Ridders' method
In numerical analysis, Ridders' method is a root-finding algorithm based on the false position method and the use of an exponential function to successively approximate a root of a continuous function formula_1. The method is due to C. Ridders. Ridders' method is simpler than Muller's method or Brent's method but with similar performance. 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 is formula_2. If the function is not well-behaved, the root remains bracketed and the length of the bracketing interval at least halves on each iteration, so convergence is guaranteed. Method. Given two values of the independent variable, formula_3 and formula_4, which are on two different sides of the root being sought, i.e.,formula_5, the method begins by evaluating the function at the midpoint formula_6. One then finds the unique exponential function formula_7 such that function formula_8 satisfies formula_9. Specifically, parameter formula_10 is determined by The false position method is then applied to the points formula_12  and formula_13, leading to a new value formula_14 between formula_15 and formula_16, which will be used as one of the two bracketing values in the next step of the iteration. The other bracketing value is taken to be formula_18 if formula_19 (well-behaved case), or otherwise whichever of formula_15 and formula_16 has function value of opposite sign to formula_22. The procedure can be terminated when a given accuracy is obtained.