Newton's Method (Iteration 반복법, Newton-Raphson 뉴튼-랩슨법)

세상의모든계산기2015.03.23 11:24

Consider the problem of finding the square root of a number. Newton's method is one of many methods of computing square roots.

For example, if one wishes to find the square root of 612, this is equivalent to finding the solution to

\,x^2 = 612

The function to use in Newton's method is then,

\,f(x) = x^2 - 612

with derivative,

f'(x) = 2x. \,

With an initial guess of 10, the sequence given by Newton's method is

\begin{matrix} x_1 & = & x_0 - \dfrac{f(x_0)}{f'(x_0)} & = & 10 - \dfrac{10^2 - 612}{2 \cdot 10} & = & 35.6 \quad\quad\quad{} \\ x_2 & = & x_1 - \dfrac{f(x_1)}{f'(x_1)} & = & 35.6 - \dfrac{35.6^2 - 612}{2 \cdot 35.6} & = & \underline{2}6.395505617978\dots \\ x_3 & = & \vdots & = & \vdots & = & \underline{24.7}90635492455\dots \\ x_4 & = & \vdots & = & \vdots & = & \underline{24.7386}88294075\dots \\ x_5 & = & \vdots & = & \vdots & = & \underline{24.7386337537}67\dots \end{matrix}

where the correct digits are underlined. With only a few iterations one can obtain a solution accurate to many decimal places.

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