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The bisection method consists roughly of starting from an interval containing all real roots of a polynomial, and divides it recursively into two parts until getting eventually intervals that contain either zero or one root. The starting interval may be of the form (-B, B), where B is an upper bound on the absolute values of the roots, such as ...
The simplest root-finding algorithm is the bisection method.Let f be a continuous function for which one knows an interval [a, b] such that f(a) and f(b) have opposite signs (a bracket).
In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b). At each step the method divides the interval in two parts/halves by computing the midpoint c = (a+b) / 2 of the interval and the value of the function f(c) at that point.
Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
b k is the current iterate, i.e., the current guess for the root of f. a k is the "contrapoint," i.e., a point such that f(a k) and f(b k) have opposite signs, so the interval [a k, b k] contains the solution. Furthermore, |f(b k)| should be less than or equal to |f(a k)|, so that b k is a better guess for the unknown solution than a k.
An interval is said to be bounded, if it is both left- and right-bounded; and is said to be unbounded otherwise. Intervals that are bounded at only one end are said to be half-bounded. The empty set is bounded, and the set of all reals is the only interval that is unbounded at both ends. Bounded intervals are also commonly known as finite ...
If the multiplicity m of the root is finite then g(x) = f(x) / f ′ (x) will have a root at the same location with multiplicity 1. Applying Newton's method to find the root of g(x) recovers quadratic convergence in many cases although it generally involves the second derivative of f(x).
Use Budan's "0_1 roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, 1). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b). If there are two or more sign variations Budan's "0_1 roots test ...