andydude Wrote:I think that both regular iteration and natural/intuitive iteration can be viewed as different techniques for applying matrix iteration in general.
Haha, that would introduce more ambiguity!
Matrix iteration would mean matrix power method for me!
Which is applicable to fixed points and non-fixed points.
If it is applied to fixed points then it is regular iteration.
But intuitive iteration (\( \operatorname{slog}^{-1}(t+\operatorname{slog}(x)) \)) is imho not equal to matrix power iteration, at least not a priori.
Quote:like parabolic and hyperbolic fixed points
Where also it is not quite clear what "hyperbolic" means (\( |f'(x_0)|>1 \)). Is there also an elliptic (\( 0<|f'(x_0)|<1 \))?
Quote:Analytic iteration of f(x) about \( x=x_0 \) where \( \mathbf{J}(\mathbf{B}[f(x + x_0)] - I)\mathbf{K} \) is invertible (which requires at the very least that \( x_0 \) is not a fixed point), is called natural/intuitive iteration.
No, analytic iteration means just that the iterates are analytic. It can not presumed uniquely to be intuitive iteration until equality is shown.
