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This is what my teacher shared with us.

My instructor asserts that the Regula Falsi method has a superlinear convergence order, specifically citing that the error decreases by a factor related to the golden ratio $(\frac{\sqrt{5} - 1}{2})$ per iteration. However, standard references like Burden and Faires' Numerical Analysis only detail convergence for the Secant and Newton-Raphson methods, omitting Regula Falsi. Online resources also lack clarity, with some suggesting it converges linearly or superlinearly under certain conditions.

My confusion arises because:

Regula Falsi maintains a bracketed root by fixing one endpoint, unlike the Secant method. Intuitively, this might restrict convergence speed.

The claim about the golden ratio factor resembles properties of the Secant method, but I’m unsure if this applies to Regula Falsi.

Mittens
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Kylie
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    A question shouldn't consist of a link, also important parts of your question should be included in the body of the question and not in the title. Title is just a short description of the problem – Jakobian Mar 03 '25 at 11:02
  • not a regular visitor. just facing some conflicts so thought of clearing my doubts – Kylie Mar 03 '25 at 11:05
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    The cited arguments are correct for the secant method. But the bare regula falsi indeed has the mentioned stalling problem, so will converge linearly. – Lutz Lehmann Mar 04 '25 at 06:36
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    My thoughts on the convergence can be found in https://math.stackexchange.com/questions/2100039/how-to-show-that-regula-falsi-has-linear-rate-of-convergence?rq=1, if the function close to the root is closely approximated by its quadratic Taylor polynomial, linear convergence follows. The convergence rate strongly depends on the initial interval, by how close the stalling interval end is to the root. – Lutz Lehmann Mar 05 '25 at 09:25
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    The way to intelligently use the same information on the function as regula falsi is Dekkers method. I've tried to motivate its steps in https://math.stackexchange.com/a/4818183/115115, sort of as prequel to https://math.stackexchange.com/a/2949893/115115 – Lutz Lehmann Mar 05 '25 at 09:51

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