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 org.apache.commons.math3.exception.DimensionMismatchException; 021import org.apache.commons.math3.exception.MaxCountExceededException; 022import org.apache.commons.math3.exception.NoBracketingException; 023import org.apache.commons.math3.exception.NumberIsTooSmallException; 024import org.apache.commons.math3.linear.Array2DRowRealMatrix; 025import org.apache.commons.math3.ode.EquationsMapper; 026import org.apache.commons.math3.ode.ExpandableStatefulODE; 027import org.apache.commons.math3.ode.sampling.NordsieckStepInterpolator; 028import org.apache.commons.math3.util.FastMath; 029 030 031/** 032 * This class implements explicit Adams-Bashforth integrators for Ordinary 033 * Differential Equations. 034 * 035 * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit 036 * multistep ODE solvers. This implementation is a variation of the classical 037 * one: it uses adaptive stepsize to implement error control, whereas 038 * classical implementations are fixed step size. The value of state vector 039 * at step n+1 is a simple combination of the value at step n and of the 040 * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous 041 * steps one wants to use for computing the next value, different formulas 042 * are available:</p> 043 * <ul> 044 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li> 045 * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li> 046 * <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> 047 * <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> 048 * <li>...</li> 049 * </ul> 050 * 051 * <p>A k-steps Adams-Bashforth method is of order k.</p> 052 * 053 * <h3>Implementation details</h3> 054 * 055 * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: 056 * <pre> 057 * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative 058 * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative 059 * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative 060 * ... 061 * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative 062 * </pre></p> 063 * 064 * <p>The definitions above use the classical representation with several previous first 065 * derivatives. Lets define 066 * <pre> 067 * 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> 068 * </pre> 069 * (we omit the k index in the notation for clarity). With these definitions, 070 * Adams-Bashforth methods can be written: 071 * <ul> 072 * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li> 073 * <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> 074 * <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> 075 * <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> 076 * <li>...</li> 077 * </ul></p> 078 * 079 * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, 080 * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with 081 * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) 082 * and r<sub>n</sub>) where r<sub>n</sub> is defined as: 083 * <pre> 084 * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> 085 * </pre> 086 * (here again we omit the k index in the notation for clarity) 087 * </p> 088 * 089 * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be 090 * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact 091 * for degree k polynomials. 092 * <pre> 093 * 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) 094 * </pre> 095 * The previous formula can be used with several values for i to compute the transform between 096 * classical representation and Nordsieck vector. The transform between r<sub>n</sub> 097 * and q<sub>n</sub> resulting from the Taylor series formulas above is: 098 * <pre> 099 * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> 100 * </pre> 101 * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built 102 * with the j (-i)<sup>j-1</sup> terms: 103 * <pre> 104 * [ -2 3 -4 5 ... ] 105 * [ -4 12 -32 80 ... ] 106 * P = [ -6 27 -108 405 ... ] 107 * [ -8 48 -256 1280 ... ] 108 * [ ... ] 109 * </pre></p> 110 * 111 * <p>Using the Nordsieck vector has several advantages: 112 * <ul> 113 * <li>it greatly simplifies step interpolation as the interpolator mainly applies 114 * Taylor series formulas,</li> 115 * <li>it simplifies step changes that occur when discrete events that truncate 116 * the step are triggered,</li> 117 * <li>it allows to extend the methods in order to support adaptive stepsize.</li> 118 * </ul></p> 119 * 120 * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: 121 * <ul> 122 * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> 123 * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> 124 * <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> 125 * </ul> 126 * where A is a rows shifting matrix (the lower left part is an identity matrix): 127 * <pre> 128 * [ 0 0 ... 0 0 | 0 ] 129 * [ ---------------+---] 130 * [ 1 0 ... 0 0 | 0 ] 131 * A = [ 0 1 ... 0 0 | 0 ] 132 * [ ... | 0 ] 133 * [ 0 0 ... 1 0 | 0 ] 134 * [ 0 0 ... 0 1 | 0 ] 135 * </pre></p> 136 * 137 * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, 138 * they only depend on k and therefore are precomputed once for all.</p> 139 * 140 * @since 2.0 141 */ 142public class AdamsBashforthIntegrator extends AdamsIntegrator { 143 144 /** Integrator method name. */ 145 private static final String METHOD_NAME = "Adams-Bashforth"; 146 147 /** 148 * Build an Adams-Bashforth integrator with the given order and step control parameters. 149 * @param nSteps number of steps of the method excluding the one being computed 150 * @param minStep minimal step (sign is irrelevant, regardless of 151 * integration direction, forward or backward), the last step can 152 * be smaller than this 153 * @param maxStep maximal step (sign is irrelevant, regardless of 154 * integration direction, forward or backward), the last step can 155 * be smaller than this 156 * @param scalAbsoluteTolerance allowed absolute error 157 * @param scalRelativeTolerance allowed relative error 158 * @exception NumberIsTooSmallException if order is 1 or less 159 */ 160 public AdamsBashforthIntegrator(final int nSteps, 161 final double minStep, final double maxStep, 162 final double scalAbsoluteTolerance, 163 final double scalRelativeTolerance) 164 throws NumberIsTooSmallException { 165 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 166 scalAbsoluteTolerance, scalRelativeTolerance); 167 } 168 169 /** 170 * Build an Adams-Bashforth integrator with the given order and step control parameters. 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 vecAbsoluteTolerance allowed absolute error 179 * @param vecRelativeTolerance allowed relative error 180 * @exception IllegalArgumentException if order is 1 or less 181 */ 182 public AdamsBashforthIntegrator(final int nSteps, 183 final double minStep, final double maxStep, 184 final double[] vecAbsoluteTolerance, 185 final double[] vecRelativeTolerance) 186 throws IllegalArgumentException { 187 super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, 188 vecAbsoluteTolerance, vecRelativeTolerance); 189 } 190 191 /** {@inheritDoc} */ 192 @Override 193 public void integrate(final ExpandableStatefulODE equations, final double t) 194 throws NumberIsTooSmallException, DimensionMismatchException, 195 MaxCountExceededException, NoBracketingException { 196 197 sanityChecks(equations, t); 198 setEquations(equations); 199 final boolean forward = t > equations.getTime(); 200 201 // initialize working arrays 202 final double[] y0 = equations.getCompleteState(); 203 final double[] y = y0.clone(); 204 final double[] yDot = new double[y.length]; 205 206 // set up an interpolator sharing the integrator arrays 207 final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); 208 interpolator.reinitialize(y, forward, 209 equations.getPrimaryMapper(), equations.getSecondaryMappers()); 210 211 // set up integration control objects 212 initIntegration(equations.getTime(), y0, t); 213 214 // compute the initial Nordsieck vector using the configured starter integrator 215 start(equations.getTime(), y, t); 216 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 217 interpolator.storeTime(stepStart); 218 final int lastRow = nordsieck.getRowDimension() - 1; 219 220 // reuse the step that was chosen by the starter integrator 221 double hNew = stepSize; 222 interpolator.rescale(hNew); 223 224 // main integration loop 225 isLastStep = false; 226 do { 227 228 double error = 10; 229 while (error >= 1.0) { 230 231 stepSize = hNew; 232 233 // evaluate error using the last term of the Taylor expansion 234 error = 0; 235 for (int i = 0; i < mainSetDimension; ++i) { 236 final double yScale = FastMath.abs(y[i]); 237 final double tol = (vecAbsoluteTolerance == null) ? 238 (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : 239 (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); 240 final double ratio = nordsieck.getEntry(lastRow, i) / tol; 241 error += ratio * ratio; 242 } 243 error = FastMath.sqrt(error / mainSetDimension); 244 245 if (error >= 1.0) { 246 // reject the step and attempt to reduce error by stepsize control 247 final double factor = computeStepGrowShrinkFactor(error); 248 hNew = filterStep(stepSize * factor, forward, false); 249 interpolator.rescale(hNew); 250 251 } 252 } 253 254 // predict a first estimate of the state at step end 255 final double stepEnd = stepStart + stepSize; 256 interpolator.shift(); 257 interpolator.setInterpolatedTime(stepEnd); 258 final ExpandableStatefulODE expandable = getExpandable(); 259 final EquationsMapper primary = expandable.getPrimaryMapper(); 260 primary.insertEquationData(interpolator.getInterpolatedState(), y); 261 int index = 0; 262 for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { 263 secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), y); 264 ++index; 265 } 266 267 // evaluate the derivative 268 computeDerivatives(stepEnd, y, yDot); 269 270 // update Nordsieck vector 271 final double[] predictedScaled = new double[y0.length]; 272 for (int j = 0; j < y0.length; ++j) { 273 predictedScaled[j] = stepSize * yDot[j]; 274 } 275 final Array2DRowRealMatrix nordsieckTmp = updateHighOrderDerivativesPhase1(nordsieck); 276 updateHighOrderDerivativesPhase2(scaled, predictedScaled, nordsieckTmp); 277 interpolator.reinitialize(stepEnd, stepSize, predictedScaled, nordsieckTmp); 278 279 // discrete events handling 280 interpolator.storeTime(stepEnd); 281 stepStart = acceptStep(interpolator, y, yDot, t); 282 scaled = predictedScaled; 283 nordsieck = nordsieckTmp; 284 interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); 285 286 if (!isLastStep) { 287 288 // prepare next step 289 interpolator.storeTime(stepStart); 290 291 if (resetOccurred) { 292 // some events handler has triggered changes that 293 // invalidate the derivatives, we need to restart from scratch 294 start(stepStart, y, t); 295 interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); 296 } 297 298 // stepsize control for next step 299 final double factor = computeStepGrowShrinkFactor(error); 300 final double scaledH = stepSize * factor; 301 final double nextT = stepStart + scaledH; 302 final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); 303 hNew = filterStep(scaledH, forward, nextIsLast); 304 305 final double filteredNextT = stepStart + hNew; 306 final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); 307 if (filteredNextIsLast) { 308 hNew = t - stepStart; 309 } 310 311 interpolator.rescale(hNew); 312 313 } 314 315 } while (!isLastStep); 316 317 // dispatch results 318 equations.setTime(stepStart); 319 equations.setCompleteState(y); 320 321 resetInternalState(); 322 323 } 324 325}