A repository of matrices with special inversion properties. We are interested in matrices A, such that solving Ax = B has computational complexity under O(n^3). These methods look at both exact and approximate direct solvers.
[NOTE: This page is a work in progress and will never be a finished resource. If you identify any method that should be included please push to the repository. Thank you.]
Inverse is a diagnol matrix with each element being the inverse of the corresponding original diagnol element.
The inverse is also is a block diagnol matrix with each block being the inverse of the corresponding block of the original matrix
These matrices have a fixed rank m << n. In such cases we endeavour to utilise the fact that less singular have to be inverted.
This method uses subsampling to of the matrix under consideration in order to approximate the eigen spectrum of the full matrix. A great resource for those interested in Nystrom's method is Nicholas Arcolano's Thesis.
Toeplitz matrices have a special structure of their elements whereby each consecutive row is equal to it's previous row but shifted one element to the right. This isn't exactly a rigourous definition but for now I will leave it to Wikipedia to explain in greater depth. Check out A Superfast Algorithm for Toeplitz Systems of Linear Equations for a full description of the solver.
The vandermonde matrix is an interesting matrix where each column is a vector raised to a higher elementwise power. Again for a more formal description check out Wikipedia. There is a closed form solution for the matrix inverse and the determinant is also easily found.
Hiearchical matrices utilise blockwise low rank matrix approximations for low memory storage and fast matrix opperations. Some methods are outlined below.
Hierarchical Off Diagnol Low Rank (HODLR) Matrices utilise a recursive off diagnol structure. There is a good paper which clearly explains this factoring and inversion by Ambikasaran et al.