Preconditioners for Sparse Iterative Methods

Synonyms

Linear equations solvers

Definition

Iterative methods for solving sparse systems of linear equations are potentially less memory and computation intensive than direct methods, but often experience slow convergence or fail to converge at all. The robustness and the speed of Krylov subspace iterative methods is improved, often dramatically, by preconditioning. Preconditioning is a technique for transforming the original system of equations into one with an improved distribution (clustering) of eigenvalues so that the transformed system can be solved in fewer iterations. A key step in preconditioning a linear system Ax = b is to find a nonsingular preconditioner matrix M such that the inverse of M is as close to the inverse of A as possible and solving a system of the form Mz = r is significantly less expensive than solving Ax = b. The system is then solved by solving $({M}^{-1}A)x = {M}^{-1}b$. This particular example shows what is known as left preconditioning. There are two...