The alternative-triangular method 683- The alternative triangle 1nethod. Making a substantiated choice of the
operator B will be justified in more detail a little latter. Recall that any
product B of "economical" operators is also an "economical" operator. This
is certainly true for "triangle" operators B 1 and B 2 for which the operator
B = B 1 B 2 would be "economical".
Any such operator R = R* > 0 is representable by
(21) R = R* > 0,
where R 1 and R 2 are "triangle" operators. The associated matrix R = (r';j)
is symmetric, that is, rij = rji· It is obvious that the appropriate matrices
R. 1 and R2 are given by:R1 = ( rij), {
r;j for j < i,
1',, Z)
() for j > i,
1',, ],! = ri~ = 0.5 rii.R2=(r0), r~ = {
0 for j < i ,
!J ,,, i) for j > i,From here and the symmetry condition rij = rji it follows that Ri = R2.
The operator B built into scheme (6) arranges itself as a product of
"triangle" operators:(22)where w > 0 is a nurnerical parameter. One succeeds in showing that B
is a self-adjoint positive operator: B = B* > 0. Due to these properties
scheme (6) with operator (22) will belong to the primary family of schemes
(12).
Indeed, it turns out that thP operators
are adjoint and positive:
since R 1 > 0 and R2 > 0:
(Ry, y) = (R1 y, y) + (R2 y, y) = 2 (R1 y, y) = 2 (R2 y, y) > 0.