Upload
daferro
View
1.944
Download
3
Embed Size (px)
Citation preview
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