Previous question, somewhat related: Bracketing root-finding methods: my modified Illinois method.
I came up with another simple bracketing method that seems to work really well, especially against functions with initially poor secant approximations or multiple order roots.
We define:
$[a_i,b_i]$ is the interval which bounds the root.
$\hat c_i=\dfrac{a_if(b_i)-b_if(a_i)}{f(b_i)-f(a_i)}$ is the estimated next root from the secant approximation.
$c_i=\dfrac{w_i\hat c_i+a_i+b_i}{w_i+2}$ is the next bound used, where $w_i$ is a weight.
Note that as $w\to\infty$, $c\to\hat c$, and as $w\to0$, $c\to\dfrac{a+b}2$, so we get a sort of continuum between the bisection and regula falsi methods.
This differs from the Illinois type methods mentioned in the link above in that we are adjusting an average of $a,b,c$ opposed to adjusting an averaging of $f(a)$ and $f(b)$, so this method can depend on the function less, making it closer to the bisection method.
The exact choice of $w$ can vary, though I've found it works reasonably well with something like
$$w_{i+1}=\begin{cases}w_i^2,&f(c_i)f(c_{i-1})<0\\w_i^{3/4},&f(c_i)f(c_{i-1})>0\end{cases}$$
Try it online, a lot of test cases from [1] give similar convergence as the Illinois method, but in general this seems to be more reliable. There is also a table of the results at the end of the output.
Is there a name for this method?
Should I expect this to converge faster, slower, or roughly the same as the Illinois method? What is the order of convergence?
For the 2nd question, I believe it should be roughly the same, but better at handling multiple order roots and really bad initial bounds. Some of the test cases suggest it may be slower in other situations though. Overall it seems to be much more reliable.
[1]: Bus, Dekker: "Two efficient algorithms with guaranteed convergence for finding a zero of a function", 1975, ACM Trans. Num. Soft. (dl.acm.org/doi/10.1145/355656.355659)[dl.acm.org/doi/10.1145/355656.355659].