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 org.apache.commons.math.linear.Array2DRowRealMatrix; 21 import org.apache.commons.math.ode.DerivativeException; 22 import org.apache.commons.math.ode.FirstOrderDifferentialEquations; 23 import org.apache.commons.math.ode.IntegratorException; 24 import org.apache.commons.math.ode.sampling.NordsieckStepInterpolator; 25 import org.apache.commons.math.ode.sampling.StepHandler; 26 import org.apache.commons.math.util.FastMath; 27 28 29 /** 30 * This class implements explicit Adams-Bashforth integrators for Ordinary 31 * Differential Equations. 32 * 33 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit 34 * multistep ODE solvers. This implementation is a variation of the classical 35 * one: it uses adaptive stepsize to implement error control, whereas 36 * classical implementations are fixed step size. The value of state vector 37 * at step n+1 is a simple combination of the value at step n and of the 38 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous 39 * steps one wants to use for computing the next value, different formulas 40 * are available:</p> 41 * <ul> 42 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li> 43 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li> 44 * <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> 45 * <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> 46 * <li>...</li> 47 * </ul> 48 * 49 * <p>A k-steps Adams-Bashforth method is of order k.</p> 50 * 51 * <h3>Implementation details</h3> 52 * 53 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 54 * <pre> 55 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 56 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 57 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 58 * ... 59 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y(k)<sub>n</sub> for k<sup>th</sup> derivative 60 * </pre></p> 61 * 62 * <p>The definitions above use the classical representation with several previous first 63 * derivatives. Lets define 64 * <pre> 65 * 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> 66 * </pre> 67 * (we omit the k index in the notation for clarity). With these definitions, 68 * Adams-Bashforth methods can be written: 69 * <ul> 70 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li> 71 * <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> 72 * <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> 73 * <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> 74 * <li>...</li> 75 * </ul></p> 76 * 77 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 78 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with 79 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 80 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 81 * <pre> 82 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 83 * </pre> 84 * (here again we omit the k index in the notation for clarity) 85 * </p> 86 * 87 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 88 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 89 * for degree k polynomials. 90 * <pre> 91 * 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) 92 * </pre> 93 * The previous formula can be used with several values for i to compute the transform between 94 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 95 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 96 * <pre> 97 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 98 * </pre> 99 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 100 * with the j (-i)<sup>j-1</sup> terms: 101 * <pre> 102 * [ -2 3 -4 5 ... ] 103 * [ -4 12 -32 80 ... ] 104 * P = [ -6 27 -108 405 ... ] 105 * [ -8 48 -256 1280 ... ] 106 * [ ... ] 107 * </pre></p> 108 * 109 * <p>Using the Nordsieck vector has several advantages: 110 * <ul> 111 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 112 * Taylor series formulas,</li> 113 * <li>it simplifies step changes that occur when discrete events that truncate 114 * the step are triggered,</li> 115 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 116 * </ul></p> 117 * 118 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: 119 * <ul> 120 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 121 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 122 * <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> 123 * </ul> 124 * where A is a rows shifting matrix (the lower left part is an identity matrix): 125 * <pre> 126 * [ 0 0 ... 0 0 | 0 ] 127 * [ ---------------+---] 128 * [ 1 0 ... 0 0 | 0 ] 129 * A = [ 0 1 ... 0 0 | 0 ] 130 * [ ... | 0 ] 131 * [ 0 0 ... 1 0 | 0 ] 132 * [ 0 0 ... 0 1 | 0 ] 133 * </pre></p> 134 * 135 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 136 * they only depend on k and therefore are precomputed once for all.</p> 137 * 138 * @version $Revision: 1073158 $ $Date: 2011-02-21 22:46:52 +0100 (lun. 21 fvr. 2011) $ 139 * @since 2.0 140 */ 141 public class AdamsBashforthIntegrator extends AdamsIntegrator { 142 143 /** Integrator method name. */ 144 private static final String METHOD_NAME = "Adams-Bashforth"; 145 146 /** 147 * Build an Adams-Bashforth integrator with the given order and step control parameters. 148 * @param nSteps number of steps of the method excluding the one being computed 149 * @param minStep minimal step (must be positive even for backward 150 * integration), the last step can be smaller than this 151 * @param maxStep maximal step (must be positive even for backward 152 * integration) 153 * @param scalAbsoluteTolerance allowed absolute error 154 * @param scalRelativeTolerance allowed relative error 155 * @exception IllegalArgumentException if order is 1 or less 156 */ 157 public AdamsBashforthIntegrator(final int nSteps, 158 final double minStep, final double maxStep, 159 final double scalAbsoluteTolerance, 160 final double scalRelativeTolerance) 161 throws IllegalArgumentException { 162 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 163 scalAbsoluteTolerance, scalRelativeTolerance); 164 } 165 166 /** 167 * Build an Adams-Bashforth integrator with the given order and step control parameters. 168 * @param nSteps number of steps of the method excluding the one being computed 169 * @param minStep minimal step (must be positive even for backward 170 * integration), the last step can be smaller than this 171 * @param maxStep maximal step (must be positive even for backward 172 * integration) 173 * @param vecAbsoluteTolerance allowed absolute error 174 * @param vecRelativeTolerance allowed relative error 175 * @exception IllegalArgumentException if order is 1 or less 176 */ 177 public AdamsBashforthIntegrator(final int nSteps, 178 final double minStep, final double maxStep, 179 final double[] vecAbsoluteTolerance, 180 final double[] vecRelativeTolerance) 181 throws IllegalArgumentException { 182 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 183 vecAbsoluteTolerance, vecRelativeTolerance); 184 } 185 186 /** {@inheritDoc} */ 187 @Override 188 public double integrate(final FirstOrderDifferentialEquations equations, 189 final double t0, final double[] y0, 190 final double t, final double[] y) 191 throws DerivativeException, IntegratorException { 192 193 final int n = y0.length; 194 sanityChecks(equations, t0, y0, t, y); 195 setEquations(equations); 196 resetEvaluations(); 197 final boolean forward = t > t0; 198 199 // initialize working arrays 200 if (y != y0) { 201 System.arraycopy(y0, 0, y, 0, n); 202 } 203 final double[] yDot = new double[n]; 204 205 // set up an interpolator sharing the integrator arrays 206 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 207 interpolator.reinitialize(y, forward); 208 209 // set up integration control objects 210 for (StepHandler handler : stepHandlers) { 211 handler.reset(); 212 } 213 setStateInitialized(false); 214 215 // compute the initial Nordsieck vector using the configured starter integrator 216 start(t0, y, t); 217 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 218 interpolator.storeTime(stepStart); 219 final int lastRow = nordsieck.getRowDimension() - 1; 220 221 // reuse the step that was chosen by the starter integrator 222 double hNew = stepSize; 223 interpolator.rescale(hNew); 224 225 // main integration loop 226 isLastStep = false; 227 do { 228 229 double error = 10; 230 while (error >= 1.0) { 231 232 stepSize = hNew; 233 234 // evaluate error using the last term of the Taylor expansion 235 error = 0; 236 for (int i = 0; i < mainSetDimension; ++i) { 237 final double yScale = FastMath.abs(y[i]); 238 final double tol = (vecAbsoluteTolerance == null) ? 239 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 240 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 241 final double ratio = nordsieck.getEntry(lastRow, i) / tol; 242 error += ratio * ratio; 243 } 244 error = FastMath.sqrt(error / mainSetDimension); 245 246 if (error >= 1.0) { 247 // reject the step and attempt to reduce error by stepsize control 248 final double factor = computeStepGrowShrinkFactor(error); 249 hNew = filterStep(stepSize * factor, forward, false); 250 interpolator.rescale(hNew); 251 252 } 253 } 254 255 // predict a first estimate of the state at step end 256 final double stepEnd = stepStart + stepSize; 257 interpolator.shift(); 258 interpolator.setInterpolatedTime(stepEnd); 259 System.arraycopy(interpolator.getInterpolatedState(), 0, y, 0, y0.length); 260 261 // evaluate the derivative 262 computeDerivatives(stepEnd, y, yDot); 263 264 // update Nordsieck vector 265 final double[] predictedScaled = new double[y0.length]; 266 for (int j = 0; j < y0.length; ++j) { 267 predictedScaled[j] = stepSize * yDot[j]; 268 } 269 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck); 270 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp); 271 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp); 272 273 // discrete events handling 274 interpolator.storeTime(stepEnd); 275 stepStart = acceptStep(interpolator, y, yDot, t); 276 scaled = predictedScaled; 277 nordsieck = nordsieckTmp; 278 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); 279 280 if (!isLastStep) { 281 282 // prepare next step 283 interpolator.storeTime(stepStart); 284 285 if (resetOccurred) { 286 // some events handler has triggered changes that 287 // invalidate the derivatives, we need to restart from scratch 288 start(stepStart, y, t); 289 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 290 } 291 292 // stepsize control for next step 293 final double factor = computeStepGrowShrinkFactor(error); 294 final double scaledH = stepSize * factor; 295 final double nextT = stepStart + scaledH; 296 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 297 hNew = filterStep(scaledH, forward, nextIsLast); 298 299 final double filteredNextT = stepStart + hNew; 300 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 301 if (filteredNextIsLast) { 302 hNew = t - stepStart; 303 } 304 305 interpolator.rescale(hNew); 306 307 } 308 309 } while (!isLastStep); 310 311 final double stopTime = stepStart; 312 resetInternalState(); 313 return stopTime; 314 315 } 316 317 } 318
