Superlinear Convergence

Abstract

In Section 3 we showed that if the capacity of the spectrum of the iteration operator vanishes, then

$$\eta (L):=\inf||p_k(L)||^{1/k}=0$$

((5.1.1))

(provided 1 ∉ σ (L)), meaning that the convergence is eventually faster than any linear rate. This is simply the definition of superlinear convergence. We shall in this section study superlinear convergence and in particular we assume always that cap(σ(L)). Recall (Definition 2.9.1) that operators with this property are called quasialgebraic. What interests us here is to establish scales of speed for the convergence of quasialgebraic operators. In order to get an initial feeling of the possible speeds, think of a self-adjoint negatively semidefinite operator A in a Hilbert space, with a countable spectrum λ¹ ≤ λ² ≤ … → 0. Then, if we interpolate from the left

$$ p_k (\lambda ): = \prod\limits_1^k {\frac{{\lambda - \lambda _j }} {{1 - \lambda _j }}} $$

((5.1.2))

we clearly have

$$ {\text{||p}}_{\text{k}} {\text{(A)|| = p}}_{\text{k}} {\text{(0)}}{\text{.}} $$

((5.1.3))