dense_coefficients Subroutine

private pure subroutine dense_coefficients(me, b, form)

Returns the dense output coefficients for the method. This is the default for methods that do not have a dense output (b is not allocated).

A method with a dense output overrides this to return the matrix b (number_of_stages x degree), defining the interpolant over the step $[t, t+h]$, with $0 \le \theta \le 1$. The coefficients can be in either of the two forms commonly used in the literature (so they can be used as published), where $f_i$ are the function evaluations stored in me%funcs by the step function (and dense_extra_stages, if the method requires them):

• dense_form_polynomial: $x(t + \theta h) = x(t) + h \sum_{i} b_i(\theta) f_i$, where $b_i(\theta) = \sum_{j=1}^{degree} b_{ij} \theta^j$. So, row i of b contains the coefficients of $\theta^1 ... \theta^{degree}$ in $b_i$.

• dense_form_dopri (the form used in DOPRI5 and DOP853): $x(t + \theta h) = x(t) + \theta(r_1 + \theta_1(r_2 + \theta(r_3 + \theta_1(r_4 + ...))))$, where $\theta_1 = 1 - \theta$ and $r_j = h \sum_{i} b_{ij} f_i$. So, column j of b contains the coefficients of $r_j$.

Type Bound

rk_class

Arguments

    pure subroutine dense_coefficients(me,b,form)
        class(rk_class),intent(in) :: me
        real(wp),dimension(:,:),allocatable,intent(out) :: b !! dense output coefficients
        integer,intent(out) :: form !! the form of `b` (`dense_form_polynomial` or `dense_form_dopri`)
        form = 0
    end subroutine dense_coefficients