| /* |
| * Licensed to the Apache Software Foundation (ASF) under one or more |
| * contributor license agreements. See the NOTICE file distributed with |
| * this work for additional information regarding copyright ownership. |
| * The ASF licenses this file to You under the Apache License, Version 2.0 |
| * (the "License"); you may not use this file except in compliance with |
| * the License. You may obtain a copy of the License at |
| * |
| * http://www.apache.org/licenses/LICENSE-2.0 |
| * |
| * Unless required by applicable law or agreed to in writing, software |
| * distributed under the License is distributed on an "AS IS" BASIS, |
| * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. |
| * See the License for the specific language governing permissions and |
| * limitations under the License. |
| */ |
| |
| package org.apache.commons.math4.legacy.ode.nonstiff; |
| |
| import org.apache.commons.math4.legacy.exception.DimensionMismatchException; |
| import org.apache.commons.math4.legacy.exception.MaxCountExceededException; |
| import org.apache.commons.math4.legacy.exception.NoBracketingException; |
| import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException; |
| import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix; |
| import org.apache.commons.math4.legacy.linear.RealMatrix; |
| import org.apache.commons.math4.legacy.ode.EquationsMapper; |
| import org.apache.commons.math4.legacy.ode.ExpandableStatefulODE; |
| import org.apache.commons.math4.legacy.ode.sampling.NordsieckStepInterpolator; |
| import org.apache.commons.math4.core.jdkmath.JdkMath; |
| |
| |
| /** |
| * This class implements explicit Adams-Bashforth integrators for Ordinary |
| * Differential Equations. |
| * |
| * <p>Adams-Bashforth methods (in fact due to Adams alone) are explicit |
| * multistep ODE solvers. This implementation is a variation of the classical |
| * one: it uses adaptive stepsize to implement error control, whereas |
| * classical implementations are fixed step size. The value of state vector |
| * at step n+1 is a simple combination of the value at step n and of the |
| * derivatives at steps n, n-1, n-2 ... Depending on the number k of previous |
| * steps one wants to use for computing the next value, different formulas |
| * are available:</p> |
| * <ul> |
| * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n</sub></li> |
| * <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (3y'<sub>n</sub>-y'<sub>n-1</sub>)/2</li> |
| * <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> |
| * <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> |
| * <li>...</li> |
| * </ul> |
| * |
| * <p>A k-steps Adams-Bashforth method is of order k.</p> |
| * |
| * <h3>Implementation details</h3> |
| * |
| * <p>We define scaled derivatives s<sub>i</sub>(n) at step n as: |
| * <div style="white-space: pre"><code> |
| * s<sub>1</sub>(n) = h y'<sub>n</sub> for first derivative |
| * s<sub>2</sub>(n) = h<sup>2</sup>/2 y''<sub>n</sub> for second derivative |
| * s<sub>3</sub>(n) = h<sup>3</sup>/6 y'''<sub>n</sub> for third derivative |
| * ... |
| * s<sub>k</sub>(n) = h<sup>k</sup>/k! y<sup>(k)</sup><sub>n</sub> for k<sup>th</sup> derivative |
| * </code></div> |
| * |
| * <p>The definitions above use the classical representation with several previous first |
| * derivatives. Lets define |
| * <div style="white-space: pre"><code> |
| * 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> |
| * </code></div> |
| * (we omit the k index in the notation for clarity). With these definitions, |
| * Adams-Bashforth methods can be written: |
| * <ul> |
| * <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n)</li> |
| * <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> |
| * <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> |
| * <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> |
| * <li>...</li> |
| * </ul> |
| * |
| * <p>Instead of using the classical representation with first derivatives only (y<sub>n</sub>, |
| * s<sub>1</sub>(n) and q<sub>n</sub>), our implementation uses the Nordsieck vector with |
| * higher degrees scaled derivatives all taken at the same step (y<sub>n</sub>, s<sub>1</sub>(n) |
| * and r<sub>n</sub>) where r<sub>n</sub> is defined as: |
| * <div style="white-space: pre"><code> |
| * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> |
| * </code></div> |
| * (here again we omit the k index in the notation for clarity) |
| * |
| * <p>Taylor series formulas show that for any index offset i, s<sub>1</sub>(n-i) can be |
| * computed from s<sub>1</sub>(n), s<sub>2</sub>(n) ... s<sub>k</sub>(n), the formula being exact |
| * for degree k polynomials. |
| * <div style="white-space: pre"><code> |
| * 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) |
| * </code></div> |
| * The previous formula can be used with several values for i to compute the transform between |
| * classical representation and Nordsieck vector. The transform between r<sub>n</sub> |
| * and q<sub>n</sub> resulting from the Taylor series formulas above is: |
| * <div style="white-space: pre"><code> |
| * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> |
| * </code></div> |
| * where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built |
| * with the (j+1) (-i)<sup>j</sup> terms with i being the row number starting from 1 and j being |
| * the column number starting from 1: |
| * <pre> |
| * [ -2 3 -4 5 ... ] |
| * [ -4 12 -32 80 ... ] |
| * P = [ -6 27 -108 405 ... ] |
| * [ -8 48 -256 1280 ... ] |
| * [ ... ] |
| * </pre> |
| * |
| * <p>Using the Nordsieck vector has several advantages: |
| * <ul> |
| * <li>it greatly simplifies step interpolation as the interpolator mainly applies |
| * Taylor series formulas,</li> |
| * <li>it simplifies step changes that occur when discrete events that truncate |
| * the step are triggered,</li> |
| * <li>it allows to extend the methods in order to support adaptive stepsize.</li> |
| * </ul> |
| * |
| * <p>The Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows: |
| * <ul> |
| * <li>y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n) + u<sup>T</sup> r<sub>n</sub></li> |
| * <li>s<sub>1</sub>(n+1) = h f(t<sub>n+1</sub>, y<sub>n+1</sub>)</li> |
| * <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> |
| * </ul> |
| * where A is a rows shifting matrix (the lower left part is an identity matrix): |
| * <pre> |
| * [ 0 0 ... 0 0 | 0 ] |
| * [ ---------------+---] |
| * [ 1 0 ... 0 0 | 0 ] |
| * A = [ 0 1 ... 0 0 | 0 ] |
| * [ ... | 0 ] |
| * [ 0 0 ... 1 0 | 0 ] |
| * [ 0 0 ... 0 1 | 0 ] |
| * </pre> |
| * |
| * <p>The P<sup>-1</sup>u vector and the P<sup>-1</sup> A P matrix do not depend on the state, |
| * they only depend on k and therefore are precomputed once for all.</p> |
| * |
| * @since 2.0 |
| */ |
| public class AdamsBashforthIntegrator extends AdamsIntegrator { |
| |
| /** Integrator method name. */ |
| private static final String METHOD_NAME = "Adams-Bashforth"; |
| |
| /** |
| * Build an Adams-Bashforth integrator with the given order and step control parameters. |
| * @param nSteps number of steps of the method excluding the one being computed |
| * @param minStep minimal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param maxStep maximal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param scalAbsoluteTolerance allowed absolute error |
| * @param scalRelativeTolerance allowed relative error |
| * @exception NumberIsTooSmallException if order is 1 or less |
| */ |
| public AdamsBashforthIntegrator(final int nSteps, |
| final double minStep, final double maxStep, |
| final double scalAbsoluteTolerance, |
| final double scalRelativeTolerance) |
| throws NumberIsTooSmallException { |
| super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, |
| scalAbsoluteTolerance, scalRelativeTolerance); |
| } |
| |
| /** |
| * Build an Adams-Bashforth integrator with the given order and step control parameters. |
| * @param nSteps number of steps of the method excluding the one being computed |
| * @param minStep minimal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param maxStep maximal step (sign is irrelevant, regardless of |
| * integration direction, forward or backward), the last step can |
| * be smaller than this |
| * @param vecAbsoluteTolerance allowed absolute error |
| * @param vecRelativeTolerance allowed relative error |
| * @exception IllegalArgumentException if order is 1 or less |
| */ |
| public AdamsBashforthIntegrator(final int nSteps, |
| final double minStep, final double maxStep, |
| final double[] vecAbsoluteTolerance, |
| final double[] vecRelativeTolerance) |
| throws IllegalArgumentException { |
| super(METHOD_NAME, nSteps, nSteps, minStep, maxStep, |
| vecAbsoluteTolerance, vecRelativeTolerance); |
| } |
| |
| /** Estimate error. |
| * <p> |
| * Error is estimated by interpolating back to previous state using |
| * the state Taylor expansion and comparing to real previous state. |
| * </p> |
| * @param previousState state vector at step start |
| * @param predictedState predicted state vector at step end |
| * @param predictedScaled predicted value of the scaled derivatives at step end |
| * @param predictedNordsieck predicted value of the Nordsieck vector at step end |
| * @return estimated normalized local discretization error |
| */ |
| private double errorEstimation(final double[] previousState, |
| final double[] predictedState, |
| final double[] predictedScaled, |
| final RealMatrix predictedNordsieck) { |
| |
| double error = 0; |
| for (int i = 0; i < mainSetDimension; ++i) { |
| final double yScale = JdkMath.abs(predictedState[i]); |
| final double tol = (vecAbsoluteTolerance == null) ? |
| (scalAbsoluteTolerance + scalRelativeTolerance * yScale) : |
| (vecAbsoluteTolerance[i] + vecRelativeTolerance[i] * yScale); |
| |
| // apply Taylor formula from high order to low order, |
| // for the sake of numerical accuracy |
| double variation = 0; |
| int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1; |
| for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { |
| variation += sign * predictedNordsieck.getEntry(k, i); |
| sign = -sign; |
| } |
| variation -= predictedScaled[i]; |
| |
| final double ratio = (predictedState[i] - previousState[i] + variation) / tol; |
| error += ratio * ratio; |
| |
| } |
| |
| return JdkMath.sqrt(error / mainSetDimension); |
| |
| } |
| |
| /** {@inheritDoc} */ |
| @Override |
| public void integrate(final ExpandableStatefulODE equations, final double t) |
| throws NumberIsTooSmallException, DimensionMismatchException, |
| MaxCountExceededException, NoBracketingException { |
| |
| sanityChecks(equations, t); |
| setEquations(equations); |
| final boolean forward = t > equations.getTime(); |
| |
| // initialize working arrays |
| final double[] y = equations.getCompleteState(); |
| final double[] yDot = new double[y.length]; |
| |
| // set up an interpolator sharing the integrator arrays |
| final NordsieckStepInterpolator interpolator = new NordsieckStepInterpolator(); |
| interpolator.reinitialize(y, forward, |
| equations.getPrimaryMapper(), equations.getSecondaryMappers()); |
| |
| // set up integration control objects |
| initIntegration(equations.getTime(), y, t); |
| |
| // compute the initial Nordsieck vector using the configured starter integrator |
| start(equations.getTime(), y, t); |
| interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); |
| interpolator.storeTime(stepStart); |
| |
| // reuse the step that was chosen by the starter integrator |
| double hNew = stepSize; |
| interpolator.rescale(hNew); |
| |
| // main integration loop |
| isLastStep = false; |
| do { |
| |
| interpolator.shift(); |
| final double[] predictedY = new double[y.length]; |
| final double[] predictedScaled = new double[y.length]; |
| Array2DRowRealMatrix predictedNordsieck = null; |
| double error = 10; |
| while (error >= 1.0) { |
| |
| // predict a first estimate of the state at step end |
| final double stepEnd = stepStart + hNew; |
| interpolator.storeTime(stepEnd); |
| final ExpandableStatefulODE expandable = getExpandable(); |
| final EquationsMapper primary = expandable.getPrimaryMapper(); |
| primary.insertEquationData(interpolator.getInterpolatedState(), predictedY); |
| int index = 0; |
| for (final EquationsMapper secondary : expandable.getSecondaryMappers()) { |
| secondary.insertEquationData(interpolator.getInterpolatedSecondaryState(index), predictedY); |
| ++index; |
| } |
| |
| // evaluate the derivative |
| computeDerivatives(stepEnd, predictedY, yDot); |
| |
| // predict Nordsieck vector at step end |
| for (int j = 0; j < predictedScaled.length; ++j) { |
| predictedScaled[j] = hNew * yDot[j]; |
| } |
| predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); |
| updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); |
| |
| // evaluate error |
| error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); |
| |
| if (error >= 1.0) { |
| // reject the step and attempt to reduce error by stepsize control |
| final double factor = computeStepGrowShrinkFactor(error); |
| hNew = filterStep(hNew * factor, forward, false); |
| interpolator.rescale(hNew); |
| |
| } |
| } |
| |
| stepSize = hNew; |
| final double stepEnd = stepStart + stepSize; |
| interpolator.reinitialize(stepEnd, stepSize, predictedScaled, predictedNordsieck); |
| |
| // discrete events handling |
| interpolator.storeTime(stepEnd); |
| System.arraycopy(predictedY, 0, y, 0, y.length); |
| stepStart = acceptStep(interpolator, y, yDot, t); |
| scaled = predictedScaled; |
| nordsieck = predictedNordsieck; |
| interpolator.reinitialize(stepEnd, stepSize, scaled, nordsieck); |
| |
| if (!isLastStep) { |
| |
| // prepare next step |
| interpolator.storeTime(stepStart); |
| |
| if (resetOccurred) { |
| // some events handler has triggered changes that |
| // invalidate the derivatives, we need to restart from scratch |
| start(stepStart, y, t); |
| interpolator.reinitialize(stepStart, stepSize, scaled, nordsieck); |
| } |
| |
| // stepsize control for next step |
| final double factor = computeStepGrowShrinkFactor(error); |
| final double scaledH = stepSize * factor; |
| final double nextT = stepStart + scaledH; |
| final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t); |
| hNew = filterStep(scaledH, forward, nextIsLast); |
| |
| final double filteredNextT = stepStart + hNew; |
| final boolean filteredNextIsLast = forward ? (filteredNextT >= t) : (filteredNextT <= t); |
| if (filteredNextIsLast) { |
| hNew = t - stepStart; |
| } |
| |
| interpolator.rescale(hNew); |
| |
| } |
| |
| } while (!isLastStep); |
| |
| // dispatch results |
| equations.setTime(stepStart); |
| equations.setCompleteState(y); |
| |
| resetInternalState(); |
| |
| } |
| |
| } |
