A Concise Introduction to Numerical Analysis by A. C. Faul

Author:A. C. Faul
Language: eng
Format: epub, pdf
Publisher: Taylor & Francis Group, LLC
Published: 2016-03-14T16:00:00+00:00

In every iteration the algorithm overshoots the solution onto the other side further away than it was initially. The distance to the solution is doubled in each iteration.

There are also cases where Newton’s method converges, but the rate of convergence is not quadratic. For example take f(x) = x2. Then for every approximation x(n) the next approximation is

Thus the distance to the root is halved in every iteration comparable to the bisection method.

Newton’s method readily generalizes to higher dimensional problems. Given a function f: ℝm → ℝm, we consider f as a vector of m functions

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