A note on finding roots of monotonic branching functions

Abstract

SUMMARYSome physical problems require ÿnding roots of branching functions. Transcendental eigenvalue problems are a common example, where certain functions are known to be branching, monotonic within branches, and have root counting algorithms. Solution conventionally starts by surrounding a required root with bounds on the same branch, and then reÿning one or both of the estimates by conventional or problem-speciÿc algorithms. Here, di culties caused by branching are removed by deÿning a new function that is free of branches and has a single root. The root of this function is found by linear interpolation, with some acceleration applied if high accuracy is wanted. Construction of the new function is completely numerical, as is the root ÿnding acceleration.