Local inversion of matrices with sparse inverses

Sparse Approximate Inverses and Target Matrices

Inversion error, condition number, and approximate inverses of uncertain matrices

The classical condition number is a very rough measure of the effect of perturbations on the inverse of a square matrix. First, it assumes that the perturbation is infinitesimally small. Second, it does not take into account the perturbation structure (e.g., Vandermonde). Similarly, the classical notion of the inverse of a matrix neglects the possibility of large, structured perturbations. We d...

Inversion of block matrices with block banded inverses: application to Kalman-Bucy filtering

We investigate the properties of block matrices with block banded inverses to derive efficient matrix inversion algorithms for such matrices. In particular, we derive the following: (1) a recursive algorithm to invert a full matrix whose inverse is structured as a block tridiagonal matrix; (2) a recursive algorithm to compute the inverse of a structured block tridiagonal matrix. These algorithm...

Matrices with banded inverses: Inversion algorithms and factorization of Gauss-Markov processes

The paper considers the inversion of full matrices whose inverses are -banded. We derive a nested inversion algorithm for such matrices. Applied to a tridiagonal matrix, the algorithm provides its explicit inverse as an element-wise product (Hadamard product) of three matrices. When related to Gauss–Markov random processes (GMrp), this result provides a closed-form factored expression for the c...

Groups of banded matrices with banded inverses

AproductADF1 : : : FN of invertible block-diagonalmatrices will be bandedwith a banded inverse. We establish this factorization with the numberN controlled by the bandwidthsw and not by the matrix size n:When A is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors Fi have w D 1 and generate that corresponding group. In the case of infinite matrices, conjectures rema...