ME 142 Engineering Computation I

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Presentation transcript:

ME 142 Engineering Computation I Root Finding & Iterative Solutions

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

Using a Graph to Find Roots x 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

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

Bisection Routine Determine subinterval of root x 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 -0.4375 -1.375 0.21875 -0.109375 -1.4375 -0.13281 0.06640625 -1.40625 0.044922 -0.005966187 -1.421875 -0.04346 0.005771637

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

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