numerics Module

Single precision version of a function that returns the convergence criterium.

Arguments

Return Value real(kind=sp)

The single precision convergence criterium.

Double precision version of a function that returns the convergence criterium.

Arguments

Return Value real(kind=dp)

The double precision convergence criterium.

Quad precision version of a function that returns the convergence criterium.

Arguments

Return Value real(kind=qp)

The quad precision convergence criterium.

Function to calculate Lambert's W-function.

Arguments

Return Value real(kind=wp)

The return value Lambert(z).

Function to invert the tortoise radius as a function of Schwarzschild radius.

Arguments

Return Value real(kind=wp)

The return value is $r_{\mathrm{schw}}(z)-2M$.

Function to invert the tortoise radius as a function of Schwarzschild radius.

Arguments

Return Value real(kind=wp)

The return value is $r_{\mathrm{schw}}(r_*) - 2M$.

Function to calculate the tortoise radius, $r_*$, as function of the Schwarzschild radius, $r_{\mathrm{schw}}$.

Arguments

Return Value real(kind=wp)

The return value is $r_* = r_{\mathrm{schw}}+2 M\log\left ( \frac{r_{\mathrm{schw}}}{2 M} - 1\right )$.

Function that calculates the higher order terms, $c_n(\ell)$ in the self-force expansion over $\ell$.

Arguments

Return Value real(kind=wp)

The return value is $$c_n(\ell)=\begin{cases} \frac{1}{(2\ell-1)(2\ell+3)}, & n=2 \\ \frac{1}{(2\ell-3)(2\ell-1)(2\ell+3)(2\ell+5)}, & n=3 \\ \frac{1}{(2\ell-5)(2\ell-3)(2\ell-1)(2\ell+3)(2\ell+5)(2\ell+7)}, & n=4 \end{cases}$$ .

Function that calculates the sum of higher order terms, $c_n(\ell)$, provided by ldep from $\ell_{\mathrm{min}}$ to $\infty$.

Arguments

Return Value real(kind=wp)

The return value is $$\sum_{\ell=\ell_{\mathrm{min}}}^{\infty} c_n(\ell) = \begin{cases} \frac{\ell_{\mathrm{min}}}{4\ell_{\mathrm{min}}^2-1}, & n=2 \\ \frac{\ell_{\mathrm{min}}}{48\ell_{\mathrm{min}}^4- 120\ell_{\mathrm{min}}^2+27}, & n=3 \\ \frac{\ell_{\mathrm{min}}}{5 (64\ell_{\mathrm{min}}^6- 560\ell_{\mathrm{min}}^4+1036\ell_{\mathrm{min}}^2-225)}, & n=4 \end{cases}$$ .

Function that fits higher order terms to data containing amplitude as function of l-values and corrects the sum over the evolved l-modes with the contribution from the not evolved l-modes.

Arguments

Return Value real(kind=wp), dimension(3)

A 1d-array of size 3 that on return contains the sum of evolved modes (first entry), the corrected sum (second entry) and the error estimate (third entry).

A simple factorial function. Use only for small values of $n$ as no consideration of efficiency has been made.

Arguments

Return Value integer(kind=ip)

The return value, $n!$.

Routine to calculate the smooth transition function, $fT$, and it's first $fT'$ and second $fT''$ derivative with respect to the computational coordinate, $\rho$.

Arguments

Routine to calculate a smooth "Gaussian" type time window function, $f_t$ and it's first $f_t'$ and second $f_t''$ time derivative.

Arguments