ROOTS OF POLYNOMIALS

BAIRSTOW METHOD

DANIEL FERNANDO RODRIGUEZ

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

The synthetic division can be extended to quadratic factors:

and even by multiplying the coefficients is obtained:

Bairstow Method

01 ...)( brxbresiduo

RxQsrxxxP nn )()()( 22

residuobxbxbxbsrxx nn

nn

233

122 ...

2100

3211

1233

122

11

2100

3211

1233

122

11

1

::

sbrbab

sbrbab

sbrbab

sbrbab

rbab

ab

sbrbba

sbrbba

sbrbba

sbrbba

rbba

ba

nnnn

nnnn

nnn

nn

nnnn

nnnn

nnn

nn

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

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

...)*()*(),(*)*,(

...)*()*(),(*)*,(

0000

1111

kkkk

kkkk

sss

brr

r

bsrbsrb

sss

brr

r

bsrbsrb

ss

br

r

bbsrb

ss

br

r

bbsrb

0000

1111

0*)*,(

0*)*,(

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

)2()1(

122

11

:

knknknkn

nnnn

nnn

nn

scrcbc

scrcbc

rcbc

bc

Thus, the system of equations can be written

Bairstow Method

021

132

bscrc

bscrc

22

200

12

110

33

321

23

221

cs

bsb

s

br

s

bc

r

bsb

r

br

r

b

cs

bsb

s

br

s

bc

r

bsb

r

br

r

b

Calculation of approximate error:

When tolerance is reached estimated coefficients

r and s is used to calculate the roots:

Bairstow Method

%100.%100. ,, s

s

r

rsara

2

42 srrx

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

a

acbbx

2

42

CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002.

http://ocw.mit.edu/OcwWeb/Mathematics

PPTX EDUARDO CARRILLO, PHD.

Bibliography