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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.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.AbstractIntegrator;
import org.apache.commons.math3.ode.ExpandableStatefulODE;
import org.apache.commons.math3.ode.FirstOrderDifferentialEquations;
import org.apache.commons.math3.util.FastMath;

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


See Also:
EulerIntegrator
ClassicalRungeKuttaIntegrator
GillIntegrator
MidpointIntegrator
Since:
1.2/**
 * 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 EulerIntegrator
 * @see ClassicalRungeKuttaIntegrator
 * @see GillIntegrator
 * @see MidpointIntegrator
 * @since 1.2
 */

public abstract class RungeKuttaIntegrator extends AbstractIntegrator {

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

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

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

    
Prototype of the step interpolator. /** Prototype of the step interpolator. */
    private final RungeKuttaStepInterpolator prototype;

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

  
Simple constructor.
Build a Runge-Kutta integrator with the given
step. The default step handler does nothing.
Params:
name – name of the method
c – time steps from Butcher array (without the first zero)
a – internal weights from Butcher array (without the first empty row)
b – propagation weights for the high order method from Butcher array
prototype – prototype of the step interpolator to use
step – integration step/** Simple constructor.
   * Build a Runge-Kutta integrator with the given
   * step. The default step handler does nothing.
   * @param name name of the method
   * @param c time steps from Butcher array (without the first zero)
   * @param a internal weights from Butcher array (without the first empty row)
   * @param b propagation weights for the high order method from Butcher array
   * @param prototype prototype of the step interpolator to use
   * @param step integration step
   */
  protected RungeKuttaIntegrator(final String name,
                                 final double[] c, final double[][] a, final double[] b,
                                 final RungeKuttaStepInterpolator prototype,
                                 final double step) {
    super(name);
    this.c          = c;
    this.a          = a;
    this.b          = b;
    this.prototype  = prototype;
    this.step       = FastMath.abs(step);
  }

  
{@inheritDoc} /** {@inheritDoc} */
  @Override
  public void integrate(final ExpandableStatefulODE equations, final double t)
      throws NumberIsTooSmallException, DimensionMismatchException,
             MaxCountExceededException, NoBracketingException {

    sanityChecks(equations, t);
    setEquations(equations);
    final boolean forward = t > equations.getTime();

    // create some internal working arrays
    final double[] y0      = equations.getCompleteState();
    final double[] y       = y0.clone();
    final int stages       = c.length + 1;
    final double[][] yDotK = new double[stages][];
    for (int i = 0; i < stages; ++i) {
      yDotK [i] = new double[y0.length];
    }
    final double[] yTmp    = y0.clone();
    final double[] yDotTmp = new double[y0.length];

    // set up an interpolator sharing the integrator arrays
    final RungeKuttaStepInterpolator interpolator = (RungeKuttaStepInterpolator) prototype.copy();
    interpolator.reinitialize(this, yTmp, yDotK, forward,
                              equations.getPrimaryMapper(), equations.getSecondaryMappers());
    interpolator.storeTime(equations.getTime());

    // set up integration control objects
    stepStart = equations.getTime();
    if (forward) {
        if (stepStart + step >= t) {
            stepSize = t - stepStart;
        } else {
            stepSize = step;
        }
    } else {
        if (stepStart - step <= t) {
            stepSize = t - stepStart;
        } else {
            stepSize = -step;
        }
    }
    initIntegration(equations.getTime(), y0, t);

    // main integration loop
    isLastStep = false;
    do {

      interpolator.shift();

      // first stage
      computeDerivatives(stepStart, y, yDotK[0]);

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

          for (int j = 0; j < y0.length; ++j) {
              double sum = a[k-1][0] * yDotK[0][j];
              for (int l = 1; l < k; ++l) {
                  sum += a[k-1][l] * yDotK[l][j];
              }
              yTmp[j] = y[j] + stepSize * sum;
          }

          computeDerivatives(stepStart + c[k-1] * stepSize, yTmp, yDotK[k]);

      }

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

      // discrete events handling
      interpolator.storeTime(stepStart + stepSize);
      System.arraycopy(yTmp, 0, y, 0, y0.length);
      System.arraycopy(yDotK[stages - 1], 0, yDotTmp, 0, y0.length);
      stepStart = acceptStep(interpolator, y, yDotTmp, t);

      if (!isLastStep) {

          // prepare next step
          interpolator.storeTime(stepStart);

          // stepsize control for next step
          final double  nextT      = stepStart + stepSize;
          final boolean nextIsLast = forward ? (nextT >= t) : (nextT <= t);
          if (nextIsLast) {
              stepSize = t - stepStart;
          }
      }

    } while (!isLastStep);

    // dispatch results
    equations.setTime(stepStart);
    equations.setCompleteState(y);

    stepStart = Double.NaN;
    stepSize  = Double.NaN;

  }

  
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(ExpandableStatefulODE, double) 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(ExpandableStatefulODE, double)}
   * 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 double[] singleStep(final FirstOrderDifferentialEquations equations,
                             final double t0, final double[] y0, final double t) {

      // create some internal working arrays
      final double[] y       = y0.clone();
      final int stages       = c.length + 1;
      final double[][] yDotK = new double[stages][];
      for (int i = 0; i < stages; ++i) {
          yDotK [i] = new double[y0.length];
      }
      final double[] yTmp    = y0.clone();

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

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

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

          equations.computeDerivatives(t0 + c[k-1] * h, yTmp, yDotK[k]);

      }

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

      return y;

  }

}