Matrix analysis
A proposal for toeplitz matrix calculations
Studies in Applied Mathematics
Topics in matrix analysis
Optimal and superoptimal circulant preconditioners
SIAM Journal on Matrix Analysis and Applications
Circulant and skewcirculant matrices for solving Toeplitz matrix problems
SIAM Journal on Matrix Analysis and Applications
A family of block preconditioners for block systems
SIAM Journal on Scientific and Statistical Computing
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Matrix computations (3rd ed.)
Applied and Computational Control, Signals and Circuits: Recent Developments
Applied and Computational Control, Signals and Circuits: Recent Developments
A Circulant Preconditioner for the Systems of LMF-Based ODE Codes
SIAM Journal on Scientific Computing
A Stability Property of T. Chan's Preconditioner
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation)
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
An Introduction to Iterative Toeplitz Solvers (Fundamentals of Algorithms)
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Block preconditioner with circulant blocks (BPCB) has been used for solving linear systems with block Toeplitz structure since 1992 [R. Chan, X. Jin, A family of block preconditioners for block systems, SIAM J. Sci. Statist. Comput. (13) (1992) 1218-1235]. In this new paper, we use BPCBs to general linear systems (with no block structure usually). The BPCBs are constructed by partitioning a general matrix into a block matrix with blocks of the same size and then applying T. Chan's optimal circulant preconditioner [T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. (9) (1988) 766-771] to each block. These BPCBs can be viewed as a generalization of T. Chan's preconditioner. It is well-known that the optimal circulant preconditioner works well for solving some structured systems such as Toeplitz systems by using the preconditioned conjugate gradient (PCG) method, but it is usually not efficient for solving general linear systems. Unlike T. Chan's preconditioner, BPCBs used here are efficient for solving some general linear systems by the PCG method. Several basic properties of BPCBs are studied. The relations of the block partition with the cost per iteration and the convergence rate of the PCG method are discussed. Numerical tests are given to compare the cost of the PCG method with different BPCBs.