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> 133 * </ul> 134 * where A is a rows shifting matrix (the lower left part is an identity matrix): 135 * <pre> 136 * [ 0 0 ... 0 0 | 0 ] 137 * [ ---------------+---] 138 * [ 1 0 ... 0 0 | 0 ] 139 * A = [ 0 1 ... 0 0 | 0 ] 140 * [ ... | 0 ] 141 * [ 0 0 ... 1 0 | 0 ] 142 * [ 0 0 ... 0 1 | 0 ] 143 * </pre> 144 * From this predicted vector, the corrected vector is computed as follows: 145 * <ul> 146 * <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> 147 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 148 * <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> 149 * </ul> 150 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 151 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 152 * represent the corrected states.</p> 153 * 154 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 155 * they only depend on k and therefore are precomputed once for all.</p> 156 * 157 * @since 2.0 158 */ 159public class AdamsMoultonIntegrator extends AdamsIntegrator { 160 161 /** Integrator method name. */ 162 private static final String METHOD_NAME = "Adams-Moulton"; 163 164 /** 165 * Build an Adams-Moulton integrator with the given order and error control parameters. 166 * @param nSteps number of steps of the method excluding the one being computed 167 * @param minStep minimal step (sign is irrelevant, regardless of 168 * integration direction, forward or backward), the last step can 169 * be smaller than this 170 * @param maxStep maximal step (sign is irrelevant, regardless of 171 * integration direction, forward or backward), the last step can 172 * be smaller than this 173 * @param scalAbsoluteTolerance allowed absolute error 174 * @param scalRelativeTolerance allowed relative error 175 * @exception NumberIsTooSmallException if order is 1 or less 176 */ 177 public AdamsMoultonIntegrator(final int nSteps, 178 final double minStep, final double maxStep, 179 final double scalAbsoluteTolerance, 180 final double scalRelativeTolerance) 181 throws NumberIsTooSmallException { 182 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 183 scalAbsoluteTolerance, scalRelativeTolerance); 184 } 185 186 /** 187 * Build an Adams-Moulton integrator with the given order and error control parameters. 188 * @param nSteps number of steps of the method excluding the one being computed 189 * @param minStep minimal step (sign is irrelevant, regardless of 190 * integration direction, forward or backward), the last step can 191 * be smaller than this 192 * @param maxStep maximal step (sign is irrelevant, regardless of 193 * integration direction, forward or backward), the last step can 194 * be smaller than this 195 * @param vecAbsoluteTolerance allowed absolute error 196 * @param vecRelativeTolerance allowed relative error 197 * @exception IllegalArgumentException if order is 1 or less 198 */ 199 public AdamsMoultonIntegrator(final int nSteps, 200 final double minStep, final double maxStep, 201 final double[] vecAbsoluteTolerance, 202 final double[] vecRelativeTolerance) 203 throws IllegalArgumentException { 204 super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 205 vecAbsoluteTolerance, vecRelativeTolerance); 206 } 207 208 /** {@inheritDoc} */ 209 @Override 210 public void integrate(final ExpandableStatefulODE equations,final double t) 211 throws NumberIsTooSmallException, DimensionMismatchException, 212 MaxCountExceededException, NoBracketingException { 213 214 sanityChecks(equations, t); 215 setEquations(equations); 216 final boolean forward = t > equations.getTime(); 217 218 // initialize working arrays 219 final double[] y0 = equations.getCompleteState(); 220 final double[] y = y0.clone(); 221 final double[] yDot = new double[y.length]; 222 final double[] yTmp = new double[y.length]; 223 final double[] predictedScaled = new double[y.length]; 224 Array2DRowRealMatrix nordsieckTmp = null; 225 226 // set up two interpolators sharing the integrator arrays 227 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 228 interpolator.reinitialize(y, forward, 229 equations.getPrimaryMapper(), equations.getSecondaryMappers()); 230 231 // set up integration control objects 232 initIntegration(equations.getTime(), y0, t); 233 234 // compute the initial Nordsieck vector using the configured starter integrator 235 start(equations.getTime(), y, t); 236 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 237 interpolator.storeTime(stepStart); 238 239 double hNew = stepSize; 240 interpolator.rescale(hNew); 241 242 isLastStep = false; 243 do { 244 245 double error = 10; 246 while (error >= 1.0) { 247 248 stepSize = hNew; 249 250 // predict a first estimate of the state at step end (P in the PECE sequence) 251 final double stepEnd = stepStart + stepSize; 252 interpolator.setInterpolatedTime(stepEnd); 253 final ExpandableStatefulODE expandable = getExpandable(); 254 final EquationsMapper primary = expandable.getPrimaryMapper(); 255 primary.insertEquationData(interpolator.getInterpolatedState(), yTmp); 256 int index = 0; 257 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { 258 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), yTmp); 259 ++index; 260 } 261 262 // evaluate a first estimate of the derivative (first E in the PECE sequence) 263 computeDerivatives(stepEnd, yTmp, yDot); 264 265 // update Nordsieck vector 266 for (int j = 0; j < y0.length; ++j) { 267 predictedScaled[j] = stepSize * yDot[j]; 268 } 269 nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck); 270 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp); 271 272 // apply correction (C in the PECE sequence) 273 error = nordsieckTmp.walkInOptimizedOrder(new Corrector(y, predictedScaled, yTmp)); 274 275 if (error >= 1.0) { 276 // reject the step and attempt to reduce error by stepsize control 277 final double factor = computeStepGrowShrinkFactor(error); 278 hNew = filterStep(stepSize * factor, forward, false); 279 interpolator.rescale(hNew); 280 } 281 } 282 283 // evaluate a final estimate of the derivative (second E in the PECE sequence) 284 final double stepEnd = stepStart + stepSize; 285 computeDerivatives(stepEnd, yTmp, yDot); 286 287 // update Nordsieck vector 288 final double[] correctedScaled = new double[y0.length]; 289 for (int j = 0; j < y0.length; ++j) { 290 correctedScaled[j] = stepSize * yDot[j]; 291 } 292 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, nordsieckTmp); 293 294 // discrete events handling 295 System.arraycopy(yTmp, 0, y, 0, y.length); 296 interpolator.reinitialize(stepEnd, stepSize, correctedScaled, nordsieckTmp); 297 interpolator.storeTime(stepStart); 298 interpolator.shift(); 299 interpolator.storeTime(stepEnd); 300 stepStart = acceptStep(interpolator, y, yDot, t); 301 scaled = correctedScaled; 302 nordsieck = nordsieckTmp; 303 304 if (!isLastStep) { 305 306 // prepare next step 307 interpolator.storeTime(stepStart); 308 309 if (resetOccurred) { 310 // some events handler has triggered changes that 311 // invalidate the derivatives, we need to restart from scratch 312 start(stepStart, y, t); 313 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 314 315 } 316 317 // stepsize control for next step 318 final double factor = computeStepGrowShrinkFactor(error); 319 final double scaledH = stepSize * factor; 320 final double nextT = stepStart + scaledH; 321 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 322 hNew = filterStep(scaledH, forward, nextIsLast); 323 324 final double filteredNextT = stepStart + hNew; 325 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 326 if (filteredNextIsLast) { 327 hNew = t - stepStart; 328 } 329 330 interpolator.rescale(hNew); 331 } 332 333 } while (!isLastStep); 334 335 // dispatch results 336 equations.setTime(stepStart); 337 equations.setCompleteState(y); 338 339 resetInternalState(); 340 341 } 342 343 /** Corrector for current state in Adams-Moulton method. 344 * <p> 345 * This visitor implements the Taylor series formula: 346 * <pre> 347 * 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> 348 * </pre> 349 * </p> 350 */ 351 private class Corrector implements RealMatrixPreservingVisitor { 352 353 /** Previous state. */ 354 private final double[] previous; 355 356 /** Current scaled first derivative. */ 357 private final double[] scaled; 358 359 /** Current state before correction. */ 360 private final double[] before; 361 362 /** Current state after correction. */ 363 private final double[] after; 364 365 /** Simple constructor. 366 * @param previous previous state 367 * @param scaled current scaled first derivative 368 * @param state state to correct (will be overwritten after visit) 369 */ 370 Corrector(final double[] previous, final double[] scaled, final double[] state) { 371 this.previous = previous; 372 this.scaled = scaled; 373 this.after = state; 374 this.before = state.clone(); 375 } 376 377 /** {@inheritDoc} */ 378 public void start(int rows, int columns, 379 int startRow, int endRow, int startColumn, int endColumn) { 380 Arrays.fill(after, 0.0); 381 } 382 383 /** {@inheritDoc} */ 384 public void visit(int row, int column, double value) { 385 if ((row & 0x1) == 0) { 386 after[column] -= value; 387 } else { 388 after[column] += value; 389 } 390 } 391 392 /** 393 * End visiting the Nordsieck vector. 394 * <p>The correction is used to control stepsize. So its amplitude is 395 * considered to be an error, which must be normalized according to 396 * error control settings. If the normalized value is greater than 1, 397 * the correction was too large and the step must be rejected.</p> 398 * @return the normalized correction, if greater than 1, the step 399 * must be rejected 400 */ 401 public double end() { 402 403 double error = 0; 404 for (int i = 0; i < after.length; ++i) { 405 after[i] += previous[i] + scaled[i]; 406 if (i < mainSetDimension) { 407 final double yScale = FastMath.max(FastMath.abs(previous[i]), FastMath.abs(after[i])); 408 final double tol = (vecAbsoluteTolerance == null) ? 409 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 410 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 411 final double ratio = (after[i] - before[i]) / tol; // (corrected-predicted)/tol 412 error += ratio * ratio; 413 } 414 } 415 416 return FastMath.sqrt(error / mainSetDimension); 417 418 } 419 } 420 421}