MA65: Unsymmetric banded system of linear equations

To solve an unsymmetric banded system of linear equations. Given a unsymmetric band matrix \(\mathbf{A} = {\{a _{ij}\}} _{n \times n }\) and an \(n\)-vector \(\mathbf{b}\), this subroutine solves the system \(\mathbf{Ax} = \mathbf{b}\).

The matrix is factorized using Gaussian elimination with row interchanges. If the lower semibandwidth is \(kl\) and the upper semibandwidth is \(ku\), that is, if \(a _{ij} = 0\) for \(i > j+kl\) or \(j > i+ku\), fill-in is limited to \(kl\) additional diagonals of the upper triangle and the computation is performed within an array of size \(n(2 kl + ku +1)\). At each pivotal step, operations are avoided on any row with a zero in the pivot column and on the columns beyond the last to have an entry in the pivot row.