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单词 Halley's Method
释义

Halley's Method

Also known as the Tangent Hyperbolas Method or Halley's Rational Formula. As in Halley's IrrationalFormula, take the second-order Taylor Polynomial

(1)

A Root of satisfies , so
(2)

Now write
(3)

giving
(4)

Using the result from Newton's Method,
(5)

gives
(6)

so the iteration function is
(7)

This satisfies where is a Root, so it is third order for simple zeros. Curiously, the third derivative
(8)

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)

Taking derivatives,
(10)
(11)
(12)

which has solutions
(13)
(14)
(15)

so at a Root, and
(16)

which is Halley's method.

See also Halley's Irrational Formula, Laguerre's Method, Newton's Method


References

Scavo, T. R. and Thoo, J. B. ``On the Geometry of Halley's Method.'' Amer. Math. Monthly 102, 417-426, 1995.

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更新时间:2025/2/22 21:39:39