Package org.hipparchus.ode.nonstiff

Class AdamsNordsieckTransformer

java.lang.Object
org.hipparchus.ode.nonstiff.AdamsNordsieckTransformer
public class AdamsNordsieckTransformer extends Object
Transformer to Nordsieck vectors for Adams integrators.

This class is used by Adams-Bashforth and Adams-Moulton integrators to convert between classical representation with several previous first derivatives and Nordsieck representation with higher order scaled derivatives.

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

With the previous definition, the classical representation of multistep methods uses first derivatives only, i.e. it handles yn, s1(n) and qn where qn is defined as: \[ q_n = [ s_1(n-1) s_1(n-2) \ldots s_1(n-(k-1)) ]^T \] (we omit the k index in the notation for clarity).

Another possible representation uses the Nordsieck vector with higher degrees scaled derivatives all taken at the same step, i.e it handles 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 at step end. 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: \[ P=\begin{bmatrix} -2 & 3 & -4 & 5 & \ldots \\ -4 & 12 & -32 & 80 & \ldots \\ -6 & 27 & -108 & 405 & \ldots \\ -8 & 48 & -256 & 1280 & \ldots \\ & & \ldots\\ \end{bmatrix} \]

Changing -i into +i in the formula above can be used to compute a similar transform between classical representation and Nordsieck vector at step start. The resulting matrix is simply the absolute value of matrix P.

For Adams-Bashforth method, the 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 ]

For Adams-Moulton method, 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
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.

We observe that both methods use similar update formulas. In both cases a P-1u vector and a P-1 A P matrix are used that do not depend on the state, they only depend on k. This class handles these transformations.

  • Method Details

    • getInstance

      public static AdamsNordsieckTransformer getInstance(int nSteps)
      Get the Nordsieck transformer for a given number of steps.
      Parameters:
      nSteps - number of steps of the multistep method (excluding the one being computed)
      Returns:
      Nordsieck transformer for the specified number of steps

    • initializeHighOrderDerivatives

      public Array2DRowRealMatrix initializeHighOrderDerivatives(double h, double[] t, double[][] y, double[][] yDot)
      Initialize the high order scaled derivatives at step start.
      Parameters:
      h - step size to use for scaling
      t - first steps times
      y - first steps states
      yDot - first steps derivatives
      Returns:
      Nordieck vector at start of first step (h2/2 y''n, h3/6 y'''n ... hk/k! y(k)n)

    • updateHighOrderDerivativesPhase1

      public Array2DRowRealMatrix updateHighOrderDerivativesPhase1(Array2DRowRealMatrix highOrder)
      Update the high order scaled derivatives for Adams integrators (phase 1).

      The complete update of high order derivatives has a form similar to: \[ r_{n+1} = (s_1(n) - s_1(n+1)) P^{-1} u + P^{-1} A P r_n \] this method computes the P-1 A P rn part.

      Parameters:
      highOrder - high order scaled derivatives (h2/2 y'', ... hk/k! y(k))
      Returns:
      updated high order derivatives
      See Also:

    • updateHighOrderDerivativesPhase2

      public void updateHighOrderDerivativesPhase2(double[] start, double[] end, Array2DRowRealMatrix highOrder)
      Update the high order scaled derivatives Adams integrators (phase 2).

      The complete update of high order derivatives has a form similar to: \[ r_{n+1} = (s_1(n) - s_1(n+1)) P^{-1} u + P^{-1} A P r_n \] this method computes the (s1(n) - s1(n+1)) P-1 u part.

      Phase 1 of the update must already have been performed.

      Parameters:
      start - first order scaled derivatives at step start
      end - first order scaled derivatives at step end
      highOrder - high order scaled derivatives, will be modified (h2/2 y'', ... hk/k! y(k))
      See Also: