Re: complex roots of complex polynomials
- From: Dweeb <dweebgoober@xxxxxxxxx>
- Date: Wed, 24 Oct 2007 06:38:22 -0700
Carel wrote:
As an exercise I wrote a little program that can extract complex rootsActually, for single variable functions the 2nd oder Newton method
from a polynomial with complex coeficients.
I used Newton's method.
works far better.
Use equation #5 from this link
http://mathworld.wolfram.com/HalleysIrrationalFormula.html
(Look what happens when the second derivative term is zero ;) )
The basic Newton's method assumes the function is well approximated by
a straight line
and then proceeds to find the root of that line. A major problem
occurs whenever the
slope of the line is near zero. This leads to chaotic and fractal
behavior and can cause
the number of iterations to increaase dramatically depending on your
starting guess.
For arbitrary functions, at least 2 function evaluations are needed
for each iteration
to approximate the first derivative.
The 2nd order method assumes that the function can be approximated by
a parabola
and then proceeds to find both roots of the parabola. A slope of zero
is no problem
and the iterations converge cubically and with far less chaotic and
fractal behavior.
The cost of this improvement is the need to make 3 function
evaluations for each iteration.
But this extra cost is more than made up for by the faster convergence
and, most importantly,
far less chaotic behavior.
Could anyone verify the result?Why do you need us to verify the result?
Just plug your "root" into the equation and see if the result is very
close to zero.
Hope this helps
.
- References:
- complex roots of complex polynomials
- From: carel
- complex roots of complex polynomials
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