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:<math>x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n)} \left({\frac{1}{1-\frac{f(x_n)f''(x_n)}{2(f'(x_n))^2}}}\right).</math>
This is referred to as Halley's formula.
This ''geometrical interpretation'' <math>h(z)</math> was derived by Gander (1978), where the equivalent iteration also was derived by applying Newton's method to
:<math>g(x)=\frac{f(x)}{\sqrt{f'(x)}}=0.</math>
We call this ''algebraic interpretation'' <math>g(x)</math> of Halley's formula.
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The algebraic interpretation of this iteration is obtained by solving
:<math>g(x)=1-\frac{\alpha}{{f(x)}}=0.</math>
This one-point second-order method is known to show a locally quadratic convergence if the root of the equation is simple.
SRA strictly implies this one-point second-order interpolation by a simple rational function.
We can notice that even third order method is a variation of Newton's method. We see the Newton's steps are multiplied by some factors. These factors are called the ''convergence factors'' of the variations, which are useful for analyzing the rate of convergence. See Gander (1978).
== References ==
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| title = The spectrum of a modified linear pencil
| volume = 46
| year = 2003
}}.
*{{citation
| last1 = Gu | first1 = Ming
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| title = A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem
| volume = 16
| year = 1995
}}.
*{{citation
| last = Gander | first = Walter
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