Comp. Appl. Math. DOI 10.1007/s40314-015-0231-6 A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems Yu-Jiang Wu 1,2,3 · Xu Li 4,5 · Jin-Yun Yuan 6 Received: 30 June 2014 / Revised: 9 April 2015 / Accepted: 10 April 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015 Abstract By utilizing the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration technique, we establish a non-alternating PHSS (NPHSS) iteration method for solv- ing large sparse non-Hermitian positive definite linear systems. The convergence analysis demonstrates that the iterative series produced by the NPHSS method converge to the unique solution of the linear system when the parameters satisfy some moderate conditions. We also give a possible optimal upper bound for the iterative spectral radius. Moreover, to reduce the computational cost, we establish an inexact variant of the NPHSS (INPHSS) iteration method whose convergence property is studied. Both theoretical and numerical results validate that the NPHSS method outperforms the PHSS method when the Hermitian part of the coefficient matrix is dominant. Communicated by Ya-xiang Yuan. B Xu Li mathlixu@163.com; lixu@lut.cn Yu-Jiang Wu myjaw@lzu.edu.cn Jin-Yun Yuan jin@ufpr.br 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China 2 Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou 730000, China 3 Institute of Industrial Mathematics, IMPA, CP: 19.081, Curitiba, PR 81531-980, Brazil 4 College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou 730050, China 5 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China 6 Department of Mathematics, Centro Politécnico, Federal University of Paraná, CP: 19.081, Curitiba, PR 81531-980, Brazil 123