unning procedure (
Conjugate gradients
)
for all
steps is not always feasible. In addition, numerical errors are likely to
destroy orthogonality of
For these reasons it makes sense to study convergence of the procedure
(
Conjugate gradients
).
According to the formula
(
Conjugate gradient residue
selection
)
According to the formula (
Error and residual
2
) we then
have
for some
th
degree polynomial
Furthermore, by tracing the recipe (
Conjugate
gradients
) directly we see that
.
The conjugate gradient acts to minimize
by manipulating coefficients of
.
Thus, using notation of the definition
(
Positive definite inner
product
),
Since
is a positive definite symmetric matrix, there exists a set of eigenvectors
and positive eigenvalues
.
Then
for some numbers
.
We
have
Therefore
To evaluate the
combination we aim to apply the proposition
(
Minimum norm
optimality of Chebyshev polynomials
). We are dealing with the minimization
over
and the proposition is formed for
(
Minimum norm
optimality of Chebyshev polynomials
)
.
However, a review of the proof of that proposition shows that normalization is
irrelevant to the argument. We would still have optimality at a polynomial
that differ from a Chebyshev polynomial by multiplicative constant. We do,
however, need to transform from
to
.
where
,
and
.
Next, we map
to
:
We now apply the proposition
(
Minimum norm
optimality of Chebyshev
polynomials
).
where the constant
is the scale to
insure
The
is a scale of
given
by
with the
consequences
Hence, we
rewrite
or, by taking square root and replacing
,
We introduce the quantity


(Condition number)

then
We use the proposition (
Chebyshev
polynomials
calculation
):
Therefore
As
grows the first term grows and the second term
vanishes.
Then
