1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 10 Roots of Polynomials

2. Objective Calculate Roots of a polynomial There are n complex or real roots If n odd, at least one real root Complex roots exist in conjugate pairs

3. Polynomial Evaluation Nested Form

4. Polynomial Evaluation 6 Multiplications 4 additions 3 Multiplications 3 additions

5. Polynomial Evaluation In general n(n+1)/2 Multiplications n Additions n Multiplications n Additions

6. Evaluating the Derivative

9. Algorithm Store Coefficients in Vector Form Evaluate starting from Most Inner Parenthesis

10. Algorithm Continue by adding terms

11. Pseudo Code p=0 df=0 DO j=n,0,-1 df = df * x + p p = p * x + a(j) ENDDO Must be evaluated First

12. Polynomial Deflation Can also be expressed in factored form

13. Polynomial Deflation Divide by any of the factors, e.g. (x+3)

14. Deflation – Division by monomial (x-t) Store Coefficients in Vector Form

15. Deflation – Division by monomial (x-t) Do i=n-1, 0, -1 EndDo

16. Polynomial Division

17. Muller’s Method

18. Muller’s - Algorithm Define Initial Guesses x o, x 1, x 2 (1) (2) For Each Iteration

19. Muller’s - Algorithm (3) (4)

20. Muller’s - Algorithm (5) (6) (7)

21. Muller’s - Algorithm (8) Compute New Estimate Two Roots: Choose sign that agrees with sign of b

22. Muller’s - Algorithm (9) Compute Error If converged FINISH

23. Muller’s - Algorithm (10) Assign new guesses (A) For Real Roots ONLY Choose the two points that are closest to x 3 (B) For ALL Roots NEXT ITERATION (11)