Bairstow Method

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Bair stow Method

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Bairstow Method Bairstow Method is an iterative method used to find both the real and com complex plex roots of a polynomial. polynom ial. It is based on the idea of synthetic division of the given polynomial by a quadratic function and can be used to find all the roots of a polynomial. Given a polynomial polynomial say, (B.1)

Bairstow's method method divides the polynom polynomial ial by a quadratic function. (B.2)

Now the quotient will be a polynomial

(B.3)

and the and  the remainder remainder is a linear function

, i.e.

(B.4)

Since the quotient division the co-efficients

and the rem remainder ainder

are obtained by standard synthetic

can be obtained by the following recurrence

relation. (B.5a)

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Bairstow Method

(B.5b)

for 

If

is an exact factor of

real/complex roots of

then the remainder

are the roots of

considered based on some guess values for determining the values of r and s such that

(B.5c)

is zero and the

. It may be noted that

is

. So Bairstow's method reduces to is zero. For finding such values Bairstow's

method uses a strategy similar to Newton Raphson's method. Since both

and

are functions of r and s we can have Taylor series expansion of

,

as:

(B.6a)

(B.6b)

For neglected, so that

,

terms

i.e. second and higher order terms may be

the improvement over guess value

may be obtained by

equating (B.6a),(B.6b) to zero i.e.

(B.7a)

(B.7b)

To solve the system of equations

, we need the partial derivatives of

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Bairstow Method

w.r.t. r and s. Bairstow has shown that these partial derivatives can be obtained by synthetic division of , which amounts to using the recurrence relation replacing

with

and

with

i.e.

(B.8a)

(B.8b)

(B.8c)

for where

(B.9)

 The system of equations (B.7a)-(B.7b) may be written as. (B.10a)

(B.10b)

These equations can be solved for to

and turn be used to improve guess value

.

Now we can calculate the percentage of approximate errors in (r,s) by

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Bairstow Method

If

or

, where

is the iteration stopping error, then we repeat the

process with the new guess i.e.

. Otherwise the roots of

can be

determined by

(B.12)

If we want to find all the roots of

then at this point we have the following three

possibilities: 1. If the quotient polynomial

is a third (or higher) order polynomial then we can

again apply the Bairstow's method to the quotient polynomial. The previous values of can serve as the starting guesses for this application. 2. If the quotient polynomial remaining two roots of

is a quadratic function then use (B.12) to obtain the .

3. If the quotient polynomial

is a linear function say

then the remaining

single root is given by Example: Find all the roots of the polynomial

by Bairstow method . With the initial values Solution: Set iteration=1

Using the recurrence relations (B.5a)-(B.5c) and (B.8a)-(B.8c) we get

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Bairstow Method

the simultaneous equations for

and

are:

on solving we get

and

Set iteration=2 

now we have to solve

On solving we get

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Bairstow Method

Now proceeding in the above manner in about ten iteration we get

Now on using

with

we get

So at this point Quotient is a quadratic equation

Roots of

are:

Roots

are

i.e Exercises: (1) Use initial approximation form

to find a quadratic factor of the

of the polynomial equation

using Bairstow method and hence find all its roots. (2) Use initial approximaton

to find a quadratic factor of the form

of the polynomial e quation

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Bairstow Method

using Bairstow method and hence find all the roots.

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