What are the criteria such that a function $f(t)$ can serve as the correction in an iteration function of the form $g (t) = t - \lambda f (t)$ where $\lambda$ is some relaxation factor? It is almost reminiscent of Newton's iteration, without the derivative.
for instance, if $f(t)=sin(t)$ then $g(t)$ has attractive fixed-points at $t=n\pi$ when $n$ is even and repulsive fixed-points when $n$ is odd
if the iteration is $g (t) = t + \lambda f (t)$ then the attractiveness and repulsiveness of the fixed-points is swapped, so in the case of $f(t)=sin(t)$ it would be repulsive when $n$ is even and attractive when $n$ is odd
Do functions which have this property have a special name?
Perhaps the iteration converges to a root $f(t)=0$ where the derivative is $0<f'(t)<2$.
and iterating the function $t-Z(t)$ doesn't converge to any points where Z'(t) is not in [0,2] .. at least for the first 20 odd-numbers zeros I checked.. this table demonstrates
the column on the left is the difference between the starting point (the $2n-1$th zero -0.1 and the 50 iterations of the iteration function, and the column on the right is the derivative of $Z$ evaluated at the $2n-1$th zero
$ \left[ \begin {array}{cc} 0.0& 0.7931604332\\ 0.0 & 1.371721287\\ 0.0& 1.382119539 \\ 0.0& 1.490610763\\ 0.0& 1.568031477\\ - 1.11005001& 2.426579069 \\ 0.0& 1.391805619\\ - 0.38497400& 2.287779010\\ - 0.59958219& 2.186311017 \\ - 0.00000004& 1.779555993\\ - 0.98459094& 2.637886209\\ - 0.48377276& 2.161778835 \\ - 0.32348014& 2.176460788\\ 0.0& 1.479402184\\ - 1.37854250& 3.515767073 \\ - 0.3298843& 2.167414624\\ {\it Float}(\infty )& 2.982497202\\ 0.0& 1.361150829\\ - 14.4201635& 3.119005954 \\ - 0.3041748& 2.294939525\end {array} \right] $
I'm sure I would find the same thing with iteration function $t+Z(t)$ .
To "Fix" this, one can take $t-tanh(f(t))$ then the derivative can be no more than 2.. see does this Newton-like iterative root finding method based on the hyperbolic tangent function have a name?
It is the set of functions whose derivative at the roots is less than 2 and greater than 0. If anyone had a great idea on how to prove this....it is a conjecture based on the empirical fact that iterating this method with the Hardy Z function results in convergence when the derivative at the starting point is 0
The crux of your method is the $\lambda$ factor. It is a variation of "trial and error". It is a perfectly valid method when used with caution. I don't know of any particular name for it but it is a very simple and natural method. The effectiveness of the method can be improved by adjusting the $\lambda$ factor depending on how close $f(t)$ is to zero and perhaps other data. One example of this is Newton's method as mentioned in a comment.