1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other iterative 1nethods 735

Having squared their norms, we find that

(25)

under the above condition 1nin II vk+l 112. The arguments of the preceding

{^7 k+l}

section serve to motivate the estimate

(26)

Thus, it remains to return to zk = A-^1!^2 vk by observing furthermore that

All this enables us to transform formula (25) into fo1·1nula (22) for later use.

Moreover, from inequality (26) it follows that

( 27)

since

No doubt, several conclusions can be drawn fron1 such reasoning. First, the
method being employed above converges in the space HA with the same rate
as the simple iteration 1nethod although it occurs in one of the subordinate
norms. Second, the minimal residual method converges in the space HA2,
that is, in a more stronger norm.
By analogy with Section 2 the i1nplicit n1ethocl of steepest descent is
described by

(28) k = 0, 1, 2, ... , given Yo EH,