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ME 142Engineering Computation I
Root Finding
&
Iterative Solutions
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Key Concepts
Find Roots by PlottingFind Roots by Bisection RoutineFind Roots by using Goal Seek
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Using a Graph to Find Roots
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How do you plot a function in Excel?
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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
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Example ProblemFind all the roots for the following
equation: Plot (approximate)
42 2 xy
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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
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Bisection Routine
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What do we mean by iterative solutions?
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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
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Example ProblemFind all the roots for the following
equation: Bisection
42 2 xy
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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
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Goal Seek Function
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What is Goal Seek?
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Goal Seek Function
Begin with initial guess “x” value
Define function, f(x), related to initial guess Use Goal Seek function to find root
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Example ProblemFind all the roots for the following
equation: Goal Seek
42 2 xy
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Goal Seek Function
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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
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Review Questions
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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
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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
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Review QuestionUsing Goal Seek
Goal Seek is the only Excel function available for solving problems that are iterative in nature.
A. True
B. False
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Homework Help ‘n Hints