Optimalf 159
This abscissa is at .499439. The TWR corresponding to thisfvalue is
1.146363. When we encounter a difference in abscissas that is less than or
equal to TOL, we will have converged to the optimalf.
Shown here are the full seven passes and the values used in each pass
so that you may better understand this technique.
PARABOLIC INTERPOLATION
Pass# x1 y1 x2 y2 x3 y3 abscissa
1 0 0 0.995 0.017722 1 0 0.5
2 0 0 0.5 1.145833 0.995 0.017722 0.499439
3 0 0 0.499439 1.146363 0.5 1.145833 0.426923
4 0 0 0.426923 1.200415 0.499439 1.146363 0.410853
5 0 0 0.410853 1.208586 0.426923 1.200415 0.387431
6 0 0 0.387431 1.218059 0.410853 1.208586 0.375727
7 0 0 0.375727 1.22172 0.387431 1.218059 0.364581
8 0 0 0.364581 1.224547 0.375727 1.22172 0.356964
9 0 0 0.356964 1.226111 0.364581 1.224547 0.350489
Convergence is extremely rapid. Typically, the more peaked the curve
for the TWR (i.e., the more plays which comprise the TWR) the faster con-
vergence is attained.
Refer now to Figure 4.13. This graphically shows the parabolic interpo-
lation process for the coin-toss example with a 2:1 payoff, where the optimal
fis .25. On the graph, notice the familiarfcurve, which peaks out at .25.
The first step here is to draw a parabola through three points: A, B, and C.
The coordinates for A are (0, 0). For C the coordinates are (1, 0). For point
B we now pick a point whose coordinates lie on the fcurve itself. Once
parabola ABC is drawn, we obtain its abscissa (the f value correspond-
ing to the peak of the parabola ABC). We find what the TWR is for thisf
value. This gives us coordinates for point D. We repeat the process, this
time drawing a parabola through points A, B, and D. Once the abscissa to
parabola ABD is found, we can find the TWR that corresponds to anfvalue
of the abscissa of parabola ABD. These coordinates (fvalue, TWR) give us
point E.
Notice how quickly we are converging to the peak of the fcurve at
f=.25. If we were to continue with the exercise in Figure 4.12, we would
next draw a parabola through points E, B, and D, and continue until we
converged upon the peak of thefcurve.
One potential problem with this technique from a computer standpoint
is that the denominator in the equation that solves for the abscissa might