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please tell me the derivative method to find out the total number of roots of any equation

please tell me the derivative method to find out the total number of roots of any equation


1 Answers

askIITians Faculty 74 Points
6 years ago
Hello student,
please find the answer to your question below
Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method, and is usually quadratic. Newton's method is also important because it readily generalizes to higher-dimensional problems. Newton-like methods with higher orders of convergence are the Householder's methods. The first one after Newton's method is Halley's method with cubic order of convergence.
[x : f(x) = 0]
The Newton–Raphson method in one variable is implemented as follows:
Given a function ƒ defined over the reals x, and its derivative ƒ', we begin with a first guess x0 for a root of the function f. Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is
[x{1} = x0 - {f(x0)}/{f'(x0)}
Geometrically, (x1, 0) is the intersection with the x-axis of the tangent to the graph of f at (x0, f (x0)).
The process is repeated as
[x{n+1} = xn -{f(xn)}/{f'(x_n)}
until a sufficiently accurate value is reached.

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