Models of parallel chaotic iteration methods

Sequential and parallel synchronous alternating iterative methods

“…Nevertheless, Bru, Elsner, and Neumann [3], with the aim of avoiding loss of time and efficiency in processor use, establish the so-called Model A.…”

“…Bru, Elsner, and Neumann [3] establish the mathematical formulation of Model A, starting with an arbitrary vector x (0) , as…”

“…If we define the error vector at the kth iteration as e (k) = x (k) − ξ, then by (2) we have that (3) e (k+1) = T (k) e (k) = T (k) T (k−1) · · · T (1) T (0) e (0) , k = 0, 1, 2, . .…”

“…See, for example, Varga [24], Bermann and Plemmons [1], Miller and Neumann [18], Csordas and Varga [9], O'Leary and White [22], Woźnicki [26], Nabben [20], and Climent and Perea [6,7]. For convergence results for the iterative scheme (2), based on the concept of multisplitting, see, for example, Bru, Elsner and Neuman [3,4] and Migallón, Penadés and Szyld [19].…”

“…In Section 3 we introduce the main results for nonstationary sequential alternating iterative methods, as well as some results for the corresponding stationary sequential alternating iterative method. On the other hand, in Section 4 we present the parallel nonstationary Model A introduced by Bru, Elsner and Neuman [3] extending some convergence results. Next, in Section 5, combining sequential alternating iterative methods introduced in Section 3 and the parallel nonstationary Model A, we introduce two new parallel iterative methods, and we establish different convergence and comparison results for it.…”