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Double false position is aimed at solving more difficult problems that can be written algebraically in the form: determine x such that = + =, if it is known that =, =. Double false position is mathematically equivalent to linear interpolation.
The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method with respect to function evaluations rather than with ...
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This means that the false position method always converges; however, only with a linear order of convergence. Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see Regula falsi § Improvements in regula falsi) such as the ITP method or the Illinois method.
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The false position method, also called the regula falsi method, is similar to the bisection method, but instead of using bisection search's middle of the interval it uses the x-intercept of the line that connects the plotted function values at the endpoints of the interval, that is
The empty string, commonly denoted by ε, is assigned to the root node. Position strings within the black term are indicated in red in the picture. At each position p of a term t, a unique subterm starts, which is commonly denoted by t| p. For example, at position 122 of the black term in the picture, the subterm a+2 has its root.