TY - JOUR
T1 - On inexact hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems
AU - Bai, Zhong Zhi
AU - Golub, Gene H.
AU - Ng, Michael K.
N1 - Funding Information:
∗ Corresponding author. E-mail addresses: [email protected] (Z.-Z. Bai), [email protected] (G.H. Golub), [email protected] (M.K. Ng). 1 Research supported by The National Basic Research Program (No. 2005CB321702), The China NNSF Outstanding Young Scientist Foundation (No. 10525102) and The National Natural Science Foundation (No. 10471146), PR China. 2 Research supported in part by Hong Kong Research Grant Council 7035/04P, 7035/05P and Hong Kong Baptist University FRGs.
PY - 2008/1/15
Y1 - 2008/1/15
N2 - We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately.
AB - We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by considering both global convergence speed and overall computation workload, and computational efficiencies of both inexact iterations are analyzed and compared deliberately.
KW - Conjugate gradient (CG) method
KW - Conjugate gradient for normal equations (CGNE) method
KW - Hermitian matrix
KW - Inexact iterations
KW - Lanczos method
KW - Skew-Hermitian matrix
UR - http://www.scopus.com/inward/record.url?scp=36049023485&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2007.02.018
DO - 10.1016/j.laa.2007.02.018
M3 - Journal article
AN - SCOPUS:36049023485
SN - 0024-3795
VL - 428
SP - 413
EP - 440
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 2-3
ER -