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If x is a simple root of the polynomial , then Laguerre's method converges cubically whenever the initial guess, , is close enough to the root . On the other hand, when x 1 {\displaystyle \ x_{1}\ } is a multiple root convergence is merely linear, with the penalty of calculating values for the polynomial and its first and second derivatives at ...
Finding one root; Finding all roots; 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 ...
[5] [page needed] It says that, if the topological degree of a function f on a rectangle is non-zero, then the rectangle must contain at least one root of f. This criterion is the basis for several root-finding methods, such as those of Stenger [6] and Kearfott. [7] However, computing the topological degree can be time-consuming.
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 ϵ > 0 {\displaystyle \epsilon >0} , a root-finding algorithm is designed to solve the following problem with the least amount of queries as possible:
This iteration procedure, like the Gauss–Seidel method for linear equations, computes one number at a time based on the already computed numbers. A variant of this procedure, like the Jacobi method, computes a vector of root approximations at a time. Both variants are effective root-finding algorithms.
The idea to combine the bisection method with the secant method goes back to Dekker (1969).. Suppose that we want to solve the equation f(x) = 0.As with the bisection method, we need to initialize Dekker's method with two points, say a 0 and b 0, such that f(a 0) and f(b 0) have opposite signs.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
The next approximation x k is now one of the roots of the p k,m, i.e. one of the solutions of p k,m (x)=0. Taking m =1 we obtain the secant method whereas m =2 gives Muller's method. Muller calculated that the sequence { x k } generated this way converges to the root ξ with an order μ m where μ m is the positive solution of x m + 1 − x m ...