Field-based version of Euler method for solving ODE.

src/main/java/org/apache/commons/math3/ode/nonstiff/EulerFieldIntegrator.java

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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.Field;
+import org.apache.commons.math3.RealFieldElement;
+
+/**
+ * This class implements a simple Euler integrator for Ordinary
+ * Differential Equations.
+ *
+ * <p>The Euler algorithm is the simplest one that can be used to
+ * integrate ordinary differential equations. It is a simple inversion
+ * of the forward difference expression :
+ * <code>f'=(f(t+h)-f(t))/h</code> which leads to
+ * <code>f(t+h)=f(t)+hf'</code>. The interpolation scheme used for
+ * dense output is the linear scheme already used for integration.</p>
+ *
+ * <p>This algorithm looks cheap because it needs only one function
+ * evaluation per step. However, as it uses linear estimates, it needs
+ * very small steps to achieve high accuracy, and small steps lead to
+ * numerical errors and instabilities.</p>
+ *
+ * <p>This algorithm is almost never used and has been included in
+ * this package only as a comparison reference for more useful
+ * integrators.</p>
+ *
+ * @see MidpointFieldIntegrator
+ * @see ClassicalRungeKuttaFieldIntegrator
+ * @see GillFieldIntegrator
+ * @see ThreeEighthesFieldIntegrator
+ * @see LutherFieldIntegrator
+ * @param <T> the type of the field elements
+ * @since 3.6
+ */
+
+public class EulerFieldIntegrator<T extends RealFieldElement<T>> extends RungeKuttaFieldIntegrator<T> {
+
+  /** Time steps Butcher array. */
+  private static final double[] STATIC_C = {
+  };
+
+  /** Internal weights Butcher array. */
+  private static final double[][] STATIC_A = {
+  };
+
+  /** Propagation weights Butcher array. */
+  private static final double[] STATIC_B = {
+    1.0
+  };
+
+  /** Simple constructor.
+   * Build an Euler integrator with the given step.
+   * @param field field to which the time and state vector elements belong
+   * @param step integration step
+   */
+  public EulerFieldIntegrator(final Field<T> field, final T step) {
+    super(field, "Euler", STATIC_C, STATIC_A, STATIC_B, new EulerFieldStepInterpolator<T>(), step);
+  }
+
+}

src/main/java/org/apache/commons/math3/ode/nonstiff/EulerFieldStepInterpolator.java

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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.RealFieldElement;
+import org.apache.commons.math3.ode.FieldEquationsMapper;
+import org.apache.commons.math3.ode.FieldODEStateAndDerivative;
+import org.apache.commons.math3.util.MathArrays;
+
+/**
+ * This class implements a linear interpolator for step.
+ *
+ * <p>This interpolator computes dense output inside the last
+ * step computed. The interpolation equation is consistent with the
+ * integration scheme :
+ * <ul>
+ *   <li>Using reference point at step start:<br>
+ *     y(t<sub>n</sub> + &theta; h) = y (t<sub>n</sub>) + &theta; h y'
+ *   </li>
+ *   <li>Using reference point at step end:<br>
+ *     y(t<sub>n</sub> + &theta; h) = y (t<sub>n</sub> + h) - (1-&theta;) h y'
+ *   </li>
+ * </ul>
+ * </p>
+ *
+ * where &theta; belongs to [0 ; 1] and where y' is the evaluation of
+ * the derivatives already computed during the step.</p>
+ *
+ * @see EulerFieldIntegrator
+ * @param <T> the type of the field elements
+ * @since 3.6
+ */
+
+class EulerFieldStepInterpolator<T extends RealFieldElement<T>>
+  extends RungeKuttaFieldStepInterpolator<T> {
+
+  /** 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.
+   */
+  EulerFieldStepInterpolator() {
+  }
+
+  /** 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
+   */
+  EulerFieldStepInterpolator(final EulerFieldStepInterpolator<T> interpolator) {
+    super(interpolator);
+  }
+
+  /** {@inheritDoc} */
+  @Override
+  protected EulerFieldStepInterpolator<T> doCopy() {
+    return new EulerFieldStepInterpolator<T>(this);
+  }
+
+
+  /** {@inheritDoc} */
+  @Override
+  protected FieldODEStateAndDerivative<T> computeInterpolatedStateAndDerivatives(final FieldEquationsMapper<T> mapper,
+                                                                                 final T time, final T theta,
+                                                                                 final T oneMinusThetaH) {
+      final T[] interpolatedState = MathArrays.buildArray(theta.getField(), previousState.length);
+      if ((previousState != null) && (theta.getReal() <= 0.5)) {
+          for (int i = 0; i < previousState.length; ++i) {
+              interpolatedState[i] = previousState[i].add(theta.multiply(h).multiply(yDotK[0][i]));
+          }
+      } else {
+          for (int i = 0; i < previousState.length; ++i) {
+              interpolatedState[i] = currentState[i].subtract(oneMinusThetaH.multiply(yDotK[0][i]));
+          }
+      }
+
+      return new FieldODEStateAndDerivative<T>(time,
+                                               interpolatedState,
+                                               yDotK[0]);
+  }
+
+}