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

2. 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

3. Using a Graph to Find Rootsx y
-10
-196 -9
-158 -8
-124 -7
-94 -6
-68 -5
-46 -4
-28 -3
-14 -2 -1 2 4 1 3 5 6 7 8 9 10

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

5. Bisection Routine Determine subinterval of rootx y
-10
-196 -9
-158 -8
-124 -7
-94 -6
-68 -5
-46 -4
-28 -3
-14 -2 -1 2
Determine subinterval of root
If f(xlow)*f(xroot)<0 , set xhi=xroot
If f(xlow)*f(xroot)>0, set xlow=xroot
Xlow
f(Xlow)
Xhigh
f(Xhigh)
Xroot
f(Xroot)
f(Xlow)*f(Xroot) -2 -4 -1 2
-1.5
-0.5
-1.25
0.875
-1.375

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

7. Goal Seek Function

8. 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