Direct multiplicative methods for sparse matrices. Unbalanced linear systems.

Direct linear time solvers for sparse matrices

In the last couple of years it has been realized that Gaussian elimination for sparse matrices arising from certain elliptic PDEs can be done in O(n log(1/ )) flops, where n is the number of unknowns and is a user-specified tolerance [Chandrasekaran and Gu]. The resulting solver will also be backward stable with an error of O( ). The techniques used to achieve this speedup has some commonality ...

A survey of direct methods for sparse linear systems

Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them. This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in...

‎Finite iterative methods for solving systems of linear matrix equations over reflexive and anti-reflexive matrices

A matrix $Pintextmd{C}^{ntimes n}$ is called a generalized reflection matrix if $P^{H}=P$ and $P^{2}=I$‎. ‎An $ntimes n$‎ ‎complex matrix $A$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $P$ if $A=PAP$ ($A=-PAP$)‎. ‎In this paper‎, ‎we introduce two iterative methods for solving the pair of matrix equations $AXB=C$ and $DXE=F$ over reflexiv...

Direct Solvers for Sparse Matrices

There is a vast variety of algorithms associated with each step. The review papers by Duff [16] (see also [15, Chapter 6]) and Heath et al. [27] can serve as excellent reference of various algorithms. Usually steps 1 and 2 involve only the graphs of the matrices, and hence only integer operations. Steps 3 and 4 involve floating-point operations. Step 3 is usually the most time-consuming part, w...

Iterative methods for sparse linear systems