A root-finding algorithm also known as the tangent hyperbolas method or Halley's rational formula. As in Halley's irrational formula, take the second-order Taylor series
(1)
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A root of satisfies , so
(2)
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Now write
(3)
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giving
(4)
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Using the result from Newton's method,
(5)
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gives
(6)
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so the iteration function is
(7)
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This satisfies where is a root, so it is third order for simple zeros. Curiously, the third derivative
(8)
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is the Schwarzian derivative. Halley's method may also be derived by applying Newton's method to . It may also be derived by using an osculating curve of the form
(9)
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Taking derivatives,
(10)
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(11)
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(12)
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which has solutions
(13)
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(14)
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(15)
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so at a root, and
(16)
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which is Halley's method.