LU Decomposition and Extended Thomas Algorithm

Let us consider the linear system Ax = LUx = b where the N x N matrix A can be factorised into a lower triangular matrix L and an upper triangular matrix U as 

The solution of the above triangular set of equations can be performed easily by forward substitution  

Finally, the vector x can be determined by solving Ux = y through backward substitution  

The equation systems commonly found in the simulation of electrochemical problems have a diagonal structure with a bandwidth 2n+l such that a j = 0 for j i + n and for i j n. By particularising the LU decomposition for diagonal systems [extended Thomas algorithm) the above expressions turn into 

---