The Bairstow method is an iterative technique for finding the roots of polynomials by calculating the coefficients of a quadratic factor. It works by taking an initial approximation of the quadratic factor and generating better approximations using partial derivatives until the remainder of dividing the polynomial by the quadratic factor is zero, giving the roots. The method calculates the partial derivatives to update the approximations in a way that avoids having to perform calculations with complex numbers directly. When applied repeatedly, it can find all roots of polynomials of third order or higher.

1. ROOTS OF POLYNOMIALSBAIRSTOW METHODDANIEL FERNANDO RODRIGUEZ

2. A method for calculating roots of polynomials can calculate peer (conjugated in the case of complex roots).Unlike Newton, calculate complex roots without having to make calculations with complex numbers. It is based on the synthetic division of the polynomial Pn (x) by the quadratic (x2 - rx - s).Bairstow Method

3. The synthetic division can be extended to quadratic factors:and even by multiplying the coefficients is obtained:Bairstow Method

4. We want to find the values of r and s that make b1 and b0 equal to zero since, in this case, the factor divided exactly quadratic polynomial. The first method works by taking an initial approximation (r0, s0) and generate approximations (rk, sk) getting better using an iterative procedure until the remainder of division by the quadratic polynomial (x2 - rkx - sk) is zero. The iterative procedure of calculation is based on the fact that both b1 and b0 are functions of r and s.Bairstow Method

5. In developing b1 (rk, sk) and b0 (rk, sk) in Taylor series around the point (r *, s *), we obtain:It takes (r *, s *) as the point where the residue is zero and Δr = r * - rk, Δs = s * - sk. Then:Bairstow Method

6. Bairstow showed that the required partial derivatives can be obtained from the bi by a second synthetic division between factor (x2 - r0x - s0) in the same way that the bi are obtained from the ai. The calculation is:Bairstow Method

7. Thus, the system of equations can be writtenBairstow Method

8. Calculation of approximate error:When tolerance is reached estimated coefficientsrand s is used to calculate the roots:Bairstow Method

9. Then:When the resulting polynomial is of third order or more, the Bairstow method should be applied to obtain a resultant function of order 2.When the result is quadratic polynomial, defines two of the roots using the quadratic equation.When the final function is first order root is determined from the clearance of the equation.Bairstow Method

10. CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002.http://ocw.mit.edu/OcwWeb/MathematicsPPTX EDUARDO CARRILLO, PHD.Bibliography