Bairstow-Method.pdf

In the name of God

Lin-Bairstow Method

Compiled by Naser Bagheri Student ID : 9016393

Lin-Bairstow Method

http://math.fullerton.edu/mathews /n2003/bairstowmetho...

Module for The Th e Lin-Bai Bairsto rstow w Met Method hod

Quadrati Quad ratic c Synt Syntheti hetic c Divi Division sion

Let the polynomial

of degree n have coefficien coefficients ts

. Then

has the familiar

form

(1)

.

Let

be a fix fixed q qu uadratic te term. Then

(2)

,

where . Here

is th the re remainder wh when is a polynomial of degree

(3)

If we we s se et

can be expressed expressed as

and can be represented by

.

and

, then

(4)

where (5)

and equati equation on (4) can can be writ writte ten n

,

is di divided by by

Lin-Bairstow Method


The terms in (6) can be expanded so that

is represented in powers of x.

(7)

The numbers

are found by comparing the coefficients of

(7). The coefficients

(8)

Set

of

in equations (1) and

and are computed recursively.

and

,

,

and then

for

.

Proof Lin-Bairstow Method Lin-Bairstow Method

Example 1. Use quadratic synthetic division to divide by

.

Solution 1.

Heuristics

In the days when "hand computations" were necessary, the quadratic synthetic division tableau (or table) was used. The coefficients

of the polynomial are entered on the

first row in descending order, the second and third rows are reserved for the intermediate computation steps ( ,

and

and

) and the bottom row contains the coefficients .

Lin-Bairstow Method


Example 2. Use the "quadratic synthetic division tableau" to divide by

.

Solution 2.

Using vector coefficients

As mentioned above, it is efficient to store the coefficients of degree n in the vector index for

and the polynomial

of a polynomial

. Notice that this is a shift of the

is written in the form

.

Given the quadratic

, the quotient

and remainder

are

of

are

and .

The recursive formulas for computing the coefficients

, and

, and then

Lin-Bairstow Method


Example 3. Use the vector form of quadratic synthetic division to divide by

.

Solution 3.

The Lin-Bairstow Method

We now build on the previous idea and develop the Lin -Bairstow's method for finding a quadratic factor

of

. Suppose that we start with the initial guess

(9)

and that

can be expressed as

.

When

u

and

v

are small, the quadratic (9) is close to a factor of

new values

so that

(10)

is closer to a factor of Observe that

u

and

than the quadratic (9). v

are functions of r and

, and .

The new values

satisfy the relations

s,

that is

. We want to find

Lin-Bairstow Method


.

The differentials of the functions

u

and

v

are used to produce the approximations

and

The new values

are to satisfy

, and .

When the quantities

are small, we replace the above approximations with

equations and obtain the linear system:

(11)

All we need to do is find the values of the partial derivatives and

and then use Cramer's rule to compute

,

,

. Let us announce that the

values of the partial derivatives are

where the coefficients

are built upon the coefficients

calculated recursively using the formulas

(12)

Set

,

and

given in (8) and are

Lin-Bairstow Method


for

.

The formulas in (12) use the coefficients

in (8). Since

, and , and

the linear system in (11) can be written as

Cramer's rule can be used to solve this linear system. The required determinants are

, and

,

and the new values

.

are computed using the formulas

,

and .

Proof Lin-Bairstow Method Lin-Bairstow Method

The iterative process is continued until good approximations to found. If the initial guesses

r

and

s

have been

are chosen small, the iteration does not tend to

wander for a long time before converging. When

, the larger powers of x can be

neglected in equation (1) and we have the approximation

.

He

th initial

fo

uld be

and

vided that

Lin-Bairstow Method


If hand calculations are done, then the quadratic synthetic division tableau can be extended to form an easy way to calculate the coefficients

.

Bairstow's method is a special case of Newton's method in two dimensions.

Algorithm (Lin-Bairstow Iteration). To find a quadratic factor of approximation

.

Computer Programs Lin-Bairstow Method Lin-Bairstow Method Mathematica Subroutine (Lin-Bairstow Iteration).

given an initial

Lin-Bairstow Method

Example 4. Given


. Start with

Lin-Bairstow method to find a quadratic factor of

and

and use the

.

Solution 4.

Research Experience for Undergraduates Lin-Bairstow Method Lin-Bairstow Method Internet hyperlinks to web sites and a bibliography of articles.

Lin-Bairstow Method

http://math.fullerton.edu/mathews/n2003/bairstowmetho...

Example 1. Use quadratic synthetic division to divide

by

Solution 1. First, construct the polynomial

Second, given

.

construct the polynomials

.

.

Lin-Bairstow Method

Third, verify that


.

Lin-Bairstow Method


We are done.

Aside. We can have Mathematica compute the quotient and remainder using the built in procedures PolynomialQuotient and PolynomialRemainder. This is just for fun!

Lin-Bairstow Method

Example

http://math.fullerton.edu/mathews /n2003/ bairstowmetho...

2. Use

the "quadratic synthetic division tableau" to divide by

Solution

.

2.

Use the "quadratic synthetic division tableau"

Since

Then simplify and get.

Thus we have

and

we use

.

Lin-Bairstow Method

http://math.fullerton.edu/mathews /n2003/ bairstowmetho...

This agrees with our previous computation in Example

We are done.

Aside. We can let Mathematica verify the result .

1.

Lin-Bairstow Method

Example


3.

Use the vector form of quadratic synthetic division to divide by

Solution

.

3.

First, construct the polynomial

Second, given

.

construct the polynomials

.

Lin-Bairstow Method

Third, verify that


.

Lin-Bairstow Method

Example


4. Given

.

Start with

Lin-Bairstow method to find a quadratic factor of Solution

and

and use the

.

4.

Enter the coefficients of the polynomial.

Enter the starting values

and

Verify that a quadratic factor has been found.

and call the subroutine Bairstow.

Lin-Bairstow Method


We are done.

Aside. We can let Mathematica find the factors too. This is just for fun.

Aside. We can let Mathematica find the roots too. This is just for fun.

/home/naser/Darss/Analize-adadi/bros/Bairsto.cpp Page 1 of 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Sun 20 Oct 2013 11:52:32 PM IRST

//********************************************** // Programmer : Naser . Bagheri (9016393) //********************************************** #include #include u si ng n am es pa ce std ; #define ESP 0.001 #define F(x) ( x)*( x)*(x) + ( x) + 10 #define a3 1 #define a2 0 #define a1 1 #define a0 10 //#define c3 0void main() int main() { double u,v,u1,v1 ,u2 ,v2 ,b3,b2 ,p,b1 ,b0 ,c2,c1,c0 ,U,V; int i=1; float c3=0; co ut <<"\nEnter the value of u: " ; sca nf ("%lf",&u); co ut <<"\nEnter the value of v: " ; sca nf ("%lf",&v); b3 =a3 ; b2 =a2 +u*b3 ; b1 =a1 +u*b2 +v*b3; b0 =a0 +u*b1 +v*b2; c2 =b3 ; c1 =b2 +u*c2 +v*c3; c0 =b1 +u*c1 +v*c2; p=c1 *c1 -c0 *c2; U=((-(b1*c1-b0 *c2 ))/( p)); V=((-(b0*c1-c0 *b1 ))/( p)); u1 =u+U; v1 =v+V; co ut <<"\n\n b0 = %lf" <
do { u =u1 ; v =v1 ; b3 =a3; b2 =a2+u*b3; b1 =a1+u*b2+v*b3 ; b0 =a0+u*b1+v*b2 ; c2 =b3; c1 =b2+u*c2+v*c3 ; c0 =b1+u*c1+v*c2 ; p =c1 *c1-c0*c2 ; U =((-(b1 *c1 -b0*c2))/(p)); - 1 -

/home/naser/Darss/Analize-adadi/bros/Bairsto.cpp Page 2 of 3 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116

V =((-(b0 *c1 -c0*b1))/(p)); u2 =u+U; v2 =v+V; cou t << "\n\n b0 = %lf" << b0 ; co ut <<"\n\n b1 = %lf" <
if (fabs(u1 - u2 ) < ESP && fabs(v1 -v2) < ESP) { cou t <<"\n\nREAL ROOT = %.3lf" << u2; cou t <<"\n\nREAL ROOT = %.3lf" << v2; i=0; } else { u1 = u2 ; v1 = v2 ; } } while (i!= 0); } /* ---------------------------------OUT PUT ----------------------------------Enter the value of u: 1.8 Enter the value of v: -4 -----------------------------------------------b0 = 3.232000 b1 = 0.240000 b2 = 1.800000 b3 = 1.000000 c0 c1 c2 c3

= = = =

2.720000 3.600000 1.000000 0.000000

u = 2.031250 v = -5.072500 b0 b1 b2 b3

= = = =

-0.194891 0.053477 2.031250 1.000000

c0 c1 c2 c3

= = = =

3.656953 4.271250 1.000000 0.000000 - 2 -

Sun 20 Oct 2013 11:52:32 PM IRST

/home/naser/Darss/Analize-adadi/bros/Bairsto.cpp Page 3 of 3 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150

u = 2.002230 v = -5.002025 b0 b1 b2 b3

= = = =

-0.001389 0.006900 2.002230 1.000000

c0 c1 c2 c3

= = = =

6.121717 5.552230 1.000000 0.000000

u = 2.000623 v = -5.000003 b0 b1 b2 b3

= = = =

0.001859 0.002490 2.000623 1.000000

c0 c1 c2 c3

= = = =

11.224619 8.108540 1.000000 0.000000

u = 2.000287 v = -4.999767 REAL ROOT = 2.000 REAL ROOT = -5.000 */

- 3 -

Sun 20 Oct 2013 11:52:32 PM IRST

Bairstow 's Method

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Applied Mathematics > Numerical Methods > Root-Finding >

Bairstow's Method

THINGS TO TRY:

Discrete Mathematics Foundations of Mathematics

A procedure for finding the quadratic factors for the complex conjugate roots of a polynomial

approximate zero

with real coefficients.

7 rows of Pascal's triangle Geometry History and Terminology

(1) Now write the original polynomial as

Number Theory

(2) Probability and Statistics

(3)

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Now use the two-dimensional Newton's method to find the simultaneous solutions.

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REFERENCES: Wolfram Web Resources » 13,192 entries Last updated: Tue Oct 8 2013

Created, developed, and nurtured by Eric Weisstein at Wolfram Research

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 277 and 283-284, 1989.

Referenced on Wolfram|Alpha: Bairstow's Method CITE THIS AS: Weisstein, Eric W. "Bairstow's Method." From MathWorld --A Wolfram Web Resource. http://mathworld.wolfram.com /BairstowsMethod.html

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Bairstow-Method.pdf

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