Newton's method and complex roots

[Delta31415]

1) the problem
I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function ( x^8-7x^7+14x^6-14x^5+27x^4-14x^3+14x^2-7x+1) to into complex form. because of that I don't know what the initial value would.

2) Relevant equations
Newton's method and z=x+yi)
3) attempts
found real roots

btw the function I wrote is the one I am solving,however that doesn't matter I just need to know how to find the complex roots for any function given in that form.

 

[BvU]

You are not very verbose with your problem statement but I assume you want to solve $f({\bf z}) = 0$ for ${\bf z} \in {\mathbb C}$ ?

That's in fact two equations (e.g $\ \operatorname { Re } (f( {\bf z} ) ) = 0 \$ and $\ \operatorname {Im} (f( {\bf z} ) )= 0$ ) and you'll have to make those explicit before embarking on Newton's method for finding zeroes of a function.

 

[Ray Vickson]

1) the problem
I understand Newton's method and I was able to find all the real roots of the function.However, I don't understand how to find the complex roots. I know that z=x+yi, and that I can plug in z for the formula. However I, don't know how to change the function ( x^8-7x^7+14x^6-14x^5+27x^4-14x^3+14x^2-7x+1) to into complex form. because of that I don't know what the initial value would.

2) Relevant equations
Newton's method and z=x+yi)
3) attempts
found real roots

btw the function I wrote is the one I am solving,however that doesn't matter I just need to know how to find the complex roots for any function given in that form.

Just replace $p(x) = x^8-7 x^7+14 x^6 -14 x^5 +27 x^4 -14 x^3 +14 x^2 -7 x +1$ by $p(z) = z^8-7 z^7+14 z^6 -14 z^5 +27 z^4 -14 z^3 +14 z^2 -7 z +1$ and use the usual Newton iteration
$$z_{k+1} = z_k - \frac{p(z_k)}{p'(z_k)}.$$
However, to converge to a non-real root you need to start at a non-real $z$, and even then there are some starting points that can yield chaotic "limit cycles" rather than single points. Google "Newtons method for complex roots" to find numerous articles on the issue.

 

[Delta31415]

Just replace $p(x) = x^8-7 x^7+14 x^6 -14 x^5 +27 x^4 -14 x^3 +14 x^2 -7 x +1$ by $p(z) = z^8-7 z^7+14 z^6 -14 z^5 +27 z^4 -14 z^3 +14 z^2 -7 z +1$ and use the usual Newton iteration
$$z_{k+1} = z_k - \frac{p(z_k)}{p'(z_k)}.$$
However, to converge to a non-real root you need to start at a non-real $z$, and even then there are some starting points that can yield chaotic "limit cycles" rather than single points. Google "Newtons method for complex roots" to find numerous articles on the issue

Thank, I was able to find the complex roots and I also looked up chaotic limit cycle the graph isn't a circle or a spiral so there isn't any orbit.right?

 

[Ray Vickson]

Thank, I was able to find the complex roots and I also looked up chaotic limit cycle the graph isn't a circle or a spiral so there isn't any orbit.right?

I don't know if it is right or not; perhaps I should have termed it differently, but the articles on the subject used such terms--possibly more accurately. As I said, there are numerous articles on the topic, and they will tell you everything you need to know. Some of them even include Matlab codes for the problem.