Package org.hipparchus.ode.nonstiff

Class AdamsMoultonIntegrator

  • All Implemented Interfaces:
    ODEIntegrator
    public class AdamsMoultonIntegrator
extends AdamsIntegrator
    This class implements implicit Adams-Moulton integrators for Ordinary Differential Equations.

    Adams-Moulton methods (in fact due to Adams alone) are implicit multistep ODE solvers. This implementation is a variation of the classical one: it uses adaptive stepsize to implement error control, whereas classical implementations are fixed step size. The value of state vector at step n+1 is a simple combination of the value at step n and of the derivatives at steps n+1, n, n-1 ... Since y'n+1 is needed to compute yn+1, another method must be used to compute a first estimate of yn+1, then compute y'n+1, then compute a final estimate of yn+1 using the following formulas. Depending on the number k of previous steps one wants to use for computing the next value, different formulas are available for the final estimate:

    • k = 1: yn+1 = yn + h y'n+1
    • k = 2: yn+1 = yn + h (y'n+1+y'n)/2
    • k = 3: yn+1 = yn + h (5y'n+1+8y'n-y'n-1)/12
    • k = 4: yn+1 = yn + h (9y'n+1+19y'n-5y'n-1+y'n-2)/24
    • ...

    A k-steps Adams-Moulton method is of order k+1.

    There must be sufficient time for the starter integrator to take several steps between the the last reset event, and the end of integration, otherwise an exception may be thrown during integration. The user can adjust the end date of integration, or the step size of the starter integrator to ensure a sufficient number of steps can be completed before the end of integration.

    Implementation details

    We define scaled derivatives si(n) at step n as: \[ \left\{\begin{align} s_1(n) &= h y'_n \text{ for first derivative}\\ s_2(n) &= \frac{h^2}{2} y_n'' \text{ for second derivative}\\ s_3(n) &= \frac{h^3}{6} y_n''' \text{ for third derivative}\\ &\cdots\\ s_k(n) &= \frac{h^k}{k!} y_n^{(k)} \text{ for } k^\mathrm{th} \text{ derivative} \end{align}\right. \]

    The definitions above use the classical representation with several previous first derivatives. Lets define \[ q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T \] (we omit the k index in the notation for clarity). With these definitions, Adams-Moulton methods can be written:

    • k = 1: yn+1 = yn + s1(n+1)
    • k = 2: yn+1 = yn + 1/2 s1(n+1) + [ 1/2 ] qn+1
    • k = 3: yn+1 = yn + 5/12 s1(n+1) + [ 8/12 -1/12 ] qn+1
    • k = 4: yn+1 = yn + 9/24 s1(n+1) + [ 19/24 -5/24 1/24 ] qn+1
    • ...

    Instead of using the classical representation with first derivatives only (yn, s1(n+1) and qn+1), our implementation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step (yn, s1(n) and rn) where rn is defined as: \[ r_n = [ s_2(n), s_3(n) \ldots s_k(n) ]^T \] (here again we omit the k index in the notation for clarity)

    Taylor series formulas show that for any index offset i, s1(n-i) can be computed from s1(n), s2(n) ... sk(n), the formula being exact for degree k polynomials. \[ s_1(n-i) = s_1(n) + \sum_{j\gt 0} (j+1) (-i)^j s_{j+1}(n) \] The previous formula can be used with several values for i to compute the transform between classical representation and Nordsieck vector. The transform between rn and qn resulting from the Taylor series formulas above is: \[ q_n = s_1(n) u + P r_n \] where u is the [ 1 1 ... 1 ]T vector and P is the (k-1)×(k-1) matrix built with the \((j+1) (-i)^j\) terms with i being the row number starting from 1 and j being the column number starting from 1: \[ P=\begin{bmatrix} -2 & 3 & -4 & 5 & \ldots \\ -4 & 12 & -32 & 80 & \ldots \\ -6 & 27 & -108 & 405 & \ldots \\ -8 & 48 & -256 & 1280 & \ldots \\ & & \ldots\\ \end{bmatrix} \]

    Using the Nordsieck vector has several advantages:

    • it greatly simplifies step interpolation as the interpolator mainly applies Taylor series formulas,
    • it simplifies step changes that occur when discrete events that truncate the step are triggered,
    • it allows to extend the methods in order to support adaptive stepsize.

    The predicted Nordsieck vector at step n+1 is computed from the Nordsieck vector at step n as follows:

    • Yn+1 = yn + s1(n) + uT rn
    • S1(n+1) = h f(tn+1, Yn+1)
    • Rn+1 = (s1(n) - S1(n+1)) P-1 u + P-1 A P rn
    where A is a rows shifting matrix (the lower left part is an identity matrix): 
            [ 0 0   ...  0 0 | 0 ]
        [ ---------------+---]
        [ 1 0   ...  0 0 | 0 ]
    A = [ 0 1   ...  0 0 | 0 ]
        [       ...      | 0 ]
        [ 0 0   ...  1 0 | 0 ]
        [ 0 0   ...  0 1 | 0 ]
    From this predicted vector, the corrected vector is computed as follows:
    • yn+1 = yn + S1(n+1) + [ -1 +1 -1 +1 ... ±1 ] rn+1
    • s1(n+1) = h f(tn+1, yn+1)
    • rn+1 = Rn+1 + (s1(n+1) - S1(n+1)) P-1 u

    where the upper case Yn+1, S1(n+1) and Rn+1 represent the predicted states whereas the lower case yn+1, sn+1 and rn+1 represent the corrected states.

    The P-1u vector and the P-1 A P matrix do not depend on the state, they only depend on k and therefore are precomputed once for all.

    • Constructor Detail

      • AdamsMoultonIntegrator

        public AdamsMoultonIntegrator​(int nSteps,
                              double minStep,
                              double maxStep,
                              double scalAbsoluteTolerance,
                              double scalRelativeTolerance)
                       throws MathIllegalArgumentException
        Build an Adams-Moulton integrator with the given order and error control parameters.
        Parameters:
        nSteps - number of steps of the method excluding the one being computed
        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
        Throws:
        MathIllegalArgumentException - if order is 1 or less

      • AdamsMoultonIntegrator

        public AdamsMoultonIntegrator​(int nSteps,
                              double minStep,
                              double maxStep,
                              double[] vecAbsoluteTolerance,
                              double[] vecRelativeTolerance)
                       throws IllegalArgumentException
        Build an Adams-Moulton integrator with the given order and error control parameters.
        Parameters:
        nSteps - number of steps of the method excluding the one being computed
        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
        vecAbsoluteTolerance - allowed absolute error
        vecRelativeTolerance - allowed relative error
        Throws:
        IllegalArgumentException - if order is 1 or less

    • Method Detail

      • errorEstimation

        protected double errorEstimation​(double[] previousState,
                                 double predictedTime,
                                 double[] predictedState,
                                 double[] predictedScaled,
                                 RealMatrix predictedNordsieck)
        Estimate error.
        Specified by:
        errorEstimation in class AdamsIntegrator
        Parameters:
        previousState - state vector at step start
        predictedTime - time at step end
        predictedState - predicted state vector at step end
        predictedScaled - predicted value of the scaled derivatives at step end
        predictedNordsieck - predicted value of the Nordsieck vector at step end
        Returns:
        estimated normalized local discretization error

      • finalizeStep

        protected org.hipparchus.ode.nonstiff.AdamsStateInterpolator finalizeStep​(double stepSize,
                                                                          double[] predictedState,
                                                                          double[] predictedScaled,
                                                                          Array2DRowRealMatrix predictedNordsieck,
                                                                          boolean isForward,
                                                                          ODEStateAndDerivative globalPreviousState,
                                                                          ODEStateAndDerivative globalCurrentState,
                                                                          EquationsMapper equationsMapper)
        Finalize the step.
        Specified by:
        finalizeStep in class AdamsIntegrator
        Parameters:
        stepSize - step size used in the scaled and Nordsieck arrays
        predictedState - predicted state at end of step
        predictedScaled - predicted first scaled derivative
        predictedNordsieck - predicted Nordsieck vector
        isForward - integration direction indicator
        globalPreviousState - start of the global step
        globalCurrentState - end of the global step
        equationsMapper - mapper for ODE equations primary and secondary components
        Returns:
        step interpolator