so I was wondering what the brief conceptual difference between the two methods of finding roots are? I understand that the Heron's Algorithm is a special case of Newton's Method, but is it just because Heron's Algorithm is just for approximating square roots?
Difference between Heron's Algorithm and Newton's Method?
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approximation
newton-raphson
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0Yes, the Heron-Algorithm is specially designed to find square-roots, the newton-method is a general method to find roots of some function $f(x)$. What exactly is your question ? – 2017-02-08
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2Heron's method happens to be what you get when you apply Newton's method to solve the equation $f(x)=x^2-r=0$. There are other distinct ways to compute square roots. Heron's method only has its special name because it predates the general Newton method. – 2017-02-08
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0How are they related exactly? – 2017-02-08
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0The Heron-algorithm can also be understood intuitively without knowing anything about newton's method. We start with a rectangular with a given area. To get a rectangular with the same area, such that the difference between the lengths of the sides gets smaller, the idea is to take one side as the arithmetic mean. The length of the other side is determined by this. This is repeated until we almost have a square with the given area. – 2017-02-08
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0As Ian pointed out, the Heron-method is known much longer than the general Newton-method. – 2017-02-08
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0Another intuition: if $x=r/x$ then $x=\sqrt{r}$; otherwise $\sqrt{r}$ is between them, and the arithmetic mean of the two is clearly between them. – 2017-02-08