We show that for any set of $n$ distinct points in the complex plane, there exists a polynomial $p$ of degree at most $n+1$ so that the corresponding Halley and Schröder map for $p$ has the given points as a super-attracting cycle. This improves the result in [
Citation: |
S. Amat
, S. Busquier
and S. Plaza
, A construction of attracting periodic orbits for some classical third-order iterative methods, J. Comput. Appl. Math., 189 (2006)
, 22-33.
doi: 10.1016/j.cam.2005.03.049.![]() ![]() ![]() |
|
J. Campbell
and J. Collins
, Specifying attracting cycles for Newton maps of polynomials, J. Difference Equ. Appl., 19 (2013)
, 1361-1379.
doi: 10.1080/10236198.2012.751987.![]() ![]() ![]() |
|
D. Hilbert
, Über die vollen Invariantensysteme, Math. Ann., 42 (1893)
, 313-373.
doi: 10.1007/BF01444162.![]() ![]() ![]() |
|
K. Kneisl
, Julia sets for the super-Newton method, Cauchy's method and Halley's method, Chaos, 11 (2001)
, 359-370.
doi: 10.1063/1.1368137.![]() ![]() ![]() |
|
S. Plaza
and N. Romero
, Attracting cycles for the relaxed Newton's method, J. Comput. Appl. Math., 235 (2011)
, 3238-3244.
doi: 10.1016/j.cam.2011.01.010.![]() ![]() ![]() |
|
S. Plaza
and V. Vergara
, Existence of attracting periodic orbits for the Newton method, Sci. Ser. A Math. Sci., 7 (2001)
, 31-36.
![]() ![]() |