The Numerical Method for Finding All the Zeros of a Function f (x) on an Interval [a,b]

In this study, we develop a simple mathematical method for finding all the roots of a function on a specified interval. Existing classical numerical methods, including the Bisection Method, Secant Method, and Newton Method, cannot find all the zeros of a function f (x) on an interval [a,b]. Popular numerical root solvers like Matlab’s ‘fzero’, Maple’s ‘fsolve’, and SageMath’s ‘find_root’ typically yield only a single zero. This thesis explores a proposed interval computation bisection method, a systematic approach based on the traditional Bisection Method and interval computation. Unlike traditional bisection, which relies on the intermediate value theorem, this approach uses interval arithmetic to compute the function value over subintervals. SageMath is used to implement the method because of its built-in support for interval arithmetic and symbolic computation, which ensures accurate root detection. The method determines whether a zero exists in each subinterval by iteratively bisecting the interval and analyzing the interval computation value of the function over the corresponding interval until the root is isolated in an interval of desired precision. This approach ensures the reliable detection of all roots, even in cases where the traditional numerical methods fail.