1. ME 142 Engineering Computation I Root Finding & Iterative Solutions

2. Key Concepts Find Roots by Plotting Find Roots by Bisection Routine Find Roots by using Goal Seek

3. Using a Graph to Find Roots

4. How do you plot a function in Excel?

5. Using a Graph to Find Roots  Approximate process  May be used as a starting process to an iterative method  May want to refine axis max/min values to focus on plot on area of interest

6. Example Problem  Find all the roots for the following equation: Plot (approximate)

7. Using a Graph to Find Roots xy -10-196 -9-158 -8-124 -7-94 -6-68 -5-46 -4-28 -3-14 -2-4 2 04 12 2-4 3-14 4-28 5-46 6-68 7-94 8-124 9-158 10-196

8. Bisection Routine

9. What do we mean by iterative solutions?

10. Bisection Routine 1.Begin with initial guess: x low, x hi 2.Calculate corresponding f(x): f(x low ), f(x hi ) f(x low )*f(x hi )<0, to ensure root lies within interval 3.Estimate Root and calculate corresponding f(x root ) 4.Determine subinterval of root If f(x low )*f(x root )<0, set x hi =x root If f(x low )*f(x root )>0, set x low =x root 5.Repeat Steps 3-4 until desired tolerance is reached

11. Example Problem  Find all the roots for the following equation: Bisection

12. Bisection Routine xy -10-196 -9-158 -8-124 -7-94 -6-68 -5-46 -4-28 -3-14 -2-4 2 Xlowf(Xlow)Xhighf(Xhigh)Xrootf(Xroot)f(Xlow)*f(Xroot) -2-42-1.5-0.52 -1.5-0.52-1.250.875-0.4375 -1.5-0.5-1.250.875-1.3750.21875-0.109375 -1.5-0.5-1.3750.21875-1.4375-0.132810.06640625 -1.4375-0.13281-1.3750.21875-1.406250.044922-0.005966187 -1.4375-0.13281-1.406250.044922-1.421875-0.043460.005771637 Determine subinterval of root If f(x low )*f(x root )<0, set x hi =x root If f(x low )*f(x root )>0, set x low =x root

13. Goal Seek Function

14. What is Goal Seek?

15. Goal Seek Function  Begin with initial guess “x” value  Define function, f(x), related to initial guess  Use Goal Seek function to find root

16. Example Problem  Find all the roots for the following equation: Goal Seek

17. Goal Seek Function

18.  Accuracy of Goal Seek function can be controlled in Excel Options  You may need to set the following: Maximum Iterations Maximum Change

19. Review Questions

20. Review Question Finding Roots by Plotting A third order polynomial has how many real roots? A.0 B.1 C.2 D.3 E.Cannot tell from info given

21. Review Question Bisection Routine Which of the following are false statements concerning the bisection method? A.The true root must be between the initial high/low guesses B.The estimated x root is calculated by averaging the x high and x low values C.The subinterval of the true route is determined by a sign comparison of f(x) values D.All of the above are true

22. Review Question Using Goal Seek Goal Seek is the only Excel function available for solving problems that are iterative in nature. A.True B.False

23. Homework Help ‘n Hints