Superoptimal Preconditioners for Functions of Matrices

@article{Bai2015SuperoptimalPF,
  title={Superoptimal Preconditioners for Functions of Matrices},
  author={Zhengjian Bai and Xiao-Qing Jin and Teng-Teng Yao},
  journal={Numerical Mathematics-theory Methods and Applications},
  year={2015},
  volume={8},
  pages={515-529}
}
For any given matrix A ∈ C, a preconditioner tU (A) called the superoptimal preconditioner was proposed in 1992 by Tyrtyshnikov. It has been shown that tU (A) is an efficient preconditioner for solving various structured systems, for instance, Toeplitz-like systems. In this paper, we construct the superoptimal preconditioners for different functions of matrices. Let f be a function of matrices from C to C. For any A ∈ C, one may construct two superoptimal preconditioners for f(A): tU (f(A)) and… 

Tables from this paper

Circulant preconditioners for functions of Hermitian Toeplitz matrices
  • S. Hon
  • Computer Science
    J. Comput. Appl. Math.
  • 2019
A Convergence Analysis of the MINRES Method for Some Hermitian Indefinite Systems
Some convergence bounds of the minimal residual (MINRES) method are studied when the method is applied for solving Hermitian indefinite linear systems. The matrices of these linear systems are

References

SHOWING 1-10 OF 26 REFERENCES
Optimal preconditioners for functions of matrices
Optimal and Superoptimal Circulant Preconditioners
TLDR
It is proved that both inherit nonsingularity and positive-definiteness from A, and fast algorithms for finding superoptimal preconditioners are suggested.
An Optimal Circulant Preconditioner for Toeplitz Systems
TLDR
The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditionser in terms of reducing the condition number of $C^{ - 1} A$ and comparably in Terms of clustering the spectrum around unity.
Stability properties of superoptimal preconditioner from numerical range
TLDR
This paper gives some sufficient and necessary conditions for the stability of superoptimal preconditioner E-U(A(n) proposed by Tyrtyshnikov (SIAM J. Matrix Anal, Appl. 1992; 13:459-473).
The circulant operator in the banach algebra of matrices
A Class of Filtering Superoptimal Preconditioners for Highly Ill-Conditioned Linear Systems
TLDR
Here, in order to improve the regularizing behaviour, the definition of superoptimal preconditioner is generalized and a particular family of preconditionsers for Toeplitz highly ill-conditioned linear systems is developed.
The spectra of super-optimal circulant preconditioned Toeplitz systems
TLDR
The solutions of Hermitian positive-definite Toeplitz systems by the preconditioned conjugate gradient method are studied and it is proved that they are clustered around one.
Conjugate Gradient Methods for Toeplitz Systems
TLDR
Some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems are surveyed, finding that the complexity of solving a large class of $n-by-n$ ToePlitz systems is reduced to $O(n \log n)$ operations.
Functions of matrices - theory and computation
A thorough and elegant treatment of the theory of matrix functions and numerical methods for computing them, including an overview of applications, new and unpublished research results, and improved
...
...