释义 |
Halley's MethodAlso 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. |