Symbolic Algorithm for Solving SLAEs with Multi-Diagonal Coefficient Matrices
Milena Veneva
TL;DR
This work tackles solving SLAEs with multi-diagonal coefficient matrices by introducing a generalized symbolic solver that extends LU-based forward reduction and backward substitution to band structures with $p=q=M$. The method adapts the Thomas approach to a $2M+1$-diagonal band, computing sequences $\mu_i$, $\alpha_i^k$, and $z_i$ and employing a symbolically guarded division to handle zero minors. A correctness theorem shows the solver remains valid under nonsingularity, with a determinant relation $\det(A)=\prod_{i=0}^{N-1} \mu_i|_{symb=0}$ and cancellations when symbolic placeholders arise. The paper derives a closed-form complexity, proving an $O(N)$ time cost for fixed $M$ and $M \ll N$, making the method a robust direct solver for banded SLAEs and PDE-related discretizations.
Abstract
This paper presents a generalised symbolic algorithm for solving systems of linear algebraic equations with multi-diagonal coefficient matrices. The algorithm is given in a pseudocode. A theorem which gives the condition for correctness of the algorithm is formulated and proven. Formula for the complexity of the multi-diagonal numerical algorithm is obtained.