# Root finding methods for Polynomials.

• mhill
In summary, there are methods available to obtain the roots of polynomials, including complex roots. One such method is the Bairstow's method, which can find both real and complex roots accurately. It involves converting the polynomial to an eigenvalue problem and has a fast convergence rate. The only challenge is finding a good starting value, but this can also be solved.
mhill
Hi , is there a method to obtain the roots of Polynomials:

$$P(x)=a_{0}+a_{1}x+a{2}x^{2}+...+a_{n}x^{n}$$

i know there are , but my problem is this if we knew that are complex roots of the form z=a+ib , would be a method to obtain the complex root with BIGGER and SMALLER real part ?? , i mean for example if a POlynomial has complex root:

1+3i 1-3i 0.6+8i 0.6-8i 0.01+34i 0.01-34i ...

my question is if we could use a root finding algorithm to check that the bigger part of the roots is '1'

Last edited:
I'm not sure what you mean. Matlab has a good method for finding the roots numerically. It transfers the roots of polynomial to an eigvenvalue problem. It converges quite fast and will give you all the roots. It is quite accurate if the eignvalues are well separated.

One method to find all roots is the one of Bairstow. This gives you not only the real, but also the complex ones. Look here for some info, and google a bit to find some more:

http://en.wikipedia.org/wiki/Bairstow's_method

The implementation is not that difficult, I've done it and it works fine. The only thing to take care of is a good starting value, but even that can be solved.

## 1. What are root finding methods for polynomials?

Root finding methods for polynomials are mathematical algorithms used to find the roots (or solutions) of polynomial equations, which are equations that contain variables raised to different powers.

## 2. Why are root finding methods important?

Root finding methods are important because they allow us to find the solutions to polynomial equations, which are used in many mathematical models and real-world applications.

## 3. What are some common root finding methods for polynomials?

Some common root finding methods for polynomials include the bisection method, the Newton-Raphson method, and the secant method. Other methods include the Muller method, the Bairstow method, and the Jenkins-Traub method.

## 4. How do root finding methods work?

Root finding methods typically involve starting with an initial guess for the solution and then iteratively refining that guess until it converges to the actual root. This is done by using the properties of polynomials and their derivatives to narrow down the possible solutions.

## 5. What are the advantages and limitations of root finding methods?

The advantages of root finding methods include their efficiency and accuracy in finding roots of polynomials. However, these methods can be limited by the complexity of the polynomial equation and may not always converge to the correct solution. Additionally, some methods may require knowledge of the polynomial's derivative or multiple initial guesses for the solution.

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