Relaxed parallel two-stage multisplitting methods^∗

Abstract

Two classes of relaxed parallel two-stage multisplitting methods based on extrapolated and AOR methods are studied for the solution of nonsingular linear systems, which are called outer relaxed or inner relaxed parallel two-stage multisplitting methods. Convergence of these methods is studied for H-matrix. Furthermore, computational results about these methods on a shared memory multiprocessor are presented. The results show that the methods we proposed are better than the corresponding existed parallel (two-stage) multisplitting methods.

Relaxed parallel two-stage multisplitting methods II: Asynchronous version

“…, a, then Algorithm 2.2 reduces to its synchronous version: Algorithm 1.2, i.e., Algorithm 2.2 studied in Ref. [13]. When outer splittings are A ¼ A À O, Algorithm 2.2 reduces to Algorithm 2 studied by Fuster et al [11], i.e., Non-stationary multisplitting AOR method.…”

“…The multisplitting iterative method was introduce by O'Leary and White [22] and was further studied by other authors; see, e.g., Bai [2], Frommer and Mayer [7,8] Gu and Wang [14,15], Neumann and Plemmons [20], Wang [26] and White [27,28]. More recently, Fuster et al [11] and Mas et al [18] studied nonstationary relaxed multisplitting methods, Szyld and Jones [24] Gu et al [12,13] and Bru et al [6] studied two-stage multisplitting methods. A set {M l : F l , G l ; N l ; E l } a l¼1 is called a two-stage multisplitting of matrix A if A ¼ M l À N l , l ¼ 1, 2, .…”

“…Mas et al [18] and Fuster et al [11] established two class of extensions of [7] and [26] to relaxed nonstationary parallel multisplitting method and proved the convergence when A is an H-matrix. Gu et al [13] proposed two class of relaxed parallel two-stage multisplitting methods.…”

“…[13], we first proposed a class of outer relaxed nonstationary two-stage multisplitting (ORNSTSM) method based on extrapolate. ALGORITHM 1.1 (Outer Relaxed Non-Stationary Two-Stage Multisplitting) Given the initial vector x 0 For k ¼ 1, 2, .…”

Global relaxed non-stationary multisplitting multi-parameter methods

“…Xu [4] studies convergence of the non-stationary parallel multisplitting SOR method. Gu et al [5] and Gu and Liu [6] studied relaxed non-stationary two-stage parallel multisplitting methods and corresponding asynchronous schemes. other hand, with the development of parallel computation in recent years, the use of parallel algorithms for the solution of large and sparse nonsingular linear systems has become widely adopted.…”