bandmatrix

A band matrix is a sparse matrix in which the nonzero entries are concentrated near the main diagonal. For an n-by-n matrix A, there are lower and upper bandwidths p and q, meaning that A can be nonzero only in diagonals at most p below the main diagonal and q above it. Equivalently, a_ij ≠ 0 only if i − j ∈ [−q, p].

Common special cases include tridiagonal (p = q = 1) and pentadiagonal (p = q = 2). Banded matrices require storage of only the (p + q + 1) diagonals, using a compact band storage format with n columns and (p + q + 1) rows, one row per diagonal. This storage preserves the band structure and reduces memory usage compared with dense representations.

Computation with band matrices exploits their sparsity. Band LU factorization uses O(n(p + q)^2) operations, and forward/backward substitution uses O(n(p + q)) operations. For tridiagonal matrices, these costs reduce to O(n). Pivoting for numerical stability may increase the effective bandwidth; many banded solvers employ limited or structured pivot strategies to balance stability and sparsity.

Applications and properties. Band matrices commonly arise from finite difference and finite element discretizations of differential equations, as well as in signal processing and dynamical systems. If the matrix is symmetric positive definite, specialized algorithms such as banded Cholesky factorization can be very efficient. The band structure is beneficial for memory and compute efficiency, but factorization can introduce fill-in that enlarges the active bandwidth, so methods are designed to mitigate this in practice.