Examples of DormandPrince54Integrator

org.apache.commons.math.ode.DormandPrince54Integrator
This class implements the 5(4) Dormand-Prince integrator for Ordinary Differential Equations.
This integrator is an embedded Runge-Kutta integrator of order 5(4) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 7 functions evaluations per step. However, since this is an fsal, the last evaluation of one step is the same as the first evaluation of the next step and hence can be avoided. So the cost is really 6 functions evaluations per step.

This method has been published (whithout the continuous output that was added by Shampine in 1986) in the following article :
```
 A family of embedded Runge-Kutta formulae J. R. Dormand and P. J. Prince Journal of Computational and Applied Mathematics volume 6, no 1, 1980, pp. 19-26 
```
@version $Revision: 620312 $ $Date: 2008-02-10 12:28:59 -0700 (Sun, 10 Feb 2008) $ @since 1.2
org.apache.commons.math.ode.nonstiff.DormandPrince54Integrator
This class implements the 5(4) Dormand-Prince integrator for Ordinary Differential Equations.
This integrator is an embedded Runge-Kutta integrator of order 5(4) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 7 functions evaluations per step. However, since this is an fsal, the last evaluation of one step is the same as the first evaluation of the next step and hence can be avoided. So the cost is really 6 functions evaluations per step.

This method has been published (whithout the continuous output that was added by Shampine in 1986) in the following article :
```
 A family of embedded Runge-Kutta formulae J. R. Dormand and P. J. Prince Journal of Computational and Applied Mathematics volume 6, no 1, 1980, pp. 19-26 
```
@version $Revision: 990655 $ $Date: 2010-08-29 23:49:40 +0200 (dim. 29 août 2010) $ @since 1.2
org.apache.commons.math3.ode.nonstiff.DormandPrince54Integrator
This class implements the 5(4) Dormand-Prince integrator for Ordinary Differential Equations.
This integrator is an embedded Runge-Kutta integrator of order 5(4) used in local extrapolation mode (i.e. the solution is computed using the high order formula) with stepsize control (and automatic step initialization) and continuous output. This method uses 7 functions evaluations per step. However, since this is an fsal, the last evaluation of one step is the same as the first evaluation of the next step and hence can be avoided. So the cost is really 6 functions evaluations per step.

This method has been published (whithout the continuous output that was added by Shampine in 1986) in the following article :
```
 A family of embedded Runge-Kutta formulae J. R. Dormand and P. J. Prince Journal of Computational and Applied Mathematics volume 6, no 1, 1980, pp. 19-26 
```
@since 1.2

Examples of org.apache.commons.math3.ode.nonstiff.DormandPrince54Integrator

    public void testInternalDifferentiation()
        throws NumberIsTooSmallException, DimensionMismatchException,
               MaxCountExceededException, NoBracketingException,
               UnknownParameterException, MismatchedEquations {
        AbstractIntegrator integ =
            new DormandPrince54Integrator(1.0e-8, 100.0, new double[] { 1.0e-4, 1.0e-4 }, new double[] { 1.0e-4, 1.0e-4 });
        double hP = 1.0e-12;
        double hY = 1.0e-12;
        SummaryStatistics residualsP0 = new SummaryStatistics();
        SummaryStatistics residualsP1 = new SummaryStatistics();
        for (double b = 2.88; b < 3.08; b += 0.001) {
            ParamBrusselator brusselator = new ParamBrusselator(b);
            brusselator.setParameter(ParamBrusselator.B, b);
            double[] z = { 1.3, b };
            double[][] dZdZ0 = new double[2][2];
            double[]   dZdP  = new double[2];


            JacobianMatrices jacob = new JacobianMatrices(brusselator, new double[] { hY, hY }, ParamBrusselator.B);
            jacob.setParameterizedODE(brusselator);
            jacob.setParameterStep(ParamBrusselator.B, hP);
            jacob.setInitialParameterJacobian(ParamBrusselator.B, new double[] { 0.0, 1.0 });


            ExpandableStatefulODE efode = new ExpandableStatefulODE(brusselator);
            efode.setTime(0);
            efode.setPrimaryState(z);
            jacob.registerVariationalEquations(efode);


            integ.setMaxEvaluations(5000);
            integ.integrate(efode, 20.0);
            jacob.getCurrentMainSetJacobian(dZdZ0);
            jacob.getCurrentParameterJacobian(ParamBrusselator.B, dZdP);
//            Assert.assertEquals(5000, integ.getMaxEvaluations());
//            Assert.assertTrue(integ.getEvaluations() > 1500);
//            Assert.assertTrue(integ.getEvaluations() < 2100);

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Examples of org.apache.commons.math3.ode.nonstiff.DormandPrince54Integrator

    public void testAnalyticalDifferentiation()
        throws MaxCountExceededException, DimensionMismatchException,
               NumberIsTooSmallException, NoBracketingException,
               UnknownParameterException, MismatchedEquations {
        AbstractIntegrator integ =
            new DormandPrince54Integrator(1.0e-8, 100.0, new double[] { 1.0e-4, 1.0e-4 }, new double[] { 1.0e-4, 1.0e-4 });
        SummaryStatistics residualsP0 = new SummaryStatistics();
        SummaryStatistics residualsP1 = new SummaryStatistics();
        for (double b = 2.88; b < 3.08; b += 0.001) {
            Brusselator brusselator = new Brusselator(b);
            double[] z = { 1.3, b };
            double[][] dZdZ0 = new double[2][2];
            double[]   dZdP  = new double[2];


            JacobianMatrices jacob = new JacobianMatrices(brusselator, Brusselator.B);
            jacob.addParameterJacobianProvider(brusselator);
            jacob.setInitialParameterJacobian(Brusselator.B, new double[] { 0.0, 1.0 });


            ExpandableStatefulODE efode = new ExpandableStatefulODE(brusselator);
            efode.setTime(0);
            efode.setPrimaryState(z);
            jacob.registerVariationalEquations(efode);


            integ.setMaxEvaluations(5000);
            integ.integrate(efode, 20.0);
            jacob.getCurrentMainSetJacobian(dZdZ0);
            jacob.getCurrentParameterJacobian(Brusselator.B, dZdP);
//            Assert.assertEquals(5000, integ.getMaxEvaluations());
//            Assert.assertTrue(integ.getEvaluations() > 350);
//            Assert.assertTrue(integ.getEvaluations() < 510);

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