I am on my research work. I need references or if perhaps, you can help me find the modeled inverse of matrices $A(4,n), \,A(6,n),\, ...\,,\,A(2r+2,n)$ where $2r+2$ is the bandwidth. I have successfully derived the model for the inverses when $i\leq j$ which is similar to W. D. Hoskins and P. J. Ponzo's formula. Can anyone help me model when $i\geq j-1$ using LU decomposition?
Examples are
$A(4,4)=\left( \begin{array}{cccc} -3 & 1 & 0 & 0 \\ 3 & -3 & 1 & 0 \\ -1 & 3 & -3 & 1 \\ 0 & -1 & 3 & -3 \\ \end{array} \right)$, $A(6,7)=\left( \begin{array}{ccccccc} 10 & -5 & 1 & 0 & 0 & 0 & 0 \\ -10 & 10 & -5 & 1 & 0 & 0 & 0 \\ 5 & -10 & 10 & -5 & 1 & 0 & 0 \\ -1 & 5 & -10 & 10 & -5 & 1 & 0 \\ 0 & -1 & 5 & -10 & 10 & -5 & 1 \\ 0 & 0 & -1 & 5 & -10 & 10 & -5 \\ 0 & 0 & 0 & -1 & 5 & -10 & 10 \\ \end{array} \right)$,...
I have downloaded a lot of materials on banded matrix but none of them has given me a formula like "Some Properties of a Class of Band Matrices" by W. D. Hoskins and P. J. Ponzo.
Pls, I need an urgent help on this.