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Aman Deep Singh Grade: 12

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

6 years ago

Answers : (1)

askIITians Faculty
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										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.
2 years ago
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