In numerical analysis, Newton’s method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
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). This x-intercept will typically be a better approximation to the function’s root than the original guess, and the method can be iterated.
Suppose f : (a, b) → ℝ is a differentiable function defined on the interval (a, b) with values in the real numbers ℝ. The formula for converging on the root can be easily derived. Suppose we have some current approximation xn. Then we can derive the…
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