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The main aim of this paper is to develop the quaternion generalized minimal residual method (QGMRES) for solving quaternion linear systems. Quaternion linear systems arise fromthree-dimensional or color imaging filtering problems. The proposed quaternion Arnoldi procedure can preserve quaternion Hessenberg form during the iterations. The main advantage is that the storage of the proposed iterative method can be reduced by comparing with the Hessenberg form constructed by the classical GMRES iterations for the real representation of quaternion linear systems. The convergence of the proposed QGMRES is also established. Numerical examples are presented to demonstrate the effectiveness of the proposed QGMRES compared with the traditional GMRES in terms of storage and computing time [ABSTRACT FROM AUTHOR]

We study the eigenvalue bounds of block two-by-two nonsingular and symmetric indefinite matrices whose (1, 1) block is symmetric positive definite and Schur complement with respect to its (2, 2) block is symmetric indefinite. A constraint preconditioner for this matrix is constructed by simply replacing the (1, 1) block by a symmetric and positive definite approximation, and the spectral properties of the preconditioned matrix are discussed. Numerical results show that, for a suitably chosen (1, 1) block-matrix, this constraint preconditioner outperforms the blockdiagonal and the block-tridiagonal ones in iteration step and computing time when they are used to accelerate the GMRES method for solving these block two-by-two symmetric positive indefinite linear systems. The new results extend the existing ones about block two-by-two matrices of symmetric negative semidefinite (2, 2) blocks to those of general symmetric (2, 2) blocks. [ABSTRACT FROM AUTHOR]

We consider the iterative solution of weighted Toeplitz least squares problems. Our approach is based on an augmented system formulation. We focus our attention on two types of preconditioners: a variant of constraint preconditioning, and the Hermitian/skew-Hermitian splitting (HSS) preconditioner. Bounds on the eigenvalues of the preconditioned matrices are given in terms of problem and algorithmic parameters, and numerical experiments are used to illustrate the performance of the preconditioners. [ABSTRACT FROM AUTHOR]

Many image processing applications require computing approximate solutions of very large, ill-conditioned linear systems. Physical assumptions of the imaging system usually dictate that the matrices in these linear systems have exploitable structure. The specific structure depends on (usually simplifying) assumptions of the physical model and other considerations such as boundary conditions. When reflexive (Neumann) boundary conditions are used, the coefficient matrix is a combination of Toeplitz and Hankel matrices. Kronecker products also occur, but this structure is not obvious from measured data. In this paper we discuss a scheme for computing a (possibly approximate) Kronecker product decomposition of structured matrices in image processing, which extends previous work by Kamm and Nagy [SIAM J. Matrix Anal. Appl., 22 (2000), pp. 155–172] to a wider class of image restoration problems. [ABSTRACT FROM AUTHOR]

Examines efficient iterative methods for the large sparse non-Hermitian positive definite systems of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. Demonstration that the Hermitian/skew-Hermitian splitting iteration method converges unconditionally to the unique solution of the system of linear equations.