Halley,s Method
المؤلف:
Ortega, J. M. and Rheinboldt, W. C
المصدر:
Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000.
الجزء والصفحة:
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12-12-2021
1440
Halley's Method
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)
|
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.
REFERENCES:
Ortega, J. M. and Rheinboldt, W. C. Iterative Solution of Nonlinear Equations in Several Variables. Philadelphia, PA: SIAM, 2000.
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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