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Suppose you have a differentiable function of x, f(x) and you are seeking the root of f(x): that is, a solution to f(x) = 0.Suppose x1 is the first approximation to the root, and suppose the exact root is at x = x1+h : that is f(x1+h) = 0.
Let f'(x) be the derivative of f(x) at x, then, by definition,
f'(x1) = limit, as h tends to 0, of {f(x1+h) - f(x1)}/h
then, since f(x1+h) = 0, f'(x1) = -f(x1)/h [approx] or h = -f'(x1)/f(x1) [approx]
and so a better estimate of the root is x2 = x1 + h = x1 - f'(x1)/f(x1).
The proof of the Newton-Raphson iterative equation involves using calculus to show that the method converges to the root of a function when certain conditions are met. By using Taylor series expansion and iterating the equation, it can be shown that the method approaches the root quadratically, making it a fast and efficient algorithm for finding roots.
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