But, in the absence of any intuition about where the zero might lie, a "guess and check" method might narrow the possibilities to a reasonably small interval by appealing to the intermediate value theorem.) The method will usually converge, provided this initial guess is close enough to the unknown zero, and that ƒ'( x 0) ≠ 0. We start the process off with some arbitrary initial value x 0. Here, f ' denotes the derivative of the function f. We know from the definition of the derivative at a given point that it is the slope of a tangent at that point. Then we can derive the formula for a better approximation, x n+1 by referring to the diagram on the right. Suppose we have some current approximation x n. The formula for converging on the root can be easily derived. Suppose ƒ : → R is a differentiable function defined on the interval with values in the real numbers R.
This x-intercept will typically be a better approximation to the function's root than the original guess, and the method can be iterated. The idea of the method is as follows: one starts with an initial guess which is reasonably close to the true root, then the function is approximated by its tangent line (which can be computed using the tools of calculus), and one computes the x-intercept of this tangent line (which is easily done with elementary algebra). We see that x n+1 is a better approximation than x n for the root x of the function f. The function ƒ is shown in blue and the tangent line is in red. 7.1 Minimization and maximization problems.6.4 Nonlinear equations over p-adic numbers.6.3 Nonlinear equations in a Banach space.5.3.3 Modified Newton's Method in case of Non-Quadratic Convergence.5.2.1 Derivative does not exist at root.
4.1 Proof of quadratic convergence for Newton's iterative method.3.3 Slow convergence for roots of multiplicity > 1.3.2 Failure of the method to converge to the root.3.1 Difficulty in calculating derivative of a function.
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