Examples of EulerIntegrator

ca.nengo.dynamics.impl.EulerIntegrator
Euler's method of numerical integration: x(t+h) ~ x(t) + h*x'(t) TODO: test TODO: should there be some means for aborting early (aside from exceptions, e.g. if output converges to constant)? @author Bryan Tripp
org.apache.commons.math.ode.EulerIntegrator
This class implements a simple Euler integrator for Ordinary Differential Equations.
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 : f'=(f(t+h)-f(t))/h which leads to f(t+h)=f(t)+hf'. The interpolation scheme used for dense output is the linear scheme already used for integration.

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.

This algorithm is almost never used and has been included in this package only as a comparison reference for more useful integrators.
@see MidpointIntegrator @see ClassicalRungeKuttaIntegrator @see GillIntegrator @see ThreeEighthesIntegrator @version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $ @since 1.2
org.apache.commons.math.ode.nonstiff.EulerIntegrator
This class implements a simple Euler integrator for Ordinary Differential Equations.
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 : f'=(f(t+h)-f(t))/h which leads to f(t+h)=f(t)+hf'. The interpolation scheme used for dense output is the linear scheme already used for integration.

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.

This algorithm is almost never used and has been included in this package only as a comparison reference for more useful integrators.
@see MidpointIntegrator @see ClassicalRungeKuttaIntegrator @see GillIntegrator @see ThreeEighthesIntegrator @version $Revision: 810196 $ $Date: 2009-09-01 21:47:46 +0200 (mar. 01 sept. 2009) $ @since 1.2

Examples of org.apache.commons.math.ode.nonstiff.EulerIntegrator


        TestProblemAbstract pb  = problems[k].copy();
        double step = (pb.getFinalTime() - pb.getInitialTime())
          * Math.pow(2.0, -i);


        FirstOrderIntegrator integ = new EulerIntegrator(step);
        TestProblemHandler handler = new TestProblemHandler(pb, integ);
        integ.addStepHandler(handler);
        EventHandler[] functions = pb.getEventsHandlers();
        for (int l = 0; l < functions.length; ++l) {
          integ.addEventHandler(functions[l],
                                     Double.POSITIVE_INFINITY, 1.0e-6 * step, 1000);
        }
        double stopTime = integ.integrate(pb, pb.getInitialTime(), pb.getInitialState(),
                                          pb.getFinalTime(), new double[pb.getDimension()]);
        if (functions.length == 0) {
            assertEquals(pb.getFinalTime(), stopTime, 1.0e-10);
        }

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Examples of org.apache.commons.math.ode.nonstiff.EulerIntegrator

    throws DerivativeException, IntegratorException {


    TestProblem1 pb  = new TestProblem1();
    double step = (pb.getFinalTime() - pb.getInitialTime()) * 0.001;


    FirstOrderIntegrator integ = new EulerIntegrator(step);
    TestProblemHandler handler = new TestProblemHandler(pb, integ);
    integ.addStepHandler(handler);
    integ.integrate(pb,
                    pb.getInitialTime(), pb.getInitialState(),
                    pb.getFinalTime(), new double[pb.getDimension()]);


   assertTrue(handler.getLastError() < 2.0e-4);
   assertTrue(handler.getMaximalValueError() < 1.0e-3);
   assertEquals(0, handler.getMaximalTimeError(), 1.0e-12);
   assertEquals("Euler", integ.getName());


  }

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Examples of org.apache.commons.math.ode.nonstiff.EulerIntegrator

    throws DerivativeException, IntegratorException {


    TestProblem1 pb  = new TestProblem1();
    double step = (pb.getFinalTime() - pb.getInitialTime()) * 0.2;


    FirstOrderIntegrator integ = new EulerIntegrator(step);
    TestProblemHandler handler = new TestProblemHandler(pb, integ);
    integ.addStepHandler(handler);
    integ.integrate(pb,
                    pb.getInitialTime(), pb.getInitialState(),
                    pb.getFinalTime(), new double[pb.getDimension()]);


    assertTrue(handler.getLastError() > 0.01);
    assertTrue(handler.getMaximalValueError() > 0.2);

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Examples of org.apache.commons.math.ode.nonstiff.EulerIntegrator

      throws DerivativeException, IntegratorException {


      TestProblem5 pb = new TestProblem5();
      double step = Math.abs(pb.getFinalTime() - pb.getInitialTime()) * 0.001;


      FirstOrderIntegrator integ = new EulerIntegrator(step);
      TestProblemHandler handler = new TestProblemHandler(pb, integ);
      integ.addStepHandler(handler);
      integ.integrate(pb, pb.getInitialTime(), pb.getInitialState(),
                      pb.getFinalTime(), new double[pb.getDimension()]);


      assertTrue(handler.getLastError() < 0.45);
      assertTrue(handler.getMaximalValueError() < 0.45);
      assertEquals(0, handler.getMaximalTimeError(), 1.0e-12);
      assertEquals("Euler", integ.getName());
  }

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Examples of org.apache.commons.math.ode.nonstiff.EulerIntegrator

  }


  public void testStepSize()
    throws DerivativeException, IntegratorException {
      final double step = 1.23456;
      FirstOrderIntegrator integ = new EulerIntegrator(step);
      integ.addStepHandler(new StepHandler() {
        public void handleStep(StepInterpolator interpolator, boolean isLast) {
            if (! isLast) {
                assertEquals(step,
                             interpolator.getCurrentTime() - interpolator.getPreviousTime(),
                             1.0e-12);
            }
        }
        public boolean requiresDenseOutput() {
            return false;
        }
        public void reset() {
        }          
      });
      integ.integrate(new FirstOrderDifferentialEquations() {
                          private static final long serialVersionUID = 0L;
                          public void computeDerivatives(double t, double[] y, double[] dot) {
                              dot[0] = 1.0;
                          }
                          public int getDimension() {

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