AdamsBashforthIntegrator.java
/*
* Licensed to the Hipparchus project under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The Hipparchus project 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
*
* https://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.hipparchus.ode.nonstiff;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.linear.Array2DRowRealMatrix;
import org.hipparchus.linear.RealMatrix;
import org.hipparchus.ode.EquationsMapper;
import org.hipparchus.ode.ODEStateAndDerivative;
import org.hipparchus.util.FastMath;
/**
* 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>
*
* <p> There must be sufficient time for the {@link #setStarterIntegrator(org.hipparchus.ode.ODEIntegrator)
* starter integrator} to take several steps between the the last reset event, and the end
* of integration, otherwise an exception may be thrown during integration. The user can
* adjust the end date of integration, or the step size of the starter integrator to
* ensure a sufficient number of steps can be completed before the end of integration.
* </p>
*
* <p><strong>Implementation details</strong></p>
*
* <p>We define scaled derivatives s<sub>i</sub>(n) at step n as:
* \[
* \left\{\begin{align}
* s_1(n) &= h y'_n \text{ for first derivative}\\
* s_2(n) &= \frac{h^2}{2} y_n'' \text{ for second derivative}\\
* s_3(n) &= \frac{h^3}{6} y_n''' \text{ for third derivative}\\
* &\cdots\\
* s_k(n) &= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative}
* \end{align}\right.
* \]</p>
*
* <p>The definitions above use the classical representation with several previous first
* derivatives. Lets define
* \[
* q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T
* \]
* (we omit the k index in the notation for clarity). With these definitions,
* Adams-Bashforth methods can be written:</p>
* \[
* \left\{\begin{align}
* k = 1: & y_{n+1} = y_n + s_1(n) \\
* k = 2: & y_{n+1} = y_n + \frac{3}{2} s_1(n) + [ \frac{-1}{2} ] q_n \\
* k = 3: & y_{n+1} = y_n + \frac{23}{12} s_1(n) + [ \frac{-16}{12} \frac{5}{12} ] q_n \\
* k = 4: & y_{n+1} = y_n + \frac{55}{24} s_1(n) + [ \frac{-59}{24} \frac{37}{24} \frac{-9}{24} ] q_n \\
* & \cdots
* \end{align}\right.
* \]
*
* <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:
* \[
* r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T
* \]
* (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.
* \[
* s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n)
* \]
* 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:
* \[
* q_n = s_1(n) u + P r_n
* \]
* where u is the [ 1 1 ... 1 ]<sup>T</sup> vector and P is the (k-1)×(k-1) matrix built
* with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being
* the column number starting from 1:
* \[
* P=\begin{bmatrix}
* -2 & 3 & -4 & 5 & \ldots \\
* -4 & 12 & -32 & 80 & \ldots \\
* -6 & 27 & -108 & 405 & \ldots \\
* -8 & 48 & -256 & 1280 & \ldots \\
* & & \ldots\\
* \end{bmatrix}
* \]
* </p>
*
* <p>Using the Nordsieck vector has several advantages:</p>
* <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>
* <p>where A is a rows shifting matrix (the lower left part is an identity matrix):</p>
* <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>
*
*/
public class AdamsBashforthIntegrator extends AdamsIntegrator {
/** Name of integration scheme. */
public 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 MathIllegalArgumentException if order is 1 or less
*/
public AdamsBashforthIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws MathIllegalArgumentException {
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);
}
/** {@inheritDoc} */
@Override
protected double errorEstimation(final double[] previousState, final double predictedTime,
final double[] predictedState,
final double[] predictedScaled,
final RealMatrix predictedNordsieck) {
final StepsizeHelper helper = getStepSizeHelper();
double error = 0;
for (int i = 0; i < helper.getMainSetDimension(); ++i) {
final double tol = helper.getTolerance(i, FastMath.abs(predictedState[i]));
// 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 FastMath.sqrt(error / helper.getMainSetDimension());
}
/** {@inheritDoc} */
@Override
protected AdamsStateInterpolator finalizeStep(final double stepSize, final double[] predictedState,
final double[] predictedScaled, final Array2DRowRealMatrix predictedNordsieck,
final boolean isForward,
final ODEStateAndDerivative globalPreviousState,
final ODEStateAndDerivative globalCurrentState,
final EquationsMapper equationsMapper) {
return new AdamsStateInterpolator(getStepSize(), globalCurrentState,
predictedScaled, predictedNordsieck, isForward,
getStepStart(), globalCurrentState, equationsMapper);
}
}