1. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Part 2 / Chapter 5

2. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2 Part 2 Roots of Equations Why? But

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3 Nonlinear Equation Solvers Bracketing Bisection False Position (Regula-Falsi ) GraphicalOpen Methods Newton Raphson Secant All Iterative

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 4 Figure PT2.1

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 5 Bracketing Methods (Or, two point methods for finding roots) Chapter 5 Two initial guesses for the root are required. These guesses must “bracket” or be on either side of the root. == > Fig. 5.1 If one root of a real and continuous function, f(x)=0, is bounded by values x=x l, x =x u then f(x l ). f(x u ) <0. ( The function changes sign on opposite sides of the root )

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6 Figure 5.2

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 7 Figure 5.3

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 8 Figure 5.4a Figure 5.4b Figure 5.4c

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 9 The Bisection Method For the arbitrary equation of one variable, f(x)=0 1.Pick x l and x u such that they bound the root of interest, check if f(x l ).f(x u ) <0. 2.Estimate the root by evaluating f[(x l +x u )/2]. 3.Find the pair If f(x l ). f[(x l +x u )/2]<0, root lies in the lower interval, then x u =(x l +x u )/2 and go to step 2.

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10 If f(x l ). f[(x l +x u )/2]>0, root lies in the upper interval, then x l = [(x l +x u )/2, go to step 2. If f(x l ). f[(x l +x u )/2]=0, then root is (x l +x u )/2 and terminate. 4.Compare  s with  a 5.If  a <  s, stop. Otherwise repeat the process.

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 11 Figure 5.6

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 12 Evaluation of Method Pros Easy Always find root Number of iterations required to attain an absolute error can be computed a priori. Cons Slow Know a and b that bound root Multiple roots No account is taken of f(x l ) and f(x u ), if f(x l ) is closer to zero, it is likely that root is closer to x l.

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13 How Many Iterations will It Take? Length of the first IntervalL o =b-a After 1 iterationL 1 =L o /2 After 2 iterationsL 2 =L o /4 After k iterationsL k =L o /2 k

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14 If the absolute magnitude of the error is and L o =2, how many iterations will you have to do to get the required accuracy in the solution?

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 15 The False-Position Method (Regula-Falsi) If a real root is bounded by x l and x u of f(x)=0, then we can approximate the solution by doing a linear interpolation between the points [x l, f(x l )] and [x u, f(x u )] to find the x r value such that l(x r )=0, l(x) is the linear approximation of f(x). == > Fig. 5.12

16. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 16 Procedure 1.Find a pair of values of x, x l and x u such that f l =f(x l ) 0. 2.Estimate the value of the root from the following formula (Refer to Box 5.1) and evaluate f(x r ).

17. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 17 3.Use the new point to replace one of the original points, keeping the two points on opposite sides of the x axis. If f(x r ) f l =f(x r ) If f(x r )>0 then x u =x r == > f u =f(x r ) If f(x r )=0 then you have found the root and need go no further!

18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 18 4.See if the new x l and x u are close enough for convergence to be declared. If they are not go back to step 2. Why this method? –Faster –Always converges for a single root.  See Sec.5.3.1, Pitfalls of the False-Position Method Note: Always check by substituting estimated root in the original equation to determine whether f(x r ) ≈ 0.