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With inspiration from other libraries
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/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * 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
 *
 *      http://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.apache.commons.math3.ode.nonstiff;

import org.apache.commons.math3.ode.sampling.StepInterpolator;

/**
 * This class implements a linear interpolator for step.
 *
 * This interpolator computes dense output inside the last
 * step computed. The interpolation equation is consistent with the
 * integration scheme :
 * 

 *   Using reference point at step start:

 *     y(t_n + θ h) = y (t_n) + θ h y'
 *   
 *   Using reference point at step end:

 *     y(t_n + θ h) = y (t_n + h) - (1-θ) h y'
 *   
 * 
 * 
 *
 * where θ belongs to [0 ; 1] and where y' is the evaluation of
 * the derivatives already computed during the step.
 *
 * @see EulerIntegrator
 * @since 1.2
 */

class EulerStepInterpolator
  extends RungeKuttaStepInterpolator {

  /** Serializable version identifier. */
  private static final long serialVersionUID = 20111120L;

  /** Simple constructor.
   * This constructor builds an instance that is not usable yet, the
   * {@link
   * org.apache.commons.math3.ode.sampling.AbstractStepInterpolator#reinitialize}
   * method should be called before using the instance in order to
   * initialize the internal arrays. This constructor is used only
   * in order to delay the initialization in some cases. The {@link
   * RungeKuttaIntegrator} class uses the prototyping design pattern
   * to create the step interpolators by cloning an uninitialized model
   * and later initializing the copy.
   */
  // CHECKSTYLE: stop RedundantModifier
  // the public modifier here is needed for serialization
  public EulerStepInterpolator() {
  }
  // CHECKSTYLE: resume RedundantModifier

  /** Copy constructor.
   * @param interpolator interpolator to copy from. The copy is a deep
   * copy: its arrays are separated from the original arrays of the
   * instance
   */
  EulerStepInterpolator(final EulerStepInterpolator interpolator) {
    super(interpolator);
  }

  /** {@inheritDoc} */
  @Override
  protected StepInterpolator doCopy() {
    return new EulerStepInterpolator(this);
  }


  /** {@inheritDoc} */
  @Override
  protected void computeInterpolatedStateAndDerivatives(final double theta,
                                          final double oneMinusThetaH) {
      if ((previousState != null) && (theta <= 0.5)) {
          for (int i = 0; i < interpolatedState.length; ++i) {
              interpolatedState[i] = previousState[i] + theta * h * yDotK[0][i];
          }
          System.arraycopy(yDotK[0], 0, interpolatedDerivatives, 0, interpolatedDerivatives.length);
      } else {
          for (int i = 0; i < interpolatedState.length; ++i) {
              interpolatedState[i] = currentState[i] - oneMinusThetaH * yDotK[0][i];
          }
          System.arraycopy(yDotK[0], 0, interpolatedDerivatives, 0, interpolatedDerivatives.length);
      }

  }

}