Civil Engineering Computation. Numerical Methods : Root Finding and Optimization

Transcription

1 Civil Engineering Computation Numerical s : Root Finding and Optimization

2 ROOT LOCATION WITH NUMERICAL METHODS 2

3 Root Location While Goal Seek provides us with a terrific tool to solve an equation, it is a bit of a black box method We really don t know how it does its job. 3

4 Root Location There may be times when you really want to locate a specific root of an equation and Goal Seek doesn t give you that type of control It is possible to program a numerical method in VBA that will give you better control and allow you to position the region that you are going to search for a root. 4

5 The Bisection method takes advantage of the fact that the function we are searching must change sign as you move through a root. The Bisection method is a search method in which the interval is always divided in half. 5

6 6 If a function changes sign over an interval, the function value at the midpoint is evaluated. The location of the root is then determined as lying within the subinterval where the sign change occurs. The subinterval then becomes the interval for the next iteration. The process is repeated until the root is known to the required precision.

7 cha01102_ch05_ qxd 12/17/10 8:01 AM Page BISECTION 135 f(m) m Root!2!4!6 xl xr xu First iteration xl xr xu Second iteration xl xr xu Third iteration xl xr xu Fourth iteration 7 FIGURE 5.5 A graphical depiction of the bisection method. This plot corresponds to the first four iterations from Example 5.3.

8 Back to the bungee jumper problem when we ignored the buoyance force we found a closed form solution for the velocity at any time t. v= 8 gcd gm tanh t cd m

9 The problem is to determine the mass of the bungee jumper (m) such that a velocity (v) of 36 m/s is attained after a time (t) of 4 s. The acceleration of gravity (g) is 9.81 m/s2 and the drag coefficient (cd) is 0.25 kg/m. v= 9 gcd gm tanh t cd m

10 Velocity Difference (m/s) Jumper Mass (kg)

11 11 From plotting the difference between the desired velocity and the actual velocity, we can make a good estimate that the mass value is between 130 and 150 kg

12 12 Since the sign of the difference in velocities goes through 0 (assuming a continuous function), the mass where the velocity difference is 0 should be between 130 and 150 kg

13 13 We can make the lower estimate for the root (the value where the function where be equal to 0) as 130 kg and the upper estimate for the root as 150 kg

14 14 Based on these estimates, the next estimate for the root will be the average of the upper and lower estimates (remember these are the root values not the function values)

15 This would mean that the next estimate for the root would be 140 kg. If the tolerance for the function evaluation at the new estimate was good enough, we could quit

16 In this case the function evaluation at 140 kg is so we will continue. We need to look at the original function evaluations and find the one whose sign our new function evaluation matches

17 17 f(130)=-0.28 f(150)=0.14 f(140)=-0.06 An easy way to find which it matches if to multiply the function evaluation at each end by the function evaluation at the midpoint It will match the one that gives a positive value for the multiplication

18 f(130)*f(140)=0.016 f(150)*f(140)= In this case, the sign of the middle point evaluation is the same as the sign of the first point It will be easier to call then xlower, xmid, and xupper

19 We will replace either the upper or lower estimate with the middle estimate. The one with the matching sign for the evaluation will be replaced

20 If the sign of f(xupper)*f(xmid) > 0 then l xupper=xmid Else l xlower=xmid 20 End if

21 In our example, xmid (140) would replace xlower (130) Now we restart the process with the new limits. l xlower=140 l xupper=150 l xmid=(xlower+xupper)/2=

22 Our new estimate for the root would be 145 which evaluates to 0.05 If this was within tolerance, we could stop. If not we check signs of the function evalutions

23 23 Since the sign of f(xmid) and f(xupper) are the same we replace xupper with xmid and begin the process again until we reach a tolerance that is acceptable

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