Product-type skew-Hermitian triangular splitting iteration methods for strongly non-Hermitian positive definite linear systems

Abstract

a b s t r a c tBy further generalizing the modified skew-Hermitian triangular splitting iteration methods studied in [L. Wang, Z.-Z. Bai, Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math. 44 (2004) 363-386], in this paper, we present a new iteration scheme, called the product-type skew-Hermitian triangular splitting iteration method, for solving the strongly non-Hermitian systems of linear equations with positive definite coefficient matrices. We discuss the convergence property and the optimal parameters of this method. Moreover, when it is applied to precondition the Krylov subspace methods, the preconditioning property of the product-type skew-Hermitian triangular splitting iteration is analyzed in detail. Numerical results show that the product-type skew-Hermitian triangular splitting iteration method can produce high-quality preconditioners for the Krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew-Hermitian parts.

Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix

“…The Euclidean norm of the ECT V (P, α) bounds the asymptotic convergence rates of many matrix splitting iteration methods such as (i) the alternating direction implicit (ADI) method [58,1,36,61], the Hermitian and skew-Hermitian splitting (HSS) method [13,10], the normal and skew-Hermitian splitting (NSS) method [14], the positive-definite and skew-Hermitian splitting (PSS) method [12], the shift-splitting preconditioning method [18] and the triangular skew-Hermitian splitting method [50,60,17,51] for solving large sparse and non-Hermitian positive-definite systems of linear equations, (ii) the preconditioned HSS (PHSS) method [15,11], the accelerated HSS (AHSS) method [9,4], the dimensional split preconditioning method [20] and the block alternating splitting implicit (BASI) method [7] for solving large sparse saddle-point linear systems, (iii) the modulus method [22,47,57,52], the modified modulus method [28], the extrapolated modulus method [38,40,37] and the modulus-based splitting methods [6] for solving large sparse linear complementarity problems, and (iv) the alternately linearized implicit (ALI) method [16], the structure-preserving doubling algorithm [35,54,43,23] and the inexact Newton methods based on doubling iteration scheme [32] for computing the minimal nonnegative solutions of large sparse nonsymmetric algebraic Riccati equations; see also [31,15,21,…”

Efficient iteration method for saddle point problems

“…Note that a nonsymmetric matrix is called dissipative if its symmetric part is positive definite [13]. When GSTS is reduced to a product type skew Hermitian triangular splitting iteration method (PSTS) studied in [14][15][16].…”

On convergence of splitting iteration methods for non-Hermitian positive-definite linear systems

“…For any non-Hermitian positive-definite linear system, the HSS and PSS methods converge unconditionally to the unique solution of the system of linear equations (1), but the HSS (or PSS) is an alternative iteration method, a Hermitian and a skew-Hermitian system of linear equations need to be solved at each iteration step, the computation is increased even though some methods for solving the skew-Hermitian systems of linear equations were presented [6,9,13]. Some splitting iteration methods for general non-Hermitian linear system [14] are only theoretical, the convergence conditions cannot be verified easily.…”