Question 10.6: It is useful to compare the effective ratios for the three l......
It is useful to compare the effective ratios for the three line searches discussed in this chapter.
Recalling Equations (10.9), (10.12), and (10.15),
\ R_m(uniform)=\frac{2}{m-1}. (10.9)
\ R_m(thirds)=(\frac{2}{3})^{m/2} (10.12)
\ R_m(golden ratio)=(\frac{-1+\sqrt{5}}{2})^{m-1} (10.15)
\ R_m(uniform)=\frac{2}{m-1},
\ R_m(thirds)=(\frac{2}{3})^{m/2}\approx (0.816)^m,
\ R_m(golden ratio)=\alpha^{m-1} \approx (0.618)^{m-1}.
These ratios are tabulated in Table 10.4 for m = 2, 3, 4, 5, 6 computations. Since the effective ratio is best kept small, it is evident that the golden ratio algorithm is, in general, the superior method.
TABLE 10.4 Comparative Effective Ratios for Search
Algorithms
\ \begin{array}{c} \hline &&Effective ratio\\ \hline Iteration m&uniform&thirds&golden ratio\\\hline 2&2.000&0.666&0.618\\3&1.000&0.543&0.382\\4&0.667&0.443&0.236\\5&0.500&0.362&0.146\\6&0.400&0.295&0.090\\\hline \end{array}