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.math4.legacy.ode.nonstiff; 019 020import java.util.Arrays; 021 022import org.apache.commons.math4.legacy.core.Field; 023import org.apache.commons.math4.legacy.core.RealFieldElement; 024import org.apache.commons.math4.legacy.exception.DimensionMismatchException; 025import org.apache.commons.math4.legacy.exception.MaxCountExceededException; 026import org.apache.commons.math4.legacy.exception.NoBracketingException; 027import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; 028import org.apache.commons.math4.legacy.linear.Array2DRowFieldMatrix; 029import org.apache.commons.math4.legacy.linear.FieldMatrixPreservingVisitor; 030import org.apache.commons.math4.legacy.ode.FieldExpandableODE; 031import org.apache.commons.math4.legacy.ode.FieldODEState; 032import org.apache.commons.math4.legacy.ode.FieldODEStateAndDerivative; 033import org.apache.commons.math4.legacy.core.MathArrays; 034 035 036/** 037 * This class implements implicit Adams-Moulton integrators for Ordinary 038 * Differential Equations. 039 * 040 * <p>Adams-Moulton methods (in fact due to Adams alone) are implicit 041 * multistep ODE solvers. This implementation is a variation of the classical 042 * one: it uses adaptive stepsize to implement error control, whereas 043 * classical implementations are fixed step size. The value of state vector 044 * at step n+1 is a simple combination of the value at step n and of the 045 * derivatives at steps n+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to 046 * compute y<sub>n+1</sub>, another method must be used to compute a first 047 * estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute 048 * a final estimate of y<sub>n+1</sub> using the following formulas. Depending 049 * on the number k of previous steps one wants to use for computing the next 050 * value, different formulas are available for the final estimate:</p> 051 * <ul> 052 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li> 053 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li> 054 * <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> 055 * <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> 056 * <li>...</li> 057 * </ul> 058 * 059 * <p>A k-steps Adams-Moulton method is of order k+1.</p> 060 * 061 * <p><b>Implementation details</b></p> 062 * 063 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 064 * <div style="white-space: pre"><code> 065 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 066 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 067 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 068 * ... 069 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 070 * </code></div> 071 * 072 * <p>The definitions above use the classical representation with several previous first 073 * derivatives. Lets define 074 * <div style="white-space: pre"><code> 075 * 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> 076 * </code></div> 077 * (we omit the k index in the notation for clarity). With these definitions, 078 * Adams-Moulton methods can be written: 079 * <ul> 080 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li> 081 * <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> 082 * <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> 083 * <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> 084 * <li>...</li> 085 * </ul> 086 * 087 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 088 * s<sub>1</sub>(n+1) and q<sub>n+1</sub>), our implementation uses the Nordsieck vector with 089 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 090 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 091 * <div style="white-space: pre"><code> 092 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 093 * </code></div> 094 * (here again we omit the k index in the notation for clarity) 095 * 096 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 097 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 098 * for degree k polynomials. 099 * <div style="white-space: pre"><code> 100 * 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) 101 * </code></div> 102 * The previous formula can be used with several values for i to compute the transform between 103 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 104 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 105 * <div style="white-space: pre"><code> 106 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 107 * </code></div> 108 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 109 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 110 * the column number starting from 1: 111 * <pre> 112 * [ -2 3 -4 5 ... ] 113 * [ -4 12 -32 80 ... ] 114 * P = [ -6 27 -108 405 ... ] 115 * [ -8 48 -256 1280 ... ] 116 * [ ... ] 117 * </pre> 118 * 119 * <p>Using the Nordsieck vector has several advantages: 120 * <ul> 121 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 122 * Taylor series formulas,</li> 123 * <li>it simplifies step changes that occur when discrete events that truncate 124 * the step are triggered,</li> 125 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 126 * </ul> 127 * 128 * <p>The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step 129 * n as follows: 130 * <ul> 131 * <li>Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 132 * <li>S<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, Y<sub>n+1</sub>)</li> 133 * <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> 134 * </ul> 135 * where A is a rows shifting matrix (the lower left part is an identity matrix): 136 * <pre> 137 * [ 0 0 ... 0 0 | 0 ] 138 * [ ---------------+---] 139 * [ 1 0 ... 0 0 | 0 ] 140 * A = [ 0 1 ... 0 0 | 0 ] 141 * [ ... | 0 ] 142 * [ 0 0 ... 1 0 | 0 ] 143 * [ 0 0 ... 0 1 | 0 ] 144 * </pre> 145 * From this predicted vector, the corrected vector is computed as follows: 146 * <ul> 147 * <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> 148 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 149 * <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> 150 * </ul> 151 * where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the 152 * predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub> 153 * represent the corrected states. 154 * 155 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 156 * they only depend on k and therefore are precomputed once for all.</p> 157 * 158 * @param <T> the type of the field elements 159 * @since 3.6 160 */ 161public class AdamsMoultonFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> { 162 163 /** Integrator method name. */ 164 private static final String METHOD_NAME = "Adams-Moulton"; 165 166 /** 167 * Build an Adams-Moulton integrator with the given order and error control parameters. 168 * @param field field to which the time and state vector elements belong 169 * @param nSteps number of steps of the method excluding the one being computed 170 * @param minStep minimal step (sign is irrelevant, regardless of 171 * integration direction, forward or backward), the last step can 172 * be smaller than this 173 * @param maxStep maximal step (sign is irrelevant, regardless of 174 * integration direction, forward or backward), the last step can 175 * be smaller than this 176 * @param scalAbsoluteTolerance allowed absolute error 177 * @param scalRelativeTolerance allowed relative error 178 * @exception NumberIsTooSmallException if order is 1 or less 179 */ 180 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps, 181 final double minStep, final double maxStep, 182 final double scalAbsoluteTolerance, 183 final double scalRelativeTolerance) 184 throws NumberIsTooSmallException { 185 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 186 scalAbsoluteTolerance, scalRelativeTolerance); 187 } 188 189 /** 190 * Build an Adams-Moulton integrator with the given order and error control parameters. 191 * @param field field to which the time and state vector elements belong 192 * @param nSteps number of steps of the method excluding the one being computed 193 * @param minStep minimal step (sign is irrelevant, regardless of 194 * integration direction, forward or backward), the last step can 195 * be smaller than this 196 * @param maxStep maximal step (sign is irrelevant, regardless of 197 * integration direction, forward or backward), the last step can 198 * be smaller than this 199 * @param vecAbsoluteTolerance allowed absolute error 200 * @param vecRelativeTolerance allowed relative error 201 * @exception IllegalArgumentException if order is 1 or less 202 */ 203 public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps, 204 final double minStep, final double maxStep, 205 final double[] vecAbsoluteTolerance, 206 final double[] vecRelativeTolerance) 207 throws IllegalArgumentException { 208 super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep, 209 vecAbsoluteTolerance, vecRelativeTolerance); 210 } 211 212 /** {@inheritDoc} */ 213 @Override 214 public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, 215 final FieldODEState<T> initialState, 216 final T finalTime) 217 throws NumberIsTooSmallException, DimensionMismatchException, 218 MaxCountExceededException, NoBracketingException { 219 220 sanityChecks(initialState, finalTime); 221 final T t0 = initialState.getTime(); 222 final T[] y = equations.getMapper().mapState(initialState); 223 setStepStart(initIntegration(equations, t0, y, finalTime)); 224 final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0; 225 226 // compute the initial Nordsieck vector using the configured starter integrator 227 start(equations, getStepStart(), finalTime); 228 229 // reuse the step that was chosen by the starter integrator 230 FieldODEStateAndDerivative<T> stepStart = getStepStart(); 231 FieldODEStateAndDerivative<T> stepEnd = 232 AdamsFieldStepInterpolator.taylor(stepStart, 233 stepStart.getTime().add(getStepSize()), 234 getStepSize(), scaled, nordsieck); 235 236 // main integration loop 237 setIsLastStep(false); 238 do { 239 240 T[] predictedY = null; 241 final T[] predictedScaled = MathArrays.buildArray(getField(), y.length); 242 Array2DRowFieldMatrix<T> predictedNordsieck = null; 243 T error = getField().getZero().add(10); 244 while (error.subtract(1.0).getReal() >= 0.0) { 245 246 // predict a first estimate of the state at step end (P in the PECE sequence) 247 predictedY = stepEnd.getState(); 248 249 // evaluate a first estimate of the derivative (first E in the PECE sequence) 250 final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY); 251 252 // update Nordsieck vector 253 for (int j = 0; j < predictedScaled.length; ++j) { 254 predictedScaled[j] = getStepSize().multiply(yDot[j]); 255 } 256 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); 257 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); 258 259 // apply correction (C in the PECE sequence) 260 error = predictedNordsieck.walkInOptimizedOrder(new Corrector(y, predictedScaled, predictedY)); 261 262 if (error.subtract(1.0).getReal() >= 0.0) { 263 // reject the step and attempt to reduce error by stepsize control 264 final T factor = computeStepGrowShrinkFactor(error); 265 rescale(filterStep(getStepSize().multiply(factor), forward, false)); 266 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), 267 getStepStart().getTime().add(getStepSize()), 268 getStepSize(), 269 scaled, 270 nordsieck); 271 } 272 } 273 274 // evaluate a final estimate of the derivative (second E in the PECE sequence) 275 final T[] correctedYDot = computeDerivatives(stepEnd.getTime(), predictedY); 276 277 // update Nordsieck vector 278 final T[] correctedScaled = MathArrays.buildArray(getField(), y.length); 279 for (int j = 0; j < correctedScaled.length; ++j) { 280 correctedScaled[j] = getStepSize().multiply(correctedYDot[j]); 281 } 282 updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck); 283 284 // discrete events handling 285 stepEnd = new FieldODEStateAndDerivative<>(stepEnd.getTime(), predictedY, correctedYDot); 286 setStepStart(acceptStep(new AdamsFieldStepInterpolator<>(getStepSize(), stepEnd, 287 correctedScaled, predictedNordsieck, forward, 288 getStepStart(), stepEnd, 289 equations.getMapper()), 290 finalTime)); 291 scaled = correctedScaled; 292 nordsieck = predictedNordsieck; 293 294 if (!isLastStep()) { 295 296 System.arraycopy(predictedY, 0, y, 0, y.length); 297 298 if (resetOccurred()) { 299 // some events handler has triggered changes that 300 // invalidate the derivatives, we need to restart from scratch 301 start(equations, getStepStart(), finalTime); 302 } 303 304 // stepsize control for next step 305 final T factor = computeStepGrowShrinkFactor(error); 306 final T scaledH = getStepSize().multiply(factor); 307 final T nextT = getStepStart().getTime().add(scaledH); 308 final boolean nextIsLast = forward ? 309 nextT.subtract(finalTime).getReal() >= 0 : 310 nextT.subtract(finalTime).getReal() <= 0; 311 T hNew = filterStep(scaledH, forward, nextIsLast); 312 313 final T filteredNextT = getStepStart().getTime().add(hNew); 314 final boolean filteredNextIsLast = forward ? 315 filteredNextT.subtract(finalTime).getReal() >= 0 : 316 filteredNextT.subtract(finalTime).getReal() <= 0; 317 if (filteredNextIsLast) { 318 hNew = finalTime.subtract(getStepStart().getTime()); 319 } 320 321 rescale(hNew); 322 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()), 323 getStepSize(), scaled, nordsieck); 324 } 325 } while (!isLastStep()); 326 327 final FieldODEStateAndDerivative<T> finalState = getStepStart(); 328 setStepStart(null); 329 setStepSize(null); 330 return finalState; 331 } 332 333 /** Corrector for current state in Adams-Moulton method. 334 * <p> 335 * This visitor implements the Taylor series formula: 336 * <pre> 337 * 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> 338 * </pre> 339 * </p> 340 */ 341 private final class Corrector implements FieldMatrixPreservingVisitor<T> { 342 343 /** Previous state. */ 344 private final T[] previous; 345 346 /** Current scaled first derivative. */ 347 private final T[] scaled; 348 349 /** Current state before correction. */ 350 private final T[] before; 351 352 /** Current state after correction. */ 353 private final T[] after; 354 355 /** Simple constructor. 356 * @param previous previous state 357 * @param scaled current scaled first derivative 358 * @param state state to correct (will be overwritten after visit) 359 */ 360 Corrector(final T[] previous, final T[] scaled, final T[] state) { 361 this.previous = previous; 362 this.scaled = scaled; 363 this.after = state; 364 this.before = state.clone(); 365 } 366 367 /** {@inheritDoc} */ 368 @Override 369 public void start(int rows, int columns, 370 int startRow, int endRow, int startColumn, int endColumn) { 371 Arrays.fill(after, getField().getZero()); 372 } 373 374 /** {@inheritDoc} */ 375 @Override 376 public void visit(int row, int column, T value) { 377 if ((row & 0x1) == 0) { 378 after[column] = after[column].subtract(value); 379 } else { 380 after[column] = after[column].add(value); 381 } 382 } 383 384 /** 385 * End visiting the Nordsieck vector. 386 * <p>The correction is used to control stepsize. So its amplitude is 387 * considered to be an error, which must be normalized according to 388 * error control settings. If the normalized value is greater than 1, 389 * the correction was too large and the step must be rejected.</p> 390 * @return the normalized correction, if greater than 1, the step 391 * must be rejected 392 */ 393 @Override 394 public T end() { 395 396 T error = getField().getZero(); 397 for (int i = 0; i < after.length; ++i) { 398 after[i] = after[i].add(previous[i].add(scaled[i])); 399 if (i < mainSetDimension) { 400 final T yScale = RealFieldElement.max(previous[i].abs(), after[i].abs()); 401 final T tol = (vecAbsoluteTolerance == null) ? 402 yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : 403 yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]); 404 final T ratio = after[i].subtract(before[i]).divide(tol); // (corrected-predicted)/tol 405 error = error.add(ratio.multiply(ratio)); 406 } 407 } 408 409 return error.divide(mainSetDimension).sqrt(); 410 } 411 } 412}