quasiseparability

Quasiseparability is a concept in numerical analysis and linear algebra that describes matrices whose structure is close to, but not exactly, a separable matrix. A separable matrix can be expressed as the outer product of two vectors, or more generally, as a sum of outer products of vectors. This structure allows for efficient storage and computation.

A quasiseparable matrix is characterized by having a special block structure where certain off-diagonal blocks are of low rank. Specifically, a matrix is quasiseparable if it can be represented as a sum of a diagonal matrix and a matrix that can be decomposed into products of vectors in a structured way. This structure often arises in the numerical solution of differential equations, especially in higher dimensions, or in problems involving certain types of integral equations.

The benefit of identifying a matrix as quasiseparable lies in the potential for significant computational savings. Algorithms designed for quasiseparable matrices can exploit their structure to reduce the complexity of operations like matrix-vector multiplication, matrix inversion, and eigenvalue decomposition from the typical O(n^2) or O(n^3) for dense matrices to O(n) or O(n log n), where n is the dimension of the matrix. This makes it feasible to solve much larger problems.

The theory of quasiseparable matrices is closely related to other low-rank matrix approximation techniques. While a truly separable matrix is a very specific case, quasiseparability allows for a broader class of matrices that still exhibit sufficient structure for efficient algorithms. This concept is an active area of research, with ongoing development of new algorithms and applications in various scientific and engineering fields.