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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 Helley's formular.
This ''geometrical interpretation'' was derived by Gander(1978), where the equivalent iteration also was derived by apply Newton's method to
:<math>g(x)=\frac{f(x)}{\sqrt{f'(x)}}=0.</math>
We call this algebraic interpretation of Halley's formula.
Similarly, we can obtain a one-point second-order iterative method to solve <math>\,f(x)=\alpha</math> using simple rational approximation by
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Thus we have
:<math>x_{n+1}=x_{n}-\frac{f(x_n)-\alpha}{f'(x_n)} \left(\frac{f(x_n)}{\alpha}\right).</math>
:<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 equation is simple.
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