4 - Matrix splitting preconditioners [T1]: direct approximation of A_n×n

Summary

The term “preconditioning” appears to have been used for the first time in 1948 by Turing [461], … The first use of the term in connection with iterative methods is found in a paper by Evans [200] … in 1968.

Michele Benzi. Journal of Computational Physics, Vol. 182 (2002)

For such problems, the coefficient matrix A is often highly nonsymmetric and non-diagonally dominant and hence many classical preconditioning techniques are not effective. For these problems, the circulant preconditioners are often the only ones that work.

Raymond Chan and Tony Chan. Journal of Numerical Linear Algebra and Applications, Vol. 1 (1992)

In ending this book with the subject of preconditioners, we find ourselves at the philosophical center of the scientific computing of the future … Nothing will be more central to computational science in the next century than the art of transforming a problem that appears intractable into another whose solution can be approximated rapidly. For Krylov subspace matrix iterations, this is preconditioning.

Lloyd Nicholas Trefethen and David Bau III. Numerical Linear Algebra. SIAM Publications (1997)

Starting from this chapter, we shall first describe various preconditioning techniques that are based on manipulation of a given matrix. These are classified into four categories: direct matrix extraction (or operator splitting type), inverse approximation (or inverse operator splitting type), multilevel Schur complements and multi-level operator splitting (multilevel methods).