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That problem isn't unique to regula falsi: Other than bisection, all of the numerical equation-solving methods can have a slow-convergence or no-convergence problem under some conditions. Sometimes, Newton's method and the secant method diverge instead of converging – and often do so under the same conditions that slow regula falsi's convergence.
The construction of the queried point c follows three steps: interpolation (similar to the regula falsi), truncation (adjusting the regula falsi similar to Regula falsi § Improvements in regula falsi) and then projection onto the minmax interval. The combination of these steps produces a simultaneously minmax optimal method with guarantees ...
Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since.
A root-finding algorithm is a numerical method or algorithm for finding a value x such that f(x) = 0, for a given function f. Here, x is a single real number. Root-finding algorithms are studied in numerical analysis.
Of course not. All that can be said is that Regula Falsi replaced False Position (or, rather, the Italian expression with that meaning). There's no reason to believe that Regula Falsi was intended as a translation for False Position. In fact, the fact that Regula Falsi is not a valid Latin translation of False Position, weighs against such a claim.
Regula falsi is another method that fits the function to a degree-two polynomial, but it uses the first derivative at two points, rather than the first and second derivative at the same point. If the method is started close enough to a non-degenerate local minimum, then it has superlinear convergence of order φ ≈ 1.618 {\displaystyle \varphi ...
In numerical analysis, the ITP method (Interpolate Truncate and Project method) is the first root-finding algorithm that achieves the superlinear convergence of the secant method [1] while retaining the optimal [2] worst-case performance of the bisection method. [3]
The class of methods is based on converting the problem of finding polynomial roots to the problem of finding eigenvalues of the companion matrix of the polynomial, [1] in principle, can use any eigenvalue algorithm to find the roots of the polynomial. However, for efficiency reasons one prefers methods that employ the structure of the matrix ...