False position method to "regula falsi" is sometimes called the Illinois algorithm. Ford (1995) summarizes and analyzes this and other similar superlinear variants of the method of false position. Put simply: When the new -value (that is, )) has the same sign as the previous one ()), meaning that the end point of the previous step will be retained, the Illinois version halves the -value of the retained end point in the next estimate computation. Suppose that in the -th iteration the bracketing interval is and that the functional value of the new calculated estimate has the same sign as . In this
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