In such cases a different method, such as bisection, should be used to obtain a better estimate for the zero to use as an initial point. This can happen, for example, if the function whose root is sought approaches zero asymptotically as x goes to ∞ or −∞. In some cases the conditions on the function that are necessary for convergence are satisfied, but the point chosen as the initial point is not in the interval where the method converges. For the following subsections, failure of the method to converge indicates that the assumptions made in the proof were not met. If the assumptions made in the proof of quadratic convergence are met, the method will converge. Newton's method is only guaranteed to converge if certain conditions are satisfied. ![]() Then the expansion of f( α) about x n is: Proof of quadratic convergence for Newton's iterative method Īccording to Taylor's theorem, any function f( x) which has a continuous second derivative can be represented by an expansion about a point that is close to a root of f( x). f ″ > 0 in U +, then, for each x 0 in U + the sequence x k is monotonically decreasing to α.But there are also some results on global convergence: for instance, given a right neighborhood U + of α, if f is twice differentiable in U + and if f ′ ≠ 0, f In practice, these results are local, and the neighborhood of convergence is not known in advance. ![]() However, even linear convergence is not guaranteed in pathological situations. Alternatively, if f ′( α) = 0 and f ′( x) ≠ 0 for x ≠ α, x in a neighborhood U of α, α being a zero of multiplicity r, and if f ∈ C r( U), then there exists a neighborhood of α such that, for all starting values x 0 in that neighborhood, the sequence of iterates converges linearly. Therefore, when there are multiple independent variables but only one dependent. Specifically, if f is twice continuously differentiable, f ′( α) = 0 and f ″( α) ≠ 0, then there exists a neighborhood of α such that, for all starting values x 0 in that neighborhood, the sequence of iterates converges linearly, with rate 1 / 2. method is similar to the multi-variable Newton-Raphson method, Eq. ![]() If the derivative is 0 at α, then the convergence is usually only linear.
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