The alternating group explicit (A.G.E.) matrix iterative method (Q1106616)

The object of the paper is to introduce a new iterative method in order to use it in large scale computations for solving matrix equations \(Ay=b\), which arise from finite difference approximations to elliptic partial differential equations. The author calls it ``alternating group explicit method'', briefly ``AGE-method''. Its efficiency is examined within the bounds of an appropriate simple model problem. (In fact, a two-point boundary value problem.) The iteration scheme with respect to this model problem reads: \[ (1)\quad (G_ 1+rI)y^{(n+)}=b-(G_ 2- rI)y^{(n)} \] \[ (G_ 2+rI)y^{(n+1)}=b-(G_ 1-rI)y^{(n+)}. \] In fact, (1) is the same as the scheme in the Peaceman-Rachford implicit alternating-direction method, and is similarly based on the splitting of the matrix A into the sum \(A=G_ 1+G_ 2\). Thus, \(G_ 1+rI\) and \(G_ 2+rI\) are nonsingular matrices with r a positive parameter. In the AGE- method \(G_ 1\) and \(G_ 2\) have the additional property that it is practical to solve the systems \((G_ 1+rI)x=c\), \((G_ 2+rI)y=d\), for any vectors c and d and for any \(r>0\), in explicit form since they consist of only (2\(\times 2)\) subsystems. The theory of the AGE-method is only a slight modification of that of the Peaceman-Rachford method. It turns out that the convergence rate is of the same order as in the SOR-method, i.e. of O(h) with h the mesh spacing. A linear two-point boundary value problem, the exact solution of which is known, is solved numerically using the AGE-method. The total number of arithmetic operations required is then compared with that required in the SOR-method. In the light of this example the superiority of the AGE-method is obvious by lower values of \(h^{-1}\), but the difference between the methods decreases when \(h^{-1}\) increases.