734 Methods for Solving Grid EquationsThe forthcoming substitutions xk = B^112 yk> C = B-^112 AB-^112 and cp =
B-^1!^2 fallow us to modify these intoleading to eqnation ( 18) with the conecting term w1.; and para1nder Tk+l
such that(20)Instead of ( 17) we eventually get one more useful estimate(21)- The method of steepest descent. The explicit method of steepest descent
is given by the formulas
Yk+1=Yk-Tk+i(Ayk-f), k=0,1,2, ... , given y 0 EH,
where(22) k=0,1,2, ... ,
arising frmn the minimum condition for the norm of the enol' zk = Yk - 1l
in the space HA, meaning
II z llA = J(Az, z).
The error zk = Yk - u satisfies the equation
By interchanging the variables vk = A^112 zk we are led to
(23)