Finding Roots of Polynomials with Complex Numbers

Date: 09/27/2001 at 00:33:24
From: Ed
Subject: Find roots of polynomials with complex numbers

Dr. Math,

In one of your articles, I read that you can find the roots of 3rd- or 
higher-degree polynomials with complex numbers. You did not explain 
how to do it, since you assumed the student did not have much 
experience using complex variables. 

I would like to learn more about this topic. Please explain to me in 
detail how to find roots of such equations, along with examples and 
applications, history. Thanks in advance.

Regards,
Ed

Date: 09/27/2001 at 09:12:51
From: Doctor Rob
Subject: Re: Find roots of polynomials with complex numbers

Thanks for writing to Ask Dr. Math, Ed.

The same methods that work for polynomials with real coefficients also 
work for those with complex coefficients. For such methods applied to 
cubic and quartic equations, see Cubic and Quartic Equations from our 
Frequently Asked Questions (FAQ):

  http://mathforum.org/dr.math/faq/faq.cubic.equations.html   

For higher-degree polynomials f(x), one can use Newton's method to 
find the roots numerically, even if they are complex. That involves
starting with a guess x[0], and using the following recursive formula:

   x[n+1] = x[n] - f(x[n])/f'(x[n]),  n = 0, 1, 2, ...

Here f'(x) is the first derivatve of f(x) with respect to x:

           d
   f(x) = SUM a[i]*x^i,
          i=0

            d
   f'(x) = SUM i*a[i]*x^(i-1).
           i=1

For example, suppose we had the polynomial

   f(x) = x^5 - (6*i)*x - 2,
   f'(x) = 5*x^4 - 6*i.

Then suppose we start with the guess x = i.  Then the recursive
formula is

   x[n+1] = x[n] - (x[n]^5-6*i*x[n]-2)/(5*x[n]^4-6*i).

The computation gives

   x[0] = i,
   x[1] = -0.2295 + 0.5246*i,
   x[2] = 0.01873 + 0.3264*i,
   x[3] = 0.0007003 + 0.33331*i,
   x[4] = 0.000685769449 + 0.333207872*i,
   x[5] = 0.0006857694535760589 + 0.3333262787490043*i.

This gives you one root of f(x) = 0. Different initial guesses can
give you different roots. Starting with x[0] = 2 + i, you would get 
x[7] = 1.457461395822838 + 0.5135279776944679*i, another root.

Of course, once you have one root r, you can divide out x - r to
reduce the degree of the polynomial, but giving a quotient that has
approximate numerical coefficients.

Newton's method was devised by Sir Isaac Newton. Sometimes it is 
called the Newton-Raphson method. Apparently Newton devised it first,
about 1671, but it was published first by Joseph Raphson in 1691.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/