Topic 4b. Open Methods for Root Finding

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1 Course Instructor Dr. Ramond C. Rump Oice: A 337 Phone: (915) E Mail: rcrump@utep.edu Topic 4b Open Methods or Root Finding EE 4386/5301 Computational Methods in EE Outline Open Methods or Root Finding Newton Raphson Method Secant Method Multiple Roots 2 1

2 Newton Raphson Method 3 We Start with An Function 4 2

3 We Make an Initial Guess or the Position o the Root Evaluate the Function at

4 Calculate the Equation o the Line Tangential to the Point on () m Calculate Where the Line Crosses the Ais

5 Evaluate the Function at Calculate the Equation o the Line Tangential to the Point on () m

6 Calculate Where the Line Crosses the Ais Evaluate the Function at

7 And so on Converged in ive iterations. 13 Algorithm 1. Derive analtical epressions or () and (), or ()/ (). 2. Determine a good initial guess i. 3. Iterate until converged 1. Calculate i. i i i 2. Calculate new estimate or root i+1. i 1 i i 3. I i < tolerance, Done! 14 7

8 Poor or Unstable Convergence 15 Notes on Newton Raphson Method Does not require bounds. Requires a good initial guess. Requires () and () to be analtical. Converges etremel ast or unctions that are near linear. Algorithm vulnerable to instabilit Can converge to the wrong root i multiple roots eist. Newton Raphson s Method Newton s Method NRM is or root inding, whereas NM is or optimization. 16 8

9 Eample #1 Let () = sin. What is the root o () in the proimit o = 4? Step 1 Derive analtical epression or ()/ () sin sin sin tan d sin cos d This means our update equation is i 1 i i i tan i Step 2 MATLAB code r = 4; tol = 1e-6; d = in; Converges to while abs(d) > tol ater 4 iterations. d = tan(r); r = r - d; end Slide 17 Secant Method 18 9

10 We Start with Some Function 19 We Make Two Initial Guesses or the Position o the Root, 1 and

11 Evaluate the Function at 1 and Calculate the Equation o the Line Connecting ( 1, 1 ) and ( 2, 2 ) rise slope m run m

12 Calculate Where the Line Crosses the Ais r r r Evaluate the Function at the New Point r 1 2 r r

13 Adjust Points to 1 and r r Calculate Where the New Line Crosses the Ais r

14 Evaluate the Function at the New Point r 1 2 r r Adjust Points to 1 and

15 And so on Algorithm 1. Determine two good initial guesses: 1 and Evaluate the unction at Iterate until converged 1. Evaluate unction at Calculate Make new irst point the old second point. 4. Calculate new and I < tolerance, Done! 30 15

16 Notes on Secant Method Does not require bounds. Does not require () or () to be analtical Requires two good initial guesses Full numerical version o Newton s method Same weaknesses as Newton Raphson method Algorithm vulnerable to instabilit Can converge to the wrong root 31 Eample #1 Let () = sin. What is the root o () in the proimit o = 4? 1. Determine two good initial guesses: 1 =4.0 and 2 = Evaluate the unction at Iterate until converged a. Evaluate unction at b. Calculate % DASHBOARD unc 1 = 4.0; 2 = 3.9; tol = 1e-6; % IMPLEMENT SECANT METHOD 1 = unc(1); d = in; while abs(d) > tol 2 = unc(2); d = (2-1)*2/(2-1); 1 = 2; 1 = 2; 2 = 2 - d; end r = 2; c. Make new irst point the old second point. 1 2 and 1 2 d. Calculate new e. I < tolerance, Done! Converges to ater 5 iterations. Slide 32 16

17 Multiple Roots 33 Problem with Multiple Roots Our update equation or both Newton Raphson method and secant method involve ()/ (). When we have multiple roots, both () and () can go to zero at the root. This causes a divide b zero problem. single root 0 undeined 0 double root triple root Slide 34 17

18 A Useul Propert We can deine an auiliar unction u() that will have the same roots as () but who s derivative u () will not go to zero at the roots. u u This is the same auiliar unction we used or bracketing methods. Slide 35 The Fi I () has multiple roots, perorm root inding on the auiliar unction u() instead. u i u i i1 i i1 i i We can write our new update equation completel in terms o () and its derivatives as ollows. i i i1 i 2 i i i i Slide 36 18

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