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 * Licensed to the Apache Software Foundation (ASF) under one or more
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 * 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
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 *      http://www.apache.org/licenses/LICENSE-2.0
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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.ode.AbstractFieldIntegrator;
import org.apache.commons.math3.ode.FieldEquationsMapper;
import org.apache.commons.math3.ode.FieldExpandableODE;
import org.apache.commons.math3.ode.FirstOrderFieldDifferentialEquations;
import org.apache.commons.math3.ode.FieldODEState;
import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
import org.apache.commons.math3.util.MathArrays;

This class implements the common part of all fixed step Runge-Kutta
integrators for Ordinary Differential Equations.
These methods are explicit Runge-Kutta methods, their Butcher
arrays are as follows :
   0  |
  c2  | a21
  c3  | a31  a32
  ... |        ...
  cs  | as1  as2  ...  ass-1
      |--------------------------
      |  b1   b2  ...   bs-1  bs


Type parameters:
<T> – the type of the field elements
See Also:
EulerFieldIntegrator
ClassicalRungeKuttaFieldIntegrator
GillFieldIntegrator
MidpointFieldIntegrator
Since:
3.6/**
 * This class implements the common part of all fixed step Runge-Kutta
 * integrators for Ordinary Differential Equations.
 *
 * <p>These methods are explicit Runge-Kutta methods, their Butcher
 * arrays are as follows :
 * <pre>
 *    0  |
 *   c2  | a21
 *   c3  | a31  a32
 *   ... |        ...
 *   cs  | as1  as2  ...  ass-1
 *       |--------------------------
 *       |  b1   b2  ...   bs-1  bs
 * </pre>
 * </p>
 *
 * @see EulerFieldIntegrator
 * @see ClassicalRungeKuttaFieldIntegrator
 * @see GillFieldIntegrator
 * @see MidpointFieldIntegrator
 * @param <T> the type of the field elements
 * @since 3.6
 */

public abstract class RungeKuttaFieldIntegrator<T extends RealFieldElement<T>>
    extends AbstractFieldIntegrator<T>
    implements FieldButcherArrayProvider<T> {

    
Time steps from Butcher array (without the first zero). /** Time steps from Butcher array (without the first zero). */
    private final T[] c;

    
Internal weights from Butcher array (without the first empty row). /** Internal weights from Butcher array (without the first empty row). */
    private final T[][] a;

    
External weights for the high order method from Butcher array. /** External weights for the high order method from Butcher array. */
    private final T[] b;

    
Integration step. /** Integration step. */
    private final T step;

    
Simple constructor.
Build a Runge-Kutta integrator with the given
step. The default step handler does nothing.
Params:
field – field to which the time and state vector elements belong
name – name of the method
step – integration step/** Simple constructor.
     * Build a Runge-Kutta integrator with the given
     * step. The default step handler does nothing.
     * @param field field to which the time and state vector elements belong
     * @param name name of the method
     * @param step integration step
     */
    protected RungeKuttaFieldIntegrator(final Field<T> field, final String name, final T step) {
        super(field, name);
        this.c    = getC();
        this.a    = getA();
        this.b    = getB();
        this.step = step.abs();
    }

    
Create a fraction.
Params:
p – numerator
q – denominator
Returns:
p/q computed in the instance field/** Create a fraction.
     * @param p numerator
     * @param q denominator
     * @return p/q computed in the instance field
     */
    protected T fraction(final int p, final int q) {
        return getField().getZero().add(p).divide(q);
    }

    
Create an interpolator.
Params:
forward – integration direction indicator
yDotK – slopes at the intermediate points
globalPreviousState – start of the global step
globalCurrentState – end of the global step
mapper – equations mapper for the all equations
Returns:
external weights for the high order method from Butcher array/** Create an interpolator.
     * @param forward integration direction indicator
     * @param yDotK slopes at the intermediate points
     * @param globalPreviousState start of the global step
     * @param globalCurrentState end of the global step
     * @param mapper equations mapper for the all equations
     * @return external weights for the high order method from Butcher array
     */
    protected abstract RungeKuttaFieldStepInterpolator<T> createInterpolator(boolean forward, T[][] yDotK,
                                                                             final FieldODEStateAndDerivative<T> globalPreviousState,
                                                                             final FieldODEStateAndDerivative<T> globalCurrentState,
                                                                             FieldEquationsMapper<T> mapper);

    
{@inheritDoc} /** {@inheritDoc} */
    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[] y0 = equations.getMapper().mapState(initialState);
        setStepStart(initIntegration(equations, t0, y0, finalTime));
        final boolean forward = finalTime.subtract(initialState.getTime()).getReal() > 0;

        // create some internal working arrays
        final int   stages = c.length + 1;
        T[]         y      = y0;
        final T[][] yDotK  = MathArrays.buildArray(getField(), stages, -1);
        final T[]   yTmp   = MathArrays.buildArray(getField(), y0.length);

        // set up integration control objects
        if (forward) {
            if (getStepStart().getTime().add(step).subtract(finalTime).getReal() >= 0) {
                setStepSize(finalTime.subtract(getStepStart().getTime()));
            } else {
                setStepSize(step);
            }
        } else {
            if (getStepStart().getTime().subtract(step).subtract(finalTime).getReal() <= 0) {
                setStepSize(finalTime.subtract(getStepStart().getTime()));
            } else {
                setStepSize(step.negate());
            }
        }

        // main integration loop
        setIsLastStep(false);
        do {

            // first stage
            y        = equations.getMapper().mapState(getStepStart());
            yDotK[0] = equations.getMapper().mapDerivative(getStepStart());

            // next stages
            for (int k = 1; k < stages; ++k) {

                for (int j = 0; j < y0.length; ++j) {
                    T sum = yDotK[0][j].multiply(a[k-1][0]);
                    for (int l = 1; l < k; ++l) {
                        sum = sum.add(yDotK[l][j].multiply(a[k-1][l]));
                    }
                    yTmp[j] = y[j].add(getStepSize().multiply(sum));
                }

                yDotK[k] = computeDerivatives(getStepStart().getTime().add(getStepSize().multiply(c[k-1])), yTmp);

            }

            // estimate the state at the end of the step
            for (int j = 0; j < y0.length; ++j) {
                T sum = yDotK[0][j].multiply(b[0]);
                for (int l = 1; l < stages; ++l) {
                    sum = sum.add(yDotK[l][j].multiply(b[l]));
                }
                yTmp[j] = y[j].add(getStepSize().multiply(sum));
            }
            final T stepEnd   = getStepStart().getTime().add(getStepSize());
            final T[] yDotTmp = computeDerivatives(stepEnd, yTmp);
            final FieldODEStateAndDerivative<T> stateTmp = new FieldODEStateAndDerivative<T>(stepEnd, yTmp, yDotTmp);

            // discrete events handling
            System.arraycopy(yTmp, 0, y, 0, y0.length);
            setStepStart(acceptStep(createInterpolator(forward, yDotK, getStepStart(), stateTmp, equations.getMapper()),
                                    finalTime));

            if (!isLastStep()) {

                // stepsize control for next step
                final T  nextT      = getStepStart().getTime().add(getStepSize());
                final boolean nextIsLast = forward ?
                                           (nextT.subtract(finalTime).getReal() >= 0) :
                                           (nextT.subtract(finalTime).getReal() <= 0);
                if (nextIsLast) {
                    setStepSize(finalTime.subtract(getStepStart().getTime()));
                }
            }

        } while (!isLastStep());

        final FieldODEStateAndDerivative<T> finalState = getStepStart();
        setStepStart(null);
        setStepSize(null);
        return finalState;

    }

    
Fast computation of a single step of ODE integration.
This method is intended for the limited use case of
very fast computation of only one step without using any of the
rich features of general integrators that may take some time
to set up (i.e. no step handlers, no events handlers, no additional
states, no interpolators, no error control, no evaluations count,
no sanity checks ...). It handles the strict minimum of computation,
so it can be embedded in outer loops.

This method is not used at all by the integrate(FieldExpandableODE, FieldODEState, RealFieldElement) method. It also completely ignores the step set at construction time, and uses only a single step to go from t0 to t. 

As this method does not use any of the state-dependent features of the integrator,
it should be reasonably thread-safe if and only if the provided differential
equations are themselves thread-safe.

Params:
equations – differential equations to integrate
t0 – initial time
y0 – initial value of the state vector at t0
t – target time for the integration (can be set to a value smaller than t0 for backward integration)
Returns:
state vector at t/** Fast computation of a single step of ODE integration.
     * <p>This method is intended for the limited use case of
     * very fast computation of only one step without using any of the
     * rich features of general integrators that may take some time
     * to set up (i.e. no step handlers, no events handlers, no additional
     * states, no interpolators, no error control, no evaluations count,
     * no sanity checks ...). It handles the strict minimum of computation,
     * so it can be embedded in outer loops.</p>
     * <p>
     * This method is <em>not</em> used at all by the {@link #integrate(FieldExpandableODE,
     * FieldODEState, RealFieldElement)} method. It also completely ignores the step set at
     * construction time, and uses only a single step to go from {@code t0} to {@code t}.
     * </p>
     * <p>
     * As this method does not use any of the state-dependent features of the integrator,
     * it should be reasonably thread-safe <em>if and only if</em> the provided differential
     * equations are themselves thread-safe.
     * </p>
     * @param equations differential equations to integrate
     * @param t0 initial time
     * @param y0 initial value of the state vector at t0
     * @param t target time for the integration
     * (can be set to a value smaller than {@code t0} for backward integration)
     * @return state vector at {@code t}
     */
    public T[] singleStep(final FirstOrderFieldDifferentialEquations<T> equations,
                          final T t0, final T[] y0, final T t) {

        // create some internal working arrays
        final T[] y       = y0.clone();
        final int stages  = c.length + 1;
        final T[][] yDotK = MathArrays.buildArray(getField(), stages, -1);
        final T[] yTmp    = y0.clone();

        // first stage
        final T h = t.subtract(t0);
        yDotK[0] = equations.computeDerivatives(t0, y);

        // next stages
        for (int k = 1; k < stages; ++k) {

            for (int j = 0; j < y0.length; ++j) {
                T sum = yDotK[0][j].multiply(a[k-1][0]);
                for (int l = 1; l < k; ++l) {
                    sum = sum.add(yDotK[l][j].multiply(a[k-1][l]));
                }
                yTmp[j] = y[j].add(h.multiply(sum));
            }

            yDotK[k] = equations.computeDerivatives(t0.add(h.multiply(c[k-1])), yTmp);

        }

        // estimate the state at the end of the step
        for (int j = 0; j < y0.length; ++j) {
            T sum = yDotK[0][j].multiply(b[0]);
            for (int l = 1; l < stages; ++l) {
                sum = sum.add(yDotK[l][j].multiply(b[l]));
            }
            y[j] = y[j].add(h.multiply(sum));
        }

        return y;

    }

}