There following are (at least) two ways one can arrive to the equation
(
Recursive equation
) starting from the
equation (
Linear equation
).
Proof
We
calculate
For both propositions (
Reduction scheme 1
)
and (
Reduction scheme 2
) we want to almost
invert the matrix of the original problem (
Linear
equation
). The same is true for the preconditioner
of the section (
Preconditioning
). It
appears that preconditioner
the section (
Preconditioning
) may be
used as a matrix
of the proposition (
Reduction scheme 2
),
provided that the
condition
holds. The reverse statement is also true. However, the preconditioned
conjugate gradient technique converges faster as the following proposition
shows.
