Class GraggBulirschStoerIntegrator

  • All Implemented Interfaces:
    FirstOrderIntegrator, ODEIntegrator

    public class GraggBulirschStoerIntegrator
    extends AdaptiveStepsizeIntegrator
    This class implements a Gragg-Bulirsch-Stoer integrator for Ordinary Differential Equations.

    The Gragg-Bulirsch-Stoer algorithm is one of the most efficient ones currently available for smooth problems. It uses Richardson extrapolation to estimate what would be the solution if the step size could be decreased down to zero.

    This method changes both the step size and the order during integration, in order to minimize computation cost. It is particularly well suited when a very high precision is needed. The limit where this method becomes more efficient than high-order embedded Runge-Kutta methods like Dormand-Prince 8(5,3) depends on the problem. Results given in the Hairer, Norsett and Wanner book show for example that this limit occurs for accuracy around 1e-6 when integrating Saltzam-Lorenz equations (the authors note this problem is extremely sensitive to the errors in the first integration steps), and around 1e-11 for a two dimensional celestial mechanics problems with seven bodies (pleiades problem, involving quasi-collisions for which automatic step size control is essential).

    This implementation is basically a reimplementation in Java of the odex fortran code by E. Hairer and G. Wanner. The redistribution policy for this code is available here, for convenience, it is reproduced below.

    Copyright (c) 2004, Ernst Hairer
    Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
    • Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
    • Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
    THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
    Since:
    1.2
    • Field Detail

      • METHOD_NAME

        private static final java.lang.String METHOD_NAME
        Integrator method name.
        See Also:
        Constant Field Values
      • maxOrder

        private int maxOrder
        maximal order.
      • sequence

        private int[] sequence
        step size sequence.
      • costPerStep

        private int[] costPerStep
        overall cost of applying step reduction up to iteration k+1, in number of calls.
      • costPerTimeUnit

        private double[] costPerTimeUnit
        cost per unit step.
      • optimalStep

        private double[] optimalStep
        optimal steps for each order.
      • coeff

        private double[][] coeff
        extrapolation coefficients.
      • performTest

        private boolean performTest
        stability check enabling parameter.
      • maxChecks

        private int maxChecks
        maximal number of checks for each iteration.
      • maxIter

        private int maxIter
        maximal number of iterations for which checks are performed.
      • stabilityReduction

        private double stabilityReduction
        stepsize reduction factor in case of stability check failure.
      • stepControl1

        private double stepControl1
        first stepsize control factor.
      • stepControl2

        private double stepControl2
        second stepsize control factor.
      • stepControl3

        private double stepControl3
        third stepsize control factor.
      • stepControl4

        private double stepControl4
        fourth stepsize control factor.
      • orderControl1

        private double orderControl1
        first order control factor.
      • orderControl2

        private double orderControl2
        second order control factor.
      • useInterpolationError

        private boolean useInterpolationError
        use interpolation error in stepsize control.
      • mudif

        private int mudif
        interpolation order control parameter.
    • Constructor Detail

      • GraggBulirschStoerIntegrator

        public GraggBulirschStoerIntegrator​(double minStep,
                                            double maxStep,
                                            double scalAbsoluteTolerance,
                                            double scalRelativeTolerance)
        Simple constructor. Build a Gragg-Bulirsch-Stoer integrator with the given step bounds. All tuning parameters are set to their default values. The default step handler does nothing.
        Parameters:
        minStep - minimal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
        maxStep - maximal step (sign is irrelevant, regardless of integration direction, forward or backward), the last step can be smaller than this
        scalAbsoluteTolerance - allowed absolute error
        scalRelativeTolerance - allowed relative error
      • GraggBulirschStoerIntegrator

        public GraggBulirschStoerIntegrator​(double minStep,
                                            double maxStep,
                                            double[] vecAbsoluteTolerance,
                                            double[] vecRelativeTolerance)
        Simple constructor. Build a Gragg-Bulirsch-Stoer integrator with the given step bounds. All tuning parameters are set to their default values. The default step handler does nothing.
        Parameters:
        minStep - minimal step (must be positive even for backward integration), the last step can be smaller than this
        maxStep - maximal step (must be positive even for backward integration)
        vecAbsoluteTolerance - allowed absolute error
        vecRelativeTolerance - allowed relative error
    • Method Detail

      • setStabilityCheck

        public void setStabilityCheck​(boolean performStabilityCheck,
                                      int maxNumIter,
                                      int maxNumChecks,
                                      double stepsizeReductionFactor)
        Set the stability check controls.

        The stability check is performed on the first few iterations of the extrapolation scheme. If this test fails, the step is rejected and the stepsize is reduced.

        By default, the test is performed, at most during two iterations at each step, and at most once for each of these iterations. The default stepsize reduction factor is 0.5.

        Parameters:
        performStabilityCheck - if true, stability check will be performed, if false, the check will be skipped
        maxNumIter - maximal number of iterations for which checks are performed (the number of iterations is reset to default if negative or null)
        maxNumChecks - maximal number of checks for each iteration (the number of checks is reset to default if negative or null)
        stepsizeReductionFactor - stepsize reduction factor in case of failure (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
      • setControlFactors

        public void setControlFactors​(double control1,
                                      double control2,
                                      double control3,
                                      double control4)
        Set the step size control factors.

        The new step size hNew is computed from the old one h by:

         hNew = h * stepControl2 / (err/stepControl1)^(1/(2k+1))
         
        where err is the scaled error and k the iteration number of the extrapolation scheme (counting from 0). The default values are 0.65 for stepControl1 and 0.94 for stepControl2.

        The step size is subject to the restriction:

         stepControl3^(1/(2k+1))/stepControl4 <= hNew/h <= 1/stepControl3^(1/(2k+1))
         
        The default values are 0.02 for stepControl3 and 4.0 for stepControl4.

        Parameters:
        control1 - first stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
        control2 - second stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
        control3 - third stepsize control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
        control4 - fourth stepsize control factor (the factor is reset to default if lower than 1.0001 or greater than 999.9)
      • setOrderControl

        public void setOrderControl​(int maximalOrder,
                                    double control1,
                                    double control2)
        Set the order control parameters.

        The Gragg-Bulirsch-Stoer method changes both the step size and the order during integration, in order to minimize computation cost. Each extrapolation step increases the order by 2, so the maximal order that will be used is always even, it is twice the maximal number of columns in the extrapolation table.

         order is decreased if w(k-1) <= w(k)   * orderControl1
         order is increased if w(k)   <= w(k-1) * orderControl2
         

        where w is the table of work per unit step for each order (number of function calls divided by the step length), and k is the current order.

        The default maximal order after construction is 18 (i.e. the maximal number of columns is 9). The default values are 0.8 for orderControl1 and 0.9 for orderControl2.

        Parameters:
        maximalOrder - maximal order in the extrapolation table (the maximal order is reset to default if order <= 6 or odd)
        control1 - first order control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
        control2 - second order control factor (the factor is reset to default if lower than 0.0001 or greater than 0.9999)
      • addEventHandler

        public void addEventHandler​(EventHandler function,
                                    double maxCheckInterval,
                                    double convergence,
                                    int maxIterationCount,
                                    UnivariateSolver solver)
        Add an event handler to the integrator.
        Specified by:
        addEventHandler in interface ODEIntegrator
        Overrides:
        addEventHandler in class AbstractIntegrator
        Parameters:
        function - event handler
        maxCheckInterval - maximal time interval between switching function checks (this interval prevents missing sign changes in case the integration steps becomes very large)
        convergence - convergence threshold in the event time search
        maxIterationCount - upper limit of the iteration count in the event time search
        solver - The root-finding algorithm to use to detect the state events.
        See Also:
        ODEIntegrator.getEventHandlers(), ODEIntegrator.clearEventHandlers()
      • initializeArrays

        private void initializeArrays()
        Initialize the integrator internal arrays.
      • setInterpolationControl

        public void setInterpolationControl​(boolean useInterpolationErrorForControl,
                                            int mudifControlParameter)
        Set the interpolation order control parameter. The interpolation order for dense output is 2k - mudif + 1. The default value for mudif is 4 and the interpolation error is used in stepsize control by default.
        Parameters:
        useInterpolationErrorForControl - if true, interpolation error is used for stepsize control
        mudifControlParameter - interpolation order control parameter (the parameter is reset to default if <= 0 or >= 7)
      • rescale

        private void rescale​(double[] y1,
                             double[] y2,
                             double[] scale)
        Update scaling array.
        Parameters:
        y1 - first state vector to use for scaling
        y2 - second state vector to use for scaling
        scale - scaling array to update (can be shorter than state)
      • tryStep

        private boolean tryStep​(double t0,
                                double[] y0,
                                double step,
                                int k,
                                double[] scale,
                                double[][] f,
                                double[] yMiddle,
                                double[] yEnd,
                                double[] yTmp)
                         throws MaxCountExceededException,
                                DimensionMismatchException
        Perform integration over one step using substeps of a modified midpoint method.
        Parameters:
        t0 - initial time
        y0 - initial value of the state vector at t0
        step - global step
        k - iteration number (from 0 to sequence.length - 1)
        scale - scaling array (can be shorter than state)
        f - placeholder where to put the state vector derivatives at each substep (element 0 already contains initial derivative)
        yMiddle - placeholder where to put the state vector at the middle of the step
        yEnd - placeholder where to put the state vector at the end
        yTmp - placeholder for one state vector
        Returns:
        true if computation was done properly, false if stability check failed before end of computation
        Throws:
        MaxCountExceededException - if the number of functions evaluations is exceeded
        DimensionMismatchException - if arrays dimensions do not match equations settings
      • extrapolate

        private void extrapolate​(int offset,
                                 int k,
                                 double[][] diag,
                                 double[] last)
        Extrapolate a vector.
        Parameters:
        offset - offset to use in the coefficients table
        k - index of the last updated point
        diag - working diagonal of the Aitken-Neville's triangle, without the last element
        last - last element