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.Field; 023import org.apache.commons.math3.RealFieldElement; 024import org.apache.commons.math3.exception.DimensionMismatchException; 025import org.apache.commons.math3.exception.MaxCountExceededException; 026import org.apache.commons.math3.exception.NoBracketingException; 027import org.apache.commons.math3.exception.NumberIsTooSmallException; 028import org.apache.commons.math3.linear.Array2DRowFieldMatrix; 029import org.apache.commons.math3.linear.FieldMatrixPreservingVisitor; 030import org.apache.commons.math3.ode.FieldExpandableODE; 031import org.apache.commons.math3.ode.FieldODEState; 032import org.apache.commons.math3.ode.FieldODEStateAndDerivative; 033import org.apache.commons.math3.util.MathArrays; 034import org.apache.commons.math3.util.MathUtils; 035 036 037/** 038 * This class implements implicit Adams-Moulton integrators for Ordinary 039 * Differential Equations. 040 * 041 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit 042 * multistep ODE solvers. This implementation is a variation of the classical 043 * one: it uses adaptive stepsize to implement error control, whereas 044 * classical implementations are fixed step size. The value of state vector 045 * at step n+1 is a simple combination of the value at step n and of the 046 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to 047 * compute y<sub>n+1</sub>, another method must be used to compute a first 048 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute 049 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending 050 * on the number k of previous steps one wants to use for computing the next 051 * value, different formulas are available for the final estimate:</p> 052 * <ul> 053 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li> 054 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li> 055 * <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> 056 * <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> 057 * <li>...</li> 058 * </ul> 059 * 060 * <p>A k-steps Adams-Moulton method is of order k+1.</p> 061 * 062 * <h3>Implementation details</h3> 063 * 064 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 065 * <pre> 066 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 067 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 068 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 069 * ... 070 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 071 * </pre></p> 072 * 073 * <p>The definitions above use the classical representation with several previous first 074 * derivatives. Lets define 075 * <pre> 076 * 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> 077 * </pre> 078 * (we omit the k index in the notation for clarity). With these definitions, 079 * Adams-Moulton methods can be written: 080 * <ul> 081 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li> 082 * <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> 083 * <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> 084 * <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> 085 * <li>...</li> 086 * </ul></p> 087 * 088 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 089 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with 090 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 091 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 092 * <pre> 093 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 094 * </pre> 095 * (here again we omit the k index in the notation for clarity) 096 * </p> 097 * 098 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 099 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 100 * for degree k polynomials. 101 * <pre> 102 * 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) 103 * </pre> 104 * The previous formula can be used with several values for i to compute the transform between 105 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 106 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 107 * <pre> 108 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 109 * </pre> 110 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 111 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 112 * the column number starting from 1: 113 * <pre> 114 * [ -2 3 -4 5 ... ] 115 * [ -4 12 -32 80 ... ] 116 * P = [ -6 27 -108 405 ... ] 117 * [ -8 48 -256 1280 ... ] 118 * [ ... ] 119 * </pre></p> 120 * 121 * <p>Using the Nordsieck vector has several advantages: 122 * <ul> 123 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 124 * Taylor series formulas,</li> 125 * <li>it simplifies step changes that occur when discrete events that truncate 126 * the step are triggered,</li> 127 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 128 * </ul></p> 129 * 130 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step 131 * n as follows: 132 * <ul> 133 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 134 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 135 * <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> 136 * </ul> 137 * where A is a rows shifting matrix (the lower left part is an identity matrix): 138 * <pre> 139 * [ 0 0 ... 0 0 | 0 ] 140 * [ ---------------+---] 141 * [ 1 0 ... 0 0 | 0 ] 142 * A = [ 0 1 ... 0 0 | 0 ] 143 * [ ... | 0 ] 144 * [ 0 0 ... 1 0 | 0 ] 145 * [ 0 0 ... 0 1 | 0 ] 146 * </pre> 147 * From this predicted vector, the corrected vector is computed as follows: 148 * <ul> 149 * <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> 150 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 151 * <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> 152 * </ul> 153 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 154 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 155 * represent the corrected states.</p> 156 * 157 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 158 * they only depend on k and therefore are precomputed once for all.</p> 159 * 160 * @param <T> the type of the field elements 161 * @since 3.6 162 */ 163public class AdamsMoultonFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> { 164 165 /** Integrator method name. */ 166 private static final String METHOD_NAME = "Adams-Moulton"; 167 168 /** 169 * Build an Adams-Moulton integrator with the given order and error control parameters. 170 * @param field field to which the time and state vector elements belong 171 * @param nSteps number of steps of the method excluding the one being computed 172 * @param minStep minimal step (sign is irrelevant, regardless of 173 * integration direction, forward or backward), the last step can 174 * be smaller than this 175 * @param maxStep maximal step (sign is irrelevant, regardless of 176 * integration direction, forward or backward), the last step can 177 * be smaller than this 178 * @param scalAbsoluteTolerance allowed absolute error 179 * @param scalRelativeTolerance allowed relative error 180 * @exception NumberIsTooSmallException if order is 1 or less 181 */ 182 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps, 183 final double minStep, final double maxStep, 184 final double scalAbsoluteTolerance, 185 final double scalRelativeTolerance) 186 throws NumberIsTooSmallException { 187 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 188 scalAbsoluteTolerance, scalRelativeTolerance); 189 } 190 191 /** 192 * Build an Adams-Moulton integrator with the given order and error control parameters. 193 * @param field field to which the time and state vector elements belong 194 * @param nSteps number of steps of the method excluding the one being computed 195 * @param minStep minimal step (sign is irrelevant, regardless of 196 * integration direction, forward or backward), the last step can 197 * be smaller than this 198 * @param maxStep maximal step (sign is irrelevant, regardless of 199 * integration direction, forward or backward), the last step can 200 * be smaller than this 201 * @param vecAbsoluteTolerance allowed absolute error 202 * @param vecRelativeTolerance allowed relative error 203 * @exception IllegalArgumentException if order is 1 or less 204 */ 205 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps, 206 final double minStep, final double maxStep, 207 final double[] vecAbsoluteTolerance, 208 final double[] vecRelativeTolerance) 209 throws IllegalArgumentException { 210 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 211 vecAbsoluteTolerance, vecRelativeTolerance); 212 } 213 214 /** {@inheritDoc} */ 215 @Override 216 public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, 217 final FieldODEState<T> initialState, 218 final T finalTime) 219 throws NumberIsTooSmallException, DimensionMismatchException, 220 MaxCountExceededException, NoBracketingException { 221 222 sanityChecks(initialState, finalTime); 223 final T t0 = initialState.getTime(); 224 final T[] y = equations.getMapper().mapState(initialState); 225 setStepStart(initIntegration(equations, t0, y, finalTime)); 226 final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0; 227 228 // compute the initial Nordsieck vector using the configured starter integrator 229 start(equations, getStepStart(), finalTime); 230 231 // reuse the step that was chosen by the starter integrator 232 FieldODEStateAndDerivative<T> stepStart = getStepStart(); 233 FieldODEStateAndDerivative<T> stepEnd = 234 AdamsFieldStepInterpolator.taylor(stepStart, 235 stepStart.getTime().add(getStepSize()), 236 getStepSize(), scaled, nordsieck); 237 238 // main integration loop 239 setIsLastStep(false); 240 do { 241 242 T[] predictedY = null; 243 final T[] predictedScaled = MathArrays.buildArray(getField(), y.length); 244 Array2DRowFieldMatrix<T> predictedNordsieck = null; 245 T error = getField().getZero().add(10); 246 while (error.subtract(1.0).getReal() >= 0.0) { 247 248 // predict a first estimate of the state at step end (P in the PECE sequence) 249 predictedY = stepEnd.getState(); 250 251 // evaluate a first estimate of the derivative (first E in the PECE sequence) 252 final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY); 253 254 // update Nordsieck vector 255 for (int j = 0; j < predictedScaled.length; ++j) { 256 predictedScaled[j] = getStepSize().multiply(yDot[j]); 257 } 258 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); 259 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); 260 261 // apply correction (C in the PECE sequence) 262 error = predictedNordsieck.walkInOptimizedOrder(new Corrector(y, predictedScaled, predictedY)); 263 264 if (error.subtract(1.0).getReal() >= 0.0) { 265 // reject the step and attempt to reduce error by stepsize control 266 final T factor = computeStepGrowShrinkFactor(error); 267 rescale(filterStep(getStepSize().multiply(factor), forward, false)); 268 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), 269 getStepStart().getTime().add(getStepSize()), 270 getStepSize(), 271 scaled, 272 nordsieck); 273 } 274 } 275 276 // evaluate a final estimate of the derivative (second E in the PECE sequence) 277 final T[] correctedYDot = computeDerivatives(stepEnd.getTime(), predictedY); 278 279 // update Nordsieck vector 280 final T[] correctedScaled = MathArrays.buildArray(getField(), y.length); 281 for (int j = 0; j < correctedScaled.length; ++j) { 282 correctedScaled[j] = getStepSize().multiply(correctedYDot[j]); 283 } 284 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck); 285 286 // discrete events handling 287 stepEnd = new FieldODEStateAndDerivative<T>(stepEnd.getTime(), predictedY, correctedYDot); 288 setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(getStepSize(), stepEnd, 289 correctedScaled, predictedNordsieck, forward, 290 getStepStart(), stepEnd, 291 equations.getMapper()), 292 finalTime)); 293 scaled = correctedScaled; 294 nordsieck = predictedNordsieck; 295 296 if (!isLastStep()) { 297 298 System.arraycopy(predictedY, 0, y, 0, y.length); 299 300 if (resetOccurred()) { 301 // some events handler has triggered changes that 302 // invalidate the derivatives, we need to restart from scratch 303 start(equations, getStepStart(), finalTime); 304 } 305 306 // stepsize control for next step 307 final T factor = computeStepGrowShrinkFactor(error); 308 final T scaledH = getStepSize().multiply(factor); 309 final T nextT = getStepStart().getTime().add(scaledH); 310 final boolean nextIsLast = forward ? 311 nextT.subtract(finalTime).getReal() >= 0 : 312 nextT.subtract(finalTime).getReal() <= 0; 313 T hNew = filterStep(scaledH, forward, nextIsLast); 314 315 final T filteredNextT = getStepStart().getTime().add(hNew); 316 final boolean filteredNextIsLast = forward ? 317 filteredNextT.subtract(finalTime).getReal() >= 0 : 318 filteredNextT.subtract(finalTime).getReal() <= 0; 319 if (filteredNextIsLast) { 320 hNew = finalTime.subtract(getStepStart().getTime()); 321 } 322 323 rescale(hNew); 324 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()), 325 getStepSize(), scaled, nordsieck); 326 327 } 328 329 } while (!isLastStep()); 330 331 final FieldODEStateAndDerivative<T> finalState = getStepStart(); 332 setStepStart(null); 333 setStepSize(null); 334 return finalState; 335 336 } 337 338 /** Corrector for current state in Adams-Moulton method. 339 * <p> 340 * This visitor implements the Taylor series formula: 341 * <pre> 342 * 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> 343 * </pre> 344 * </p> 345 */ 346 private class Corrector implements FieldMatrixPreservingVisitor<T> { 347 348 /** Previous state. */ 349 private final T[] previous; 350 351 /** Current scaled first derivative. */ 352 private final T[] scaled; 353 354 /** Current state before correction. */ 355 private final T[] before; 356 357 /** Current state after correction. */ 358 private final T[] after; 359 360 /** Simple constructor. 361 * @param previous previous state 362 * @param scaled current scaled first derivative 363 * @param state state to correct (will be overwritten after visit) 364 */ 365 Corrector(final T[] previous, final T[] scaled, final T[] state) { 366 this.previous = previous; 367 this.scaled = scaled; 368 this.after = state; 369 this.before = state.clone(); 370 } 371 372 /** {@inheritDoc} */ 373 public void start(int rows, int columns, 374 int startRow, int endRow, int startColumn, int endColumn) { 375 Arrays.fill(after, getField().getZero()); 376 } 377 378 /** {@inheritDoc} */ 379 public void visit(int row, int column, T value) { 380 if ((row & 0x1) == 0) { 381 after[column] = after[column].subtract(value); 382 } else { 383 after[column] = after[column].add(value); 384 } 385 } 386 387 /** 388 * End visiting the Nordsieck vector. 389 * <p>The correction is used to control stepsize. So its amplitude is 390 * considered to be an error, which must be normalized according to 391 * error control settings. If the normalized value is greater than 1, 392 * the correction was too large and the step must be rejected.</p> 393 * @return the normalized correction, if greater than 1, the step 394 * must be rejected 395 */ 396 public T end() { 397 398 T error = getField().getZero(); 399 for (int i = 0; i < after.length; ++i) { 400 after[i] = after[i].add(previous[i].add(scaled[i])); 401 if (i < mainSetDimension) { 402 final T yScale = MathUtils.max(previous[i].abs(), after[i].abs()); 403 final T tol = (vecAbsoluteTolerance == null) ? 404 yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : 405 yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]); 406 final T ratio = after[i].subtract(before[i]).divide(tol); // (corrected-predicted)/tol 407 error = error.add(ratio.multiply(ratio)); 408 } 409 } 410 411 return error.divide(mainSetDimension).sqrt(); 412 413 } 414 } 415 416}