732 Methods for Solving Grid Equations- The minimal residual method. Until now the bounds / 1 and / 2 of the
operator A were known before giving special investigations. But it may
happen that these constants are either too rough or, generally speaking,
indeterminate in advance. In 1nastering the difficulties involved, variational
iterative 111ethods such as the n1ethod of conjugate gradient, the 111ini1nal
residual 1nethod and the method of steepest descent can be employed in
the further elaboration on this subject.
We confine ourselves here to the minimal residual method and the
n1ethod of steepest descent relating to two-layer schemes. As usual, the
explicit scheme is considered first:
( 13) k = 0, 1, 2, ... , given y 0 EH,
or( 13')where rk is the residual.
The only difference between the methods we have 1nentioned above
lies in the selection rules for the para1neter Tk+I · In the 1ninimal residual
1nethod the choice
(14) where rk =A Yk - f,
is stipulated by the 1ninimum condition for the norm II rk+l II of the residual.
In this context, several questions are yet to be answered. The equation for
the residual
( 15)in1plies that
T'k+l - T'k
----+Ark= 0, k=0,1,2, ... ,The right-hand side of relation (16) is the polynomial P 2 (rk+i) of degree
2 with respect to the parameter T1;+i· Equating the derivatives P~(rk+i)
to zero reveals Tk+l in c01nplete agreement with (14). Because the second
derivative for that value of the parameter Tk+l is positive, the quantity
11 rk+l II of interest turns out to be 1ninin1aL