Classes of stable three-layer schemes 431we rep resent scheme ( 1) in the form( 11)
y(O) = Yo , Y( r) = Y1 ,where A= A(tn) =An, B = B(tn) = Bn and R = R(tn) = Rn· Taking
the inner product of (11) and 2rya t = r(y 1 +Yr), we find that( 12) 2 T ( B y~ ) y~ ) + T^2 ( ( R - ~ A) ( Yt - yt) ) Yt + Yt)
- ~ (A ( y + :i)), ii - :i)) = 2 T ( !p, Ya t ).
Let A and R be self-adjoint operators. ThenR-0.5A= (R-0.5A)*.
By Lemma l of Section 2 we thus have(13) ((R-~A)(Yi-Yr),Yi+Yi) = ((R-~A)(Yi,Yi)
- ((R-~A) (Yr, Yr),
(14) (A (y + :iJ), y - :iJ) = (Ay, ii) - (A :i), :iJ).
At the final stage we add and subtract (Ay, y) on the right-hand side of
( 14). The outcome of this is(15) (A (y + :iJ), ii - :iJ) = [(Ay, ii)+ (Ay, y)] - [(Ay, y) + (Ay, :iJ)].
Lemma 1 Let A= A* be a self-adjoint operator. Then
(16) (Av, v) + {Az, z) = ~ (A(v + z), v + z) + ~ (A(v - z), v - z)
for any vectors v and z of the space H.
Proof Since A= A*, we have (Av, z) = (v, Az) = (Az, v) and
((A(v + z), v + z)) + ((A(v - z), v - z))
= [(Av, v) + 2(Av, z) + (Az, z)] +[(Av, v) - 2(Av, z) + (Az, z)]
= 2 [(Av, v) + (Az, z)].