he reference is
[Xu]
.
Let
be a decomposition of
:
Let
be a symmetric positive definite operator.
We define operators
and
via the
relationships


(Definition of Q i)



(Definition of P i)



(Definition of A i)

For
we
have
Thus


(Projection permuation)

Let
be a symmetric positive definite operator that (on motivational level) almost
inverts
:
In context of the section
(
Preconditioning
) we introduce the
operator


(Parallel subspace preconditioner)

We introduce the numbers
as follows.
We introduce the notation
and
note
Definition
(Strengthened
CauchySchwartz inequality) We define the matrix
where


(Definition of Epsilon)

Proposition
(Magnitude of matrix Epsilon)
1.
.
2. If
then
.
Proposition
(Estimate for K1 one)
1.
.
2.
.
3. If
for some
then
.
Proposition
(Estimate for K1 two) For any index set
,
Proof
is tedious and straightforward.
