AdamsMoultonIntegrator.java
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* The Hipparchus project licenses this file to You under the Apache License, Version 2.0
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*
* https://www.apache.org/licenses/LICENSE-2.0
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* Unless required by applicable law or agreed to in writing, software
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package org.hipparchus.ode.nonstiff;
import java.util.Arrays;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathIllegalStateException;
import org.hipparchus.linear.Array2DRowRealMatrix;
import org.hipparchus.linear.RealMatrix;
import org.hipparchus.linear.RealMatrixPreservingVisitor;
import org.hipparchus.ode.EquationsMapper;
import org.hipparchus.ode.LocalizedODEFormats;
import org.hipparchus.ode.ODEStateAndDerivative;
import org.hipparchus.util.FastMath;
/**
* This class implements implicit Adams-Moulton integrators for Ordinary
* Differential Equations.
*
* <p>Adams-Moulton methods (in fact due to Adams alone) are implicit
* 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+1, n, n-1 ... Since y'<sub>n+1</sub> is needed to
* compute y<sub>n+1</sub>, another method must be used to compute a first
* estimate of y<sub>n+1</sub>, then compute y'<sub>n+1</sub>, then compute
* a final estimate of y<sub>n+1</sub> using the following formulas. Depending
* on the number k of previous steps one wants to use for computing the next
* value, different formulas are available for the final estimate:</p>
* <ul>
* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + h y'<sub>n+1</sub></li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + h (y'<sub>n+1</sub>+y'<sub>n</sub>)/2</li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + h (5y'<sub>n+1</sub>+8y'<sub>n</sub>-y'<sub>n-1</sub>)/12</li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + h (9y'<sub>n+1</sub>+19y'<sub>n</sub>-5y'<sub>n-1</sub>+y'<sub>n-2</sub>)/24</li>
* <li>...</li>
* </ul>
*
* <p>A k-steps Adams-Moulton method is of order k+1.</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-Moulton methods can be written:</p>
* <ul>
* <li>k = 1: y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1)</li>
* <li>k = 2: y<sub>n+1</sub> = y<sub>n</sub> + 1/2 s<sub>1</sub>(n+1) + [ 1/2 ] q<sub>n+1</sub></li>
* <li>k = 3: y<sub>n+1</sub> = y<sub>n</sub> + 5/12 s<sub>1</sub>(n+1) + [ 8/12 -1/12 ] q<sub>n+1</sub></li>
* <li>k = 4: y<sub>n+1</sub> = y<sub>n</sub> + 9/24 s<sub>1</sub>(n+1) + [ 19/24 -5/24 1/24 ] q<sub>n+1</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+1) and q<sub>n+1</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>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 predicted 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>
* From this predicted vector, the corrected vector is computed as follows:
* <ul>
* <li>y<sub>n+1</sub> = y<sub>n</sub> + S<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</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> = R<sub>n+1</sub> + (s<sub>1</sub>(n+1) - S<sub>1</sub>(n+1)) P<sup>-1</sup> u</li>
* </ul>
* <p>where the upper case Y<sub>n+1</sub>, S<sub>1</sub>(n+1) and R<sub>n+1</sub> represent the
* predicted states whereas the lower case y<sub>n+1</sub>, s<sub>n+1</sub> and r<sub>n+1</sub>
* represent the corrected states.</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>
*
*/
public class AdamsMoultonIntegrator extends AdamsIntegrator {
/** Name of integration scheme. */
public static final String METHOD_NAME = "Adams-Moulton";
/**
* Build an Adams-Moulton integrator with the given order and error 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 AdamsMoultonIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws MathIllegalArgumentException {
super(METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
}
/**
* Build an Adams-Moulton integrator with the given order and error 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 AdamsMoultonIntegrator(final int nSteps,
final double minStep, final double maxStep,
final double[] vecAbsoluteTolerance,
final double[] vecRelativeTolerance)
throws IllegalArgumentException {
super(METHOD_NAME, nSteps, nSteps + 1, 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 double error = predictedNordsieck.walkInOptimizedOrder(new Corrector(previousState, predictedScaled, predictedState));
if (Double.isNaN(error)) {
throw new MathIllegalStateException(LocalizedODEFormats.NAN_APPEARING_DURING_INTEGRATION,
predictedTime);
}
return error;
}
/** {@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) {
final double[] correctedYDot = computeDerivatives(globalCurrentState.getTime(), predictedState);
// update Nordsieck vector
final double[] correctedScaled = new double[predictedState.length];
for (int j = 0; j < correctedScaled.length; ++j) {
correctedScaled[j] = getStepSize() * correctedYDot[j];
}
updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck);
final ODEStateAndDerivative updatedStepEnd =
equationsMapper.mapStateAndDerivative(globalCurrentState.getTime(),
predictedState, correctedYDot);
return new AdamsStateInterpolator(getStepSize(), updatedStepEnd,
correctedScaled, predictedNordsieck, isForward,
getStepStart(), updatedStepEnd,
equationsMapper);
}
/** Corrector for current state in Adams-Moulton method.
* <p>
* This visitor implements the Taylor series formula:
* <pre>
* Y<sub>n+1</sub> = y<sub>n</sub> + s<sub>1</sub>(n+1) + [ -1 +1 -1 +1 ... ±1 ] r<sub>n+1</sub>
* </pre>
* </p>
*/
private class Corrector implements RealMatrixPreservingVisitor {
/** Previous state. */
private final double[] previous;
/** Current scaled first derivative. */
private final double[] scaled;
/** Current state before correction. */
private final double[] before;
/** Current state after correction. */
private final double[] after;
/** Simple constructor.
* <p>
* All arrays will be stored by reference to caller arrays.
* </p>
* @param previous previous state
* @param scaled current scaled first derivative
* @param state state to correct (will be overwritten after visit)
*/
Corrector(final double[] previous, final double[] scaled, final double[] state) {
this.previous = previous; // NOPMD - array reference storage is intentional and documented here
this.scaled = scaled; // NOPMD - array reference storage is intentional and documented here
this.after = state; // NOPMD - array reference storage is intentional and documented here
this.before = state.clone();
}
/** {@inheritDoc} */
@Override
public void start(int rows, int columns,
int startRow, int endRow, int startColumn, int endColumn) {
Arrays.fill(after, 0.0);
}
/** {@inheritDoc} */
@Override
public void visit(int row, int column, double value) {
if ((row & 0x1) == 0) {
after[column] -= value;
} else {
after[column] += value;
}
}
/**
* End visiting the Nordsieck vector.
* <p>The correction is used to control stepsize. So its amplitude is
* considered to be an error, which must be normalized according to
* error control settings. If the normalized value is greater than 1,
* the correction was too large and the step must be rejected.</p>
* @return the normalized correction, if greater than 1, the step
* must be rejected
*/
@Override
public double end() {
final StepsizeHelper helper = getStepSizeHelper();
double error = 0;
for (int i = 0; i < after.length; ++i) {
after[i] += previous[i] + scaled[i];
if (i < helper.getMainSetDimension()) {
final double tol = helper.getTolerance(i, FastMath.max(FastMath.abs(previous[i]), FastMath.abs(after[i])));
final double ratio = (after[i] - before[i]) / tol; // (corrected-predicted)/tol
error += ratio * ratio;
}
}
return FastMath.sqrt(error / helper.getMainSetDimension());
}
}
}