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.math3.ode.nonstiff; 19 20 import org.apache.commons.math3.Field; 21 import org.apache.commons.math3.RealFieldElement; 22 import org.apache.commons.math3.exception.DimensionMismatchException; 23 import org.apache.commons.math3.exception.MaxCountExceededException; 24 import org.apache.commons.math3.exception.NoBracketingException; 25 import org.apache.commons.math3.exception.NumberIsTooSmallException; 26 import org.apache.commons.math3.linear.Array2DRowFieldMatrix; 27 import org.apache.commons.math3.linear.FieldMatrix; 28 import org.apache.commons.math3.ode.FieldExpandableODE; 29 import org.apache.commons.math3.ode.FieldODEState; 30 import org.apache.commons.math3.ode.FieldODEStateAndDerivative; 31 import org.apache.commons.math3.util.MathArrays; 32 33 34 /** 35 * This class implements explicit Adams-Bashforth integrators for Ordinary 36 * Differential Equations. 37 * 38 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit 39 * multistep ODE solvers. This implementation is a variation of the classical 40 * one: it uses adaptive stepsize to implement error control, whereas 41 * classical implementations are fixed step size. The value of state vector 42 * at step n+1 is a simple combination of the value at step n and of the 43 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous 44 * steps one wants to use for computing the next value, different formulas 45 * are available:</p> 46 * <ul> 47 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li> 48 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li> 49 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (23y'<sub>n</sub>-16y'<sub>n-1</sub>+5y'<sub>n-2</sub>)/12</li> 50 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (55y'<sub>n</sub>-59y'<sub>n-1</sub>+37y'<sub>n-2</sub>-9y'<sub>n-3</sub>)/24</li> 51 * <li>...</li> 52 * </ul> 53 * 54 * <p>A k-steps Adams-Bashforth method is of order k.</p> 55 * 56 * <h3>Implementation details</h3> 57 * 58 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 59 * <pre> 60 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 61 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 62 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 63 * ... 64 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 65 * </pre></p> 66 * 67 * <p>The definitions above use the classical representation with several previous first 68 * derivatives. Lets define 69 * <pre> 70 * 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> 71 * </pre> 72 * (we omit the k index in the notation for clarity). With these definitions, 73 * Adams-Bashforth methods can be written: 74 * <ul> 75 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li> 76 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 3/2 s<sub>1</sub>(n) + [ -1/2 ] q<sub>n</sub></li> 77 * <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 23/12 s<sub>1</sub>(n) + [ -16/12 5/12 ] q<sub>n</sub></li> 78 * <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 55/24 s<sub>1</sub>(n) + [ -59/24 37/24 -9/24 ] q<sub>n</sub></li> 79 * <li>...</li> 80 * </ul></p> 81 * 82 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 83 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with 84 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 85 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 86 * <pre> 87 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 88 * </pre> 89 * (here again we omit the k index in the notation for clarity) 90 * </p> 91 * 92 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 93 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 94 * for degree k polynomials. 95 * <pre> 96 * 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) 97 * </pre> 98 * The previous formula can be used with several values for i to compute the transform between 99 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 100 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 101 * <pre> 102 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 103 * </pre> 104 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 105 * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being 106 * the column number starting from 1: 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 Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: 125 * <ul> 126 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 127 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 128 * <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> 129 * </ul> 130 * where A is a rows shifting matrix (the lower left part is an identity matrix): 131 * <pre> 132 * [ 0 0 ... 0 0 | 0 ] 133 * [ ---------------+---] 134 * [ 1 0 ... 0 0 | 0 ] 135 * A = [ 0 1 ... 0 0 | 0 ] 136 * [ ... | 0 ] 137 * [ 0 0 ... 1 0 | 0 ] 138 * [ 0 0 ... 0 1 | 0 ] 139 * </pre></p> 140 * 141 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 142 * they only depend on k and therefore are precomputed once for all.</p> 143 * 144 * @param <T> the type of the field elements 145 * @since 3.6 146 */ 147 public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> { 148 149 /** Integrator method name. */ 150 private static final String METHOD_NAME = "Adams-Bashforth"; 151 152 /** 153 * Build an Adams-Bashforth integrator with the given order and step control parameters. 154 * @param field field to which the time and state vector elements belong 155 * @param nSteps number of steps of the method excluding the one being computed 156 * @param minStep minimal step (sign is irrelevant, regardless of 157 * integration direction, forward or backward), the last step can 158 * be smaller than this 159 * @param maxStep maximal step (sign is irrelevant, regardless of 160 * integration direction, forward or backward), the last step can 161 * be smaller than this 162 * @param scalAbsoluteTolerance allowed absolute error 163 * @param scalRelativeTolerance allowed relative error 164 * @exception NumberIsTooSmallException if order is 1 or less 165 */ AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, final double minStep, final double maxStep, final double scalAbsoluteTolerance, final double scalRelativeTolerance)166 public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, 167 final double minStep, final double maxStep, 168 final double scalAbsoluteTolerance, 169 final double scalRelativeTolerance) 170 throws NumberIsTooSmallException { 171 super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep, 172 scalAbsoluteTolerance, scalRelativeTolerance); 173 } 174 175 /** 176 * Build an Adams-Bashforth integrator with the given order and step control parameters. 177 * @param field field to which the time and state vector elements belong 178 * @param nSteps number of steps of the method excluding the one being computed 179 * @param minStep minimal step (sign is irrelevant, regardless of 180 * integration direction, forward or backward), the last step can 181 * be smaller than this 182 * @param maxStep maximal step (sign is irrelevant, regardless of 183 * integration direction, forward or backward), the last step can 184 * be smaller than this 185 * @param vecAbsoluteTolerance allowed absolute error 186 * @param vecRelativeTolerance allowed relative error 187 * @exception IllegalArgumentException if order is 1 or less 188 */ AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, final double minStep, final double maxStep, final double[] vecAbsoluteTolerance, final double[] vecRelativeTolerance)189 public AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, 190 final double minStep, final double maxStep, 191 final double[] vecAbsoluteTolerance, 192 final double[] vecRelativeTolerance) 193 throws IllegalArgumentException { 194 super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep, 195 vecAbsoluteTolerance, vecRelativeTolerance); 196 } 197 198 /** Estimate error. 199 * <p> 200 * Error is estimated by interpolating back to previous state using 201 * the state Taylor expansion and comparing to real previous state. 202 * </p> 203 * @param previousState state vector at step start 204 * @param predictedState predicted state vector at step end 205 * @param predictedScaled predicted value of the scaled derivatives at step end 206 * @param predictedNordsieck predicted value of the Nordsieck vector at step end 207 * @return estimated normalized local discretization error 208 */ errorEstimation(final T[] previousState, final T[] predictedState, final T[] predictedScaled, final FieldMatrix<T> predictedNordsieck)209 private T errorEstimation(final T[] previousState, 210 final T[] predictedState, 211 final T[] predictedScaled, 212 final FieldMatrix<T> predictedNordsieck) { 213 214 T error = getField().getZero(); 215 for (int i = 0; i < mainSetDimension; ++i) { 216 final T yScale = predictedState[i].abs(); 217 final T tol = (vecAbsoluteTolerance == null) ? 218 yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : 219 yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]); 220 221 // apply Taylor formula from high order to low order, 222 // for the sake of numerical accuracy 223 T variation = getField().getZero(); 224 int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1; 225 for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { 226 variation = variation.add(predictedNordsieck.getEntry(k, i).multiply(sign)); 227 sign = -sign; 228 } 229 variation = variation.subtract(predictedScaled[i]); 230 231 final T ratio = predictedState[i].subtract(previousState[i]).add(variation).divide(tol); 232 error = error.add(ratio.multiply(ratio)); 233 234 } 235 236 return error.divide(mainSetDimension).sqrt(); 237 238 } 239 240 /** {@inheritDoc} */ 241 @Override integrate(final FieldExpandableODE<T> equations, final FieldODEState<T> initialState, final T finalTime)242 public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, 243 final FieldODEState<T> initialState, 244 final T finalTime) 245 throws NumberIsTooSmallException, DimensionMismatchException, 246 MaxCountExceededException, NoBracketingException { 247 248 sanityChecks(initialState, finalTime); 249 final T t0 = initialState.getTime(); 250 final T[] y = equations.getMapper().mapState(initialState); 251 setStepStart(initIntegration(equations, t0, y, finalTime)); 252 final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0; 253 254 // compute the initial Nordsieck vector using the configured starter integrator 255 start(equations, getStepStart(), finalTime); 256 257 // reuse the step that was chosen by the starter integrator 258 FieldODEStateAndDerivative<T> stepStart = getStepStart(); 259 FieldODEStateAndDerivative<T> stepEnd = 260 AdamsFieldStepInterpolator.taylor(stepStart, 261 stepStart.getTime().add(getStepSize()), 262 getStepSize(), scaled, nordsieck); 263 264 // main integration loop 265 setIsLastStep(false); 266 do { 267 268 T[] predictedY = null; 269 final T[] predictedScaled = MathArrays.buildArray(getField(), y.length); 270 Array2DRowFieldMatrix<T> predictedNordsieck = null; 271 T error = getField().getZero().add(10); 272 while (error.subtract(1.0).getReal() >= 0.0) { 273 274 // predict a first estimate of the state at step end 275 predictedY = stepEnd.getState(); 276 277 // evaluate the derivative 278 final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY); 279 280 // predict Nordsieck vector at step end 281 for (int j = 0; j < predictedScaled.length; ++j) { 282 predictedScaled[j] = getStepSize().multiply(yDot[j]); 283 } 284 predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); 285 updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); 286 287 // evaluate error 288 error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); 289 290 if (error.subtract(1.0).getReal() >= 0.0) { 291 // reject the step and attempt to reduce error by stepsize control 292 final T factor = computeStepGrowShrinkFactor(error); 293 rescale(filterStep(getStepSize().multiply(factor), forward, false)); 294 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), 295 getStepStart().getTime().add(getStepSize()), 296 getStepSize(), 297 scaled, 298 nordsieck); 299 300 } 301 } 302 303 // discrete events handling 304 setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(getStepSize(), stepEnd, 305 predictedScaled, predictedNordsieck, forward, 306 getStepStart(), stepEnd, 307 equations.getMapper()), 308 finalTime)); 309 scaled = predictedScaled; 310 nordsieck = predictedNordsieck; 311 312 if (!isLastStep()) { 313 314 System.arraycopy(predictedY, 0, y, 0, y.length); 315 316 if (resetOccurred()) { 317 // some events handler has triggered changes that 318 // invalidate the derivatives, we need to restart from scratch 319 start(equations, getStepStart(), finalTime); 320 } 321 322 // stepsize control for next step 323 final T factor = computeStepGrowShrinkFactor(error); 324 final T scaledH = getStepSize().multiply(factor); 325 final T nextT = getStepStart().getTime().add(scaledH); 326 final boolean nextIsLast = forward ? 327 nextT.subtract(finalTime).getReal() >= 0 : 328 nextT.subtract(finalTime).getReal() <= 0; 329 T hNew = filterStep(scaledH, forward, nextIsLast); 330 331 final T filteredNextT = getStepStart().getTime().add(hNew); 332 final boolean filteredNextIsLast = forward ? 333 filteredNextT.subtract(finalTime).getReal() >= 0 : 334 filteredNextT.subtract(finalTime).getReal() <= 0; 335 if (filteredNextIsLast) { 336 hNew = finalTime.subtract(getStepStart().getTime()); 337 } 338 339 rescale(hNew); 340 stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()), 341 getStepSize(), scaled, nordsieck); 342 343 } 344 345 } while (!isLastStep()); 346 347 final FieldODEStateAndDerivative<T> finalState = getStepStart(); 348 setStepStart(null); 349 setStepSize(null); 350 return finalState; 351 352 } 353 354 } 355