Given a symmetric penta-diagonal matrix A
$A = \begin{bmatrix} c_{1} & b_{1} & a_{1} & 0 & \cdots & 0 \\ b_{1} & c_{2} & b_{2} & a_{2} & \cdots & 0 \\ a_{1} & b_{2} & c_{3} & b_{3} & a_{3} & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & \cdots & a_{L-3} & b_{L-2} & c_{L-1} & b_{L-1} \\ 0 & \cdots & 0 & a_{L-2} & b_{L-1} & c_{L} \end{bmatrix}$,
where $c_{i}$,$b_{i}$ and $a_{i}$ $\in \mathbb{R}$ or $ \mathbb{C}$. The matrix is also considered to be invertable as $G A = I$ then $G = A^{-1}$ which reads
$G = \begin{bmatrix} G_{11} & G_{12} & G_{13} & \dots & G_{1L} \\ G_{21} & G_{22} & G_{23} & \dots & G_{2L} \\ \vdots \\ G_{L1} & G_{L2} & G_{L3} & \dots & G_{LL} \end{bmatrix}$.
My questions is: is there an efficient algorithm to compute just the diagonal elements $G_{i,i}$?
I know how to do it efficiently for a tri-diagonal matrix in $O(N)$ steps, but I'm unsure if such speedup is possible for the penta-diagonal case.