How can one diagonalise this N-by-N pentadiagonal matrix (where $r$ is some real constant)?
$$ \tiny \begin{pmatrix} r^2 +r & -2r -1 & 1 & & & & & & \\ -2r^2 -r & r^2 +4r +1 & -2r -2 & 1 & & & & & \\ r^2 & -2r^2 -2r & r^2 +4r +1 & -2r -2 & 1 & & & & \\ & r^2 & -2r^2 -2r & r^2 +4r +1 & -2r -2 & 1 & & & \\ & & r^2 & -2r^2 -2r & r^2 +4r +1 & -2r -2 & 1 & & \\ \\ & & & \ddots & \ddots & \ddots & \ddots & \ddots & \\ \\ & & & & r^2 & -2r^2 -2r & r^2 +4r +1 & -2r -2 & 1 \\ & & & & & r^2 & -2r^2 -2r & r^2 +4r +1 & r-2 \\ & & & & & & r^2 & -r^2 -2r & r+1 \\ \end{pmatrix} \tag{1} $$
Re-written for clarity
$$ \begin{pmatrix} h & e & 1 & & & & & \\ g & a & b & 1 & & & & \\ r^2 & c & a & b & 1 & & & \\ & r^2 & c & a & b & 1 & & \\ & & \ddots & \ddots & \ddots & \ddots & \ddots & \\ & & & r^2 & c & a & b & 1 \\ & & & & r^2 & c & a & k \\ & & & & & r^2 & l & m \\ \end{pmatrix} \tag{2} $$
where $a=r^2+4r+1$, $b=-2r-1$, $c=-2r^2-2r$, $h=r^2+r$, $e=-2r-1$, $g=-2r^2-1$, $k=r-2$, $l=-r^2-2r$ and $m=r+1$.
I have been ttempting to solve it by hand but I cannot yet find a workable method - it does seem however that some recursive method must exist as it does for Toeplitz matrices.
This problem arose by studying the system $$ \begin{matrix} & \frac{df_1}{dt} = -rf_1+f_2 \\ & \vdots \\ & \frac{df_n}{dt} = rf_{n-1}-(r+1)f_n+f_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = rf_{N-1}-f_N \end{matrix} $$ and setting $\frac{d^{2}f_n}{dt^2} = 0$ for all $n$.
Help will be greatly appreciated.