Ridders' method: Difference between revisions

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==Method==

Press et al. (2007) describe the method as follows. Given two values of the independent variable, ''x''₁₀ and ''x''₂₁, which are on two different sides of the root being sought, the method begins by evaluating the function at the midpoint ''x''₃₁ between the two points. One then finds the unique exponential function of the form ''e''^''ax'' which, when multiplied by ''f'', transforms the function at the three points into a straight line. The false position method is then applied to the transformed values, leading to a new value ''x''₄₃, between ''x''₁₀ and ''x''₂, which can be used as one of the two bracketing values in the next step of the iteration.

The other bracketing value is taken to be ''x''₃ if f(''x''₃) has the opposite sign to f(''x''₄), or otherwise whichever of ''x''₁ and ''x''₂ has f(x) of opposite sign to f(''x''₄).

The method can be summarized by the formula (equation 9.2.4 from Press et al.)