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In Object Oriented Technologies and Java programming, implementing mathematical methods like Regula-Falsi can be achieved using classes, objects, and encapsulation, even though the original method might traditionally be implemented in procedural languages like C. The Regula-Falsi method, also known as the false position method, is used for finding the root of a function in a given interval. It is an iterative technique that improves the approximation of the root by using a linear interpolation formula between two initial guesses. In Java, this can be elegantly implemented by creating a class that represents the method and encapsulates the function, interval, tolerance, and maximum number of iterations. We start by defining a Java class named `RegulaFalsi`. This class can include private member variables to store the endpoints of the interval, the tolerance level for the root approximation, and the maximum iterations allowed to prevent infinite looping. The function whose root is to be found can be represented as a method within the class, or we can use a functional interface to allow flexibility for different mathematical functions. For example, we can define a method `function(double x)` which returns the value of the function at a given point `x`. The main logic of the Regula-Falsi method is to calculate the new approximation of the root using the formula: $$ x_r = x_1 - \frac{f(x_1) \cdot (x_0 - x_1)}{f(x_0) - f(x_1)} $$ Here, `x0` and `x1` are the initial guesses, `f(x0)` and `f(x1)` are the function values at these points, and `xr` is the next approximation of the root. This formula is derived from the equation of a straight line passing through the points `(x0, f(x0))` and `(x1, f(x1))`. The value of `xr` replaces either `x0` or `x1` depending on the sign of the function at these points, ensuring that the root lies within the new interval. In Java, we implement this iterative logic using a _____ ______ __________ _____ ________ _____ ________ ___ __________ _________ _________.
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