Other iterative methods^741
(for more detail see Section 2, formula (14)) governing the minimum con-
dition for 11 rk+l 112. No property of the self-adjointness of the operator A
applies here and below. In what follows minor changes will appear in con-
nection with the necessity of proving its convergence. In particular, f will
stand in place of Tk+l on the right-hand side of identity (16) and will result
in the relationsthereby assuring 11rk+l11 < p II 1'" II and jnstifying the estimate
11 A Yn - f 11 < Pn 11 A Yo - f 11 ·
This is certainly true for the minimal residual method (13)-(14) under
conditions (34). Here Yn is a solution of problem (13) and pis specified by
formula (39). The fastest 1nove method is useless in that case because the
operator A is non-self-adjoint in such a setting.
Observe that under conditions (34) the estimateis valid for the in1plicit 1ninimal correction n1ethod with B
incorporated.B* > 0
- Gibrid ( c01nbined) methods. In 1nastering the difficulties involved in
solving difference elliptic equations, son1e consensus of opinion is to bring
together direct and iterative 1nethods in son1e or other aspects as well as
to combine iterative methods of various types (two-step methods). All the
tricks and turns will be clarified for the iteration scheme
( 42) Yo EH given,
yielding
( 43)where wk is the correction satisfying the equation Bkwk
Ayk - f is the residual.