diff options
Diffstat (limited to 'src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthFieldIntegrator.java')
| -rw-r--r-- | src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthFieldIntegrator.java | 354 |
1 files changed, 354 insertions, 0 deletions
diff --git a/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthFieldIntegrator.java b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthFieldIntegrator.java new file mode 100644 index 0000000..bec3343 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/ode/nonstiff/AdamsBashforthFieldIntegrator.java @@ -0,0 +1,354 @@ +/* + * 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.math3.ode.nonstiff; + +import org.apache.commons.math3.Field; +import org.apache.commons.math3.RealFieldElement; +import org.apache.commons.math3.exception.DimensionMismatchException; +import org.apache.commons.math3.exception.MaxCountExceededException; +import org.apache.commons.math3.exception.NoBracketingException; +import org.apache.commons.math3.exception.NumberIsTooSmallException; +import org.apache.commons.math3.linear.Array2DRowFieldMatrix; +import org.apache.commons.math3.linear.FieldMatrix; +import org.apache.commons.math3.ode.FieldExpandableODE; +import org.apache.commons.math3.ode.FieldODEState; +import org.apache.commons.math3.ode.FieldODEStateAndDerivative; +import org.apache.commons.math3.util.MathArrays; + + +/** + * 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: + * <pre> + * 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 + * </pre></p> + * + * <p>The definitions above use the classical representation with several previous first + * derivatives. Lets define + * <pre> + * 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> + * </pre> + * (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> + * + * <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: + * <pre> + * r<sub>n</sub> = [ s<sub>2</sub>(n), s<sub>3</sub>(n) ... s<sub>k</sub>(n) ]<sup>T</sup> + * </pre> + * (here again we omit the k index in the notation for clarity) + * </p> + * + * <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. + * <pre> + * 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) + * </pre> + * 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: + * <pre> + * q<sub>n</sub> = s<sub>1</sub>(n) u + P r<sub>n</sub> + * </pre> + * 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> + * + * <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> + * + * <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> + * + * <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> + * + * @param <T> the type of the field elements + * @since 3.6 + */ +public class AdamsBashforthFieldIntegrator<T extends RealFieldElement<T>> extends AdamsFieldIntegrator<T> { + + /** 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 field field to which the time and state vector elements belong + * @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 AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, + final double minStep, final double maxStep, + final double scalAbsoluteTolerance, + final double scalRelativeTolerance) + throws NumberIsTooSmallException { + super(field, METHOD_NAME, nSteps, nSteps, minStep, maxStep, + scalAbsoluteTolerance, scalRelativeTolerance); + } + + /** + * Build an Adams-Bashforth integrator with the given order and step control parameters. + * @param field field to which the time and state vector elements belong + * @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 AdamsBashforthFieldIntegrator(final Field<T> field, final int nSteps, + final double minStep, final double maxStep, + final double[] vecAbsoluteTolerance, + final double[] vecRelativeTolerance) + throws IllegalArgumentException { + super(field, 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 T errorEstimation(final T[] previousState, + final T[] predictedState, + final T[] predictedScaled, + final FieldMatrix<T> predictedNordsieck) { + + T error = getField().getZero(); + for (int i = 0; i < mainSetDimension; ++i) { + final T yScale = predictedState[i].abs(); + final T tol = (vecAbsoluteTolerance == null) ? + yScale.multiply(scalRelativeTolerance).add(scalAbsoluteTolerance) : + yScale.multiply(vecRelativeTolerance[i]).add(vecAbsoluteTolerance[i]); + + // apply Taylor formula from high order to low order, + // for the sake of numerical accuracy + T variation = getField().getZero(); + int sign = predictedNordsieck.getRowDimension() % 2 == 0 ? -1 : 1; + for (int k = predictedNordsieck.getRowDimension() - 1; k >= 0; --k) { + variation = variation.add(predictedNordsieck.getEntry(k, i).multiply(sign)); + sign = -sign; + } + variation = variation.subtract(predictedScaled[i]); + + final T ratio = predictedState[i].subtract(previousState[i]).add(variation).divide(tol); + error = error.add(ratio.multiply(ratio)); + + } + + return error.divide(mainSetDimension).sqrt(); + + } + + /** {@inheritDoc} */ + @Override + public FieldODEStateAndDerivative<T> integrate(final FieldExpandableODE<T> equations, + final FieldODEState<T> initialState, + final T finalTime) + throws NumberIsTooSmallException, DimensionMismatchException, + MaxCountExceededException, NoBracketingException { + + sanityChecks(initialState, finalTime); + final T t0 = initialState.getTime(); + final T[] y = equations.getMapper().mapState(initialState); + setStepStart(initIntegration(equations, t0, y, finalTime)); + final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0; + + // compute the initial Nordsieck vector using the configured starter integrator + start(equations, getStepStart(), finalTime); + + // reuse the step that was chosen by the starter integrator + FieldODEStateAndDerivative<T> stepStart = getStepStart(); + FieldODEStateAndDerivative<T> stepEnd = + AdamsFieldStepInterpolator.taylor(stepStart, + stepStart.getTime().add(getStepSize()), + getStepSize(), scaled, nordsieck); + + // main integration loop + setIsLastStep(false); + do { + + T[] predictedY = null; + final T[] predictedScaled = MathArrays.buildArray(getField(), y.length); + Array2DRowFieldMatrix<T> predictedNordsieck = null; + T error = getField().getZero().add(10); + while (error.subtract(1.0).getReal() >= 0.0) { + + // predict a first estimate of the state at step end + predictedY = stepEnd.getState(); + + // evaluate the derivative + final T[] yDot = computeDerivatives(stepEnd.getTime(), predictedY); + + // predict Nordsieck vector at step end + for (int j = 0; j < predictedScaled.length; ++j) { + predictedScaled[j] = getStepSize().multiply(yDot[j]); + } + predictedNordsieck = updateHighOrderDerivativesPhase1(nordsieck); + updateHighOrderDerivativesPhase2(scaled, predictedScaled, predictedNordsieck); + + // evaluate error + error = errorEstimation(y, predictedY, predictedScaled, predictedNordsieck); + + if (error.subtract(1.0).getReal() >= 0.0) { + // reject the step and attempt to reduce error by stepsize control + final T factor = computeStepGrowShrinkFactor(error); + rescale(filterStep(getStepSize().multiply(factor), forward, false)); + stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), + getStepStart().getTime().add(getStepSize()), + getStepSize(), + scaled, + nordsieck); + + } + } + + // discrete events handling + setStepStart(acceptStep(new AdamsFieldStepInterpolator<T>(getStepSize(), stepEnd, + predictedScaled, predictedNordsieck, forward, + getStepStart(), stepEnd, + equations.getMapper()), + finalTime)); + scaled = predictedScaled; + nordsieck = predictedNordsieck; + + if (!isLastStep()) { + + System.arraycopy(predictedY, 0, y, 0, y.length); + + if (resetOccurred()) { + // some events handler has triggered changes that + // invalidate the derivatives, we need to restart from scratch + start(equations, getStepStart(), finalTime); + } + + // stepsize control for next step + final T factor = computeStepGrowShrinkFactor(error); + final T scaledH = getStepSize().multiply(factor); + final T nextT = getStepStart().getTime().add(scaledH); + final boolean nextIsLast = forward ? + nextT.subtract(finalTime).getReal() >= 0 : + nextT.subtract(finalTime).getReal() <= 0; + T hNew = filterStep(scaledH, forward, nextIsLast); + + final T filteredNextT = getStepStart().getTime().add(hNew); + final boolean filteredNextIsLast = forward ? + filteredNextT.subtract(finalTime).getReal() >= 0 : + filteredNextT.subtract(finalTime).getReal() <= 0; + if (filteredNextIsLast) { + hNew = finalTime.subtract(getStepStart().getTime()); + } + + rescale(hNew); + stepEnd = AdamsFieldStepInterpolator.taylor(getStepStart(), getStepStart().getTime().add(getStepSize()), + getStepSize(), scaled, nordsieck); + + } + + } while (!isLastStep()); + + final FieldODEStateAndDerivative<T> finalState = getStepStart(); + setStepStart(null); + setStepSize(null); + return finalState; + + } + +} |