ME 142Engineering Computation I
Root Finding
&
Iterative Solutions
Key Concepts
Find Roots by PlottingFind Roots by Bisection RoutineFind Roots by using Goal Seek
Using a Graph to Find Roots
How do you plot a function in Excel?
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
Example ProblemFind all the roots for the following
equation: Plot (approximate)
42 2 xy
Using a Graph to Find Roots42 2 xy
x y-10 -196-9 -158-8 -124-7 -94-6 -68-5 -46-4 -28-3 -14-2 -4-1 20 41 22 -43 -144 -285 -466 -687 -948 -1249 -15810 -196
-15 -10 -5 0 5 10 15
-250
-200
-150
-100
-50
0
50
y
y
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-5
-4
-3
-2
-1
0
1
2
3
4
5
y
y
Bisection Routine
What do we mean by iterative solutions?
Bisection Routine
1. Begin with initial guess: xlow, xhi
2. Calculate corresponding f(x): f(xlow), f(xhi) f(xlow)*f(xhi)<0, to ensure root lies within interval
3. Estimate Root and calculate corresponding f(xroot)
4. Determine subinterval of root If f(xlow)*f(xroot)<0 , set xhi=xroot
If f(xlow)*f(xroot)>0, set xlow=xroot
5. Repeat Steps 3-4 until desired tolerance is reached
2hilow
root
xxx
Example ProblemFind all the roots for the following
equation: Bisection
42 2 xy
Bisection Routinex y-10 -196-9 -158-8 -124-7 -94-6 -68-5 -46-4 -28-3 -14-2 -4-1 2
Xlow f(Xlow) Xhigh f(Xhigh) Xroot f(Xroot) f(Xlow)*f(Xroot)
-2 -4 -1 2 -1.5 -0.5 2
-1.5 -0.5 -1 2 -1.25 0.875 -0.4375
-1.5 -0.5 -1.25 0.875 -1.375 0.21875 -0.109375
-1.5 -0.5 -1.375 0.21875 -1.4375 -0.13281 0.06640625
-1.4375 -0.13281 -1.375 0.21875 -1.40625 0.044922 -0.005966187
-1.4375 -0.13281 -1.40625 0.044922 -1.421875 -0.04346 0.005771637
2hilow
root
xxx
Determine subinterval of rootIf f(xlow)*f(xroot)<0 , set xhi=xroot
If f(xlow)*f(xroot)>0, set xlow=xroot
42 2 xy
Goal Seek Function
What is Goal Seek?
Goal Seek Function
Begin with initial guess “x” value
Define function, f(x), related to initial guess Use Goal Seek function to find root
Example ProblemFind all the roots for the following
equation: Goal Seek
42 2 xy
Goal Seek Function
Goal Seek Function
Accuracy of Goal Seek function can be controlled in Excel Options
You may need to set the following: Maximum Iterations Maximum Change
Review Questions
Review QuestionFinding 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
Review QuestionBisection 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 xroot is calculated by averaging the xhigh and xlow 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
Review QuestionUsing Goal Seek
Goal Seek is the only Excel function available for solving problems that are iterative in nature.
A. True
B. False
Homework Help ‘n Hints