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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.
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.
However, most root-finding algorithms do not guarantee that they will find all roots of a function, and if such an algorithm does not find any root, that does not necessarily mean that no root exists. Most numerical root-finding methods are iterative methods, producing a sequence of numbers that ideally converges towards a root as a limit.
Given a continuous function defined from [,] to such that () (), where at the cost of one query one can access the values of () on any given .And, given a pre-specified target precision >, a root-finding algorithm is designed to solve the following problem with the least amount of queries as possible:
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.
The main objective of interval arithmetic is to provide a simple way of calculating upper and lower bounds of a function's range in one or more variables. These endpoints are not necessarily the true supremum or infimum of a range since the precise calculation of those values can be difficult or impossible; the bounds only need to contain the function's range as a subset.
Then for each interval (A(x), M(x)) in the list, the algorithm remove it from the list; if the number of sign variations of the coefficients of A is zero, there is no root in the interval, and one passes to the next interval. If the number of sign variations is one, the interval defined by () and () is an isolating interval.
Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...
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