Roots of Equations Roots of Polynomials

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1 Roots of Equations Roots of Polynomials

2 Content Roots of Polynomials Birge Vieta Method Lin Bairstow Method

3 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.

4 Roots of Polynomials 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 residue where either, or the degree of is less than the degree of. Example: 10 8 divided by

5 Roots of Polynomials Reminder Theorem The reminder obtained by dividing by is the value of. Proof. Dividing by using the division algorithm: evaluate on

6 Roots of Polynomials Factor Theorem Every 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:

7 Birge Vieta Method To find the real roots of by the Birge Vieta method, using estimate and using synthetic division form as follows:

8 Birge Vieta Method then compute improved estimate of root by Newton Raphson iterative formula.

9 Birge Vieta Method Summary of Birge Vieta Method 1. 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

10 Birge Vieta Method 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 3 If, go to failure to converge exit. 5. Replace by That is, replace by,

11 Birge Vieta Method 6. If, set and return to Step 2. If, set and go to Step Calculate the th (last) root of original equation by solving linear equation, i.e.,. 8. Output. Write out roots, of.

12 Birge Vieta Method Example: Find the roots of the following polynomial:

13 Birge Vieta Method Solution:

14 Birge Vieta Method

15 Birge Vieta Method

16 Birge Vieta Method Dividing by we obtain the quadratic polynomial:

17 Birge Vieta Method The roots of this polynomial are then computed by the quadratic formula: Result:

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20 Lin Bairstow Method 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 differentialcorrection procedure.

21 Lin Bairstow Method 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:

22 Lin Bairstow Method 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 and Differentiating,,, with respect to and, respectively, we find,, where

23 Lin Bairstow Method The number of computations required in each iteration of the Bairstow method can be reduced using the relation and the differentialcorrection equations and can be simplified by this relation to the form The terms can be calculated using synthetic division by quadratic form as follows:

24 Lin Bairstow Method Summary of Birge Vieta Method 1. 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.

25 Lin Bairstow Method 4. Divide by, and compute,. then 5. Calculate partial derivatives,,,. then

26 Lin Bairstow Method 6. Solve differential correction equations for,. 7. Calculate improvised values of roots,,.

27 Lin Bairstow Method 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 Write output. Write out coefficients and roots.

28 Lin Bairstow Method Example: Calculate roots of:

29 Lin Bairstow Method Solution: Using synthetic division by quadratic form:

30 Lin Bairstow Method

31 Lin Bairstow Method

32 Lin Bairstow Method

33 Lin Bairstow Method

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36 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:

37 Computer Program 5 (by team) Submit a computer program that compute the roots of a polynomial by the following methods: a) Birge Vieta Method b) Lin Bairstow Method Hand over: Computational algorithm (printed) Source Code (printed and file) Executable (file)

38 Roots of Equations Roots of Polynomials