“Halley's Method”的概念、定义、习题解题方法及翻译-数学辞典

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,

which has solutions

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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