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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements.  See the NOTICE file distributed with
* 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
*
*
* Unless required by applicable law or agreed to in writing, software
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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/*
* This is not the original file distributed by the Apache Software Foundation
* It has been modified by the Hipparchus project
*/

package org.hipparchus.ode.nonstiff;

import java.util.Arrays;

import org.hipparchus.Field;
import org.hipparchus.CalculusFieldElement;
import org.hipparchus.exception.MathIllegalArgumentException;
import org.hipparchus.exception.MathIllegalStateException;
import org.hipparchus.linear.Array2DRowFieldMatrix;
import org.hipparchus.linear.FieldMatrix;
import org.hipparchus.linear.FieldMatrixPreservingVisitor;
import org.hipparchus.ode.FieldEquationsMapper;
import org.hipparchus.ode.FieldODEStateAndDerivative;
import org.hipparchus.ode.LocalizedODEFormats;
import org.hipparchus.util.MathArrays;
import org.hipparchus.util.MathUtils;

/**
* This class implements implicit Adams-Moulton integrators for Ordinary
* Differential Equations.
*
* 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.FieldODEIntegrator)
* 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>
*
* <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-Moulton methods can be written:
* <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>
*
* <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:
* <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) + &sum;<sub>j&gt;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)&times;(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 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 ... &plusmn;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>
* 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>
*
* @param <T> the type of the field elements
*/

/** Integrator method name. */
private static final String METHOD_NAME = "Adams-Moulton";

/**
* Build an Adams-Moulton integrator with the given order and error 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 MathIllegalArgumentException if order is 1 or less
*/
public AdamsMoultonFieldIntegrator(final Field<T> field, final int nSteps,
final double minStep, final double maxStep,
final double scalAbsoluteTolerance,
final double scalRelativeTolerance)
throws MathIllegalArgumentException {
super(field, METHOD_NAME, nSteps, nSteps + 1, minStep, maxStep,
scalAbsoluteTolerance, scalRelativeTolerance);
}

/**
* Build an Adams-Moulton integrator with the given order and error 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 AdamsMoultonFieldIntegrator(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 + 1, minStep, maxStep,
vecAbsoluteTolerance, vecRelativeTolerance);
}

/** {@inheritDoc} */
@Override
protected double errorEstimation(final T[] previousState, final T predictedTime,
final T[] predictedState, final T[] predictedScaled,
final FieldMatrix<T> predictedNordsieck) {
final double error = predictedNordsieck.walkInOptimizedOrder(new Corrector(previousState, predictedScaled, predictedState)).getReal();
if (Double.isNaN(error)) {
throw new MathIllegalStateException(LocalizedODEFormats.NAN_APPEARING_DURING_INTEGRATION,
predictedTime.getReal());
}
return error;
}

/** {@inheritDoc} */
@Override
protected AdamsFieldStateInterpolator<T> finalizeStep(final T stepSize, final T[] predictedY,
final T[] predictedScaled, final Array2DRowFieldMatrix<T> predictedNordsieck,
final boolean isForward,
final FieldODEStateAndDerivative<T> globalPreviousState,
final FieldODEStateAndDerivative<T> globalCurrentState,
final FieldEquationsMapper<T> equationsMapper) {

final T[] correctedYDot = computeDerivatives(globalCurrentState.getTime(), predictedY);

// update Nordsieck vector
final T[] correctedScaled = MathArrays.buildArray(getField(), predictedY.length);
for (int j = 0; j < correctedScaled.length; ++j) {
correctedScaled[j] = getStepSize().multiply(correctedYDot[j]);
}
updateHighOrderDerivativesPhase2(predictedScaled, correctedScaled, predictedNordsieck);

final FieldODEStateAndDerivative<T> updatedStepEnd =
equationsMapper.mapStateAndDerivative(globalCurrentState.getTime(), predictedY, correctedYDot);
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 ... &plusmn;1 ] r<sub>n+1</sub>
* </pre>
* </p>
*/
private class Corrector implements FieldMatrixPreservingVisitor<T> {

/** Previous state. */
private final T[] previous;

/** Current scaled first derivative. */
private final T[] scaled;

/** Current state before correction. */
private final T[] before;

/** Current state after correction. */
private final T[] 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 T[] previous, final T[] scaled, final T[] state) { // NOPMD - array reference storage is intentional and documented here
this.previous = previous;
this.scaled   = scaled;
this.after    = state;
this.before   = state.clone();
}

/** {@inheritDoc} */
@Override
public void start(int rows, int columns,
int startRow, int endRow, int startColumn, int endColumn) {
Arrays.fill(after, getField().getZero());
}

/** {@inheritDoc} */
@Override
public void visit(int row, int column, T value) {
if ((row & 0x1) == 0) {
after[column] = after[column].subtract(value);
} else {
}
}

/**
* 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 T end() {

final StepsizeHelper helper = getStepSizeHelper();
T error = getField().getZero();
for (int i = 0; i < after.length; ++i) {
if (i < helper.getMainSetDimension()) {
final T tol   = helper.getTolerance(i, MathUtils.max(previous[i].abs(), after[i].abs()));
final T ratio = after[i].subtract(before[i]).divide(tol); // (corrected-predicted)/tol