roots of equations roots of polynomials - tec · 2015-03-19 · suppose that initial estimated n 4,...

Roots of Equations‐ Roots of Polynomials ‐

Content

• Roots of Polynomials• Birge‐Vieta Method• Lin‐Bairstow Method

Roots of Polynomials

• The methods used to get all roots (real and complex) of a polynomial are:• Birge‐Vieta method• Lin‐Bairstow method

• A polynomial of degree " " is of the form⋯

Fundamental Theorem of Algebra• Every algebraic (polynomial) equation with complex coefficients has at least one real or complex root.


Division Algorithm• If and are two polynomials of and , then polynomials and can be found which satisfy the relation

• is the dividend• is the quotient• is the divisor• is the residuewhere either , or the degree of is less than the degree of .

• Example:10 8 divided by 310 8 4 2 3 14


Reminder Theorem• The reminder obtained by dividing by is the value of

.• Proof. Dividing by using the division algorithm:

evaluate on


Factor TheoremEvery polynomial equation of the form:

⋯has at most distinct roots , .If is such a root, i.e., if , then by the remainder theorem:

simultaneously, if is a root of (which is the root of ) such that

This process is continued until we obtain

and by successive substitution of we obtain:⋯

Birge‐Vieta Method

To find the real roots of⋯

by the Birge‐Vieta method, using estimate

and using synthetic division form as follows:


then compute improved estimate of root by Newton‐Raphson iterative formula.


Summary of Birge‐Vieta Method1. Data input and initialization. Read parameters (degree), (maximum number of

iterations), (convergence term), (initial estimate of root), (if , offset by increment ), and coefficients ( , ) of . Set root counter

.2. Calculate degree of current polynomial , where . Set initial

estimate of root . Reset Newton‐Raphson iteration counter .3. a) Calculate nested terms

b) Calculate derivatives


4. Calculate improved estimate of root , by Newton‐Raphson.

where and ′

Test convergence of root (also test if )If , test iteration counter .If , set , set , return to Step 3If , go to “failure to converge” exit.

5. Replace by⋯

That is, replace by ,


6. If , set and return to Step 2.If , set and go to Step 7.

7. Calculate the th (last) root of original equation by solving linear equation , i.e., .

8. Output. Write out roots , of .


Example:Find the roots of the following polynomial:


Solution:


• Dividing by we obtain the quadratic polynomial:


• The roots of this polynomial are then computed by the quadratic formula:

• Result:

Lin‐BairstowMethod

• The Lin‐Bairstow method is an iterative procedure for calculating the roots (real or complex) of a real‐coefficient polynomial equation while requiring only the manipulation of real numbers in the computations.

• The method is based on successive extractions of quadratic factors , , … of the original polynomial of degree and

from succeeding factor polynomials of degree .• Each quadratic factor is determined by an interactive differential‐correction procedure.


If is divided by a trial quadratic factor , where and are arbitrary real constants, we obtain

In expanded form, this equation can be written as⋯

⋯

where are waste.Equating coefficients we obtain:


Suppose that initial estimated , of the roots of system of equations are known. If these initial values are increased respectively by small changes and , then first‐order approximations of the resulting changes in the functions , and

, respectively are given by the total differential equations,,

If we define

andDifferentiating , , … , with respect to and , respectively, we find

, ,

where


The number of computations required in each iteration of the Bairstowmethod can be reduced using the relation and the differential‐correction equations and can be simplified by this relation to the form

The terms can be calculated using synthetic‐division‐by‐ quadratic form as follows:


Summary of Birge‐Vieta Method1. Input and initialization. Read parameters: degree = , initial values

, , convergence criteria = . Read coefficients , of . Set index ( number of quadratic factors extracted).

Set index ( root‐pairs counter).2. Calculate degree of current polynomial. reset Newton

interaction counter . Reset , to initial values , .3. Test degree . If , go to step 4. If , go to step 3b. If , go

to step 3a.a) Calculate root of linear equation ; go to Step 10.b) Calculate root pair , of ; go to Step 10.


4. Divide by , and compute , .

then

5. Calculate partial derivatives , , , .

then


6. Solve differential‐correction equations for , .

7. Calculate improvised values of roots , , .


8. Test for convergence of differential corrections.a) If both and , calculate root pair , of the quadratic

, go to Step 9.b) If either and , text index . If , increase by 1

and go to step 3. If , go to " failure to convergence“ exit.

9. Replace by , i.e. replace by , . Increment quadratic factor counter by 1. Increase root‐pair counter by 2. Return to Step 2.

10.Write output. Write out coefficients and roots .


Example:Calculate roots of:


Solution:Using synthetic‐division‐by‐quadratic form:

Homework 6 (Individual)

1. Calculate the roots of the following polynomial function by the method of Birge‐Vieta:

2. Calculate the roots of the following polynomial function by the method of Lin‐Bairstow:

Computer Program 5 (by team)

• Submit a computer program that compute the roots of a polynomial by the following methods:a) Birge‐Vieta Methodb) Lin‐Bairstow Method

• Hand over:• Computational algorithm (printed)• Source Code (printed and file)• Executable (file)

Roots of Equations‐ Roots of Polynomials ‐

roots of equations roots of polynomials - tec · 2015-03-19 · suppose that initial estimated n 4,...

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