    001    /*
    002     * Licensed to the Apache Software Foundation (ASF) under one or more
    003     * contributor license agreements.  See the NOTICE file distributed with
    004     * this work for additional information regarding copyright ownership.
    005     * The ASF licenses this file to You under the Apache License, Version 2.0
    006     * (the "License"); you may not use this file except in compliance with
    007     * the License.  You may obtain a copy of the License at
    008     *
    009     *      http://www.apache.org/licenses/LICENSE-2.0
    010     *
    011     * Unless required by applicable law or agreed to in writing, software
    012     * distributed under the License is distributed on an "AS IS" BASIS,
    013     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
    014     * See the License for the specific language governing permissions and
    015     * limitations under the License.
    016     */
    017
    018    package org.apache.commons.math3.ode.nonstiff;
    019
    020
    021    /**
    022     * This class implements a simple Euler integrator for Ordinary
    023     * Differential Equations.
    024     *
    025     * <p>The Euler algorithm is the simplest one that can be used to
    026     * integrate ordinary differential equations. It is a simple inversion
    027     * of the forward difference expression :
    028     * <code>f'=(f(t+h)-f(t))/h</code> which leads to
    029     * <code>f(t+h)=f(t)+hf'</code>. The interpolation scheme used for
    030     * dense output is the linear scheme already used for integration.</p>
    031     *
    032     * <p>This algorithm looks cheap because it needs only one function
    033     * evaluation per step. However, as it uses linear estimates, it needs
    034     * very small steps to achieve high accuracy, and small steps lead to
    035     * numerical errors and instabilities.</p>
    036     *
    037     * <p>This algorithm is almost never used and has been included in
    038     * this package only as a comparison reference for more useful
    039     * integrators.</p>
    040     *
    041     * @see MidpointIntegrator
    042     * @see ClassicalRungeKuttaIntegrator
    043     * @see GillIntegrator
    044     * @see ThreeEighthesIntegrator
    045     * @version $Id: EulerIntegrator.java 1416643 2012-12-03 19:37:14Z tn $
    046     * @since 1.2
    047     */
    048
    049    public class EulerIntegrator extends RungeKuttaIntegrator {
    050
    051      /** Time steps Butcher array. */
    052      private static final double[] STATIC_C = {
    053      };
    054
    055      /** Internal weights Butcher array. */
    056      private static final double[][] STATIC_A = {
    057      };
    058
    059      /** Propagation weights Butcher array. */
    060      private static final double[] STATIC_B = {
    061        1.0
    062      };
    063
    064      /** Simple constructor.
    065       * Build an Euler integrator with the given step.
    066       * @param step integration step
    067       */
    068      public EulerIntegrator(final double step) {
    069        super("Euler", STATIC_C, STATIC_A, STATIC_B, new EulerStepInterpolator(), step);
    070      }
    071
    072    }