Matrices with low-rank off-diagonal blocks are a versatile tool to perform matrix compression and to speed up various matrix operations, such as the solution of linear systems. Often, the underlying block partitioning is described by a hierarchical partitioning of the row and column indices, thus giving rise to hierarchical low-rank structures. The goal of this chapter is to provide a brief int...

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎‎direct iteration methods for solving a linear system where the‎‎$i$-th iteration satisfies the first $i$ equations‎, ‎therefore a‎ ‎system of $m$ equations is solved in at most $m$ steps‎. ‎In this‎‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎‎rank linear equations‎, ‎w...

Today’s focus on sustainability within industry presents a modeling challenge that may be dealt with using dynamic programming over an infinite time horizon. However, the curse of dimensionality often results in a large number of states in these models. These large-scale models require numerically stable solution methods. The best method for infinite-horizon dynamic programming depends on both ...

A Class of Splitting Preconditioners for the iterative solution of Linear Systems arising from Interior Point Methods for Linear Programming Problems needs to find a base by a sophisticated process based in a lying rectangular LU factorization, that involves reordering columns many times until certain conditions are satisfied. At the same time, it is necessary to prevent excessive fill-in in L ...

The Kaczmarz method [1], [2], [3] was initially proposed as a row-based technique for reconstructing signals by finding the solutions to overdetermined linear systems. Its usefulness has seen wide application in irregular sampling and tomography [4], [5], [6]. In recent years, several modifications to the Kaczmarz update iterations have improved the recovery capabilities [7], [8], [9], [10], [1...

Sparsified Randomization Monte Carlo (SRMC) algorithms for solving systems of linear algebraic equations introduced in our previous paper [34] are discussed here in a broader context. In particular, I present new randomized solvers for large systems of linear equations, randomized singular value (SVD) decomposition for large matrices and their use for solving inverse problems, and stochastic si...

This research introduces a row compression and nested product decomposition of an n × n hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n)) operations, and is proven backward stable. T...

1 Preliminaries 1.1 Linear Algebra In this section, we review some definitions and concepts related to linear algebra which will be useful in describing the iterative methods later. Definition 1 The row rank of a matrix A is the maximum number of linearly independent rows in A. The column rank of a matrix A is the maximum number of linearly independent columns in A. In other words, row rank of ...

This work is concerned with the numerical solution of large-scale linear matrix equations A1XB T 1 + · · ·+ AKXB K = C. The most straightforward approach computes X ∈ Rm×n from the solution of an mn×mn linear system, typically limiting the feasible values of m,n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low-rank matrix. It combines ...