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# Conclusions

We have also run the random tests against several classical methods for finding zeros. Table 7 shows the results for Newton's method and table 8 shows the results for the Secant method. These are shown for comparison, and we can see that both are successful less often and require more time.

Table 7: Newton's method, x-f(x)/f'(x)
 66641 random equations method failures converged to a root 34.36% converged to a non-root 2.95% failed 65.64% diverged 53.24% average time 1.595 outside domain 19.94% time per root 4.644 too many iterations 23.87% iterator fails .00%

Table 8: Secant method (2-point method)
 66641 random equations method failures converged to a root 37.40% converged to a non-root 4.46% failed 62.60% diverged 72.81% average time 2.122 outside domain 7.19% time per root 5.674 too many iterations 15.53% iterator fails .01%

It is appropriate to run the witness examples that we proposed in the introduction with this method. We think the results speak for themselves. The results for each equation are shown in tables 9, 10 and 11, where the columns are for each iterator and the rows have the value of each iteration. In all cases, at least one of the iteration gives the values which were considered difficult to find. Table 12 shows other examples, the first two taken from [10] and the rest have been collected by the Maple development group. All these problems had an iterator which was successful.

Table: Partial inverse iterators for solving
 ex0.001 1.23 1.23 1.23 1.23 .254976... 5.94285... 2.71884... -.348812...+.00109473...i 3.34053... 2.72100... .365255...-1.09527...i 2.77297... 2.72101... -1.31325...-.616913...i 2.72159... converged -2.14588...+3.31938...i 2.72101... -10.0449...-.0715578...i 2.72101... -23.2443...+31.2573...i converged diverged converged

Table: Partial inverse iterators for solving
 0.6 0.6 .632004... .571793... .668499... .546785... .710223... .524475... .757788... .504453... .811228... .486381... .868933... .469981... .925654... .455027... .970858... .441327... .994075... .428723... .999439... .417083... .999883... .406296... .999905... .396264... converged .386907... .378155... .369947... .362231... .354961... .348096... .341601... .335445... too many iterations

Table: Partial inverse iterators for solving
 0.6 0.6 0.6 0.6 3.14629356... 3.13689175... 2.43573211...i -2.43573211...i 3.14159299... 3.14159230... 2.52588218...i -2.52588218...i 3.14159300... 3.14159231... 2.52864327...i -2.52864327...i converged converged 2.52872814...i -2.52872814...i converged converged

Table 12: Examples from various sources, all examples are started from x0=1.23
 equation iterator result ex-6x converged to 2.83314... converged to .20448... -W(-1/6) is 0.20448... converged to .20448... diverged too many iterations too many iterations converged to 0.83102... converged to 3.45613... converged to 50.14201... diverged converged to 3.45613... diverged converged to 50.14201... falls outside valid domain e-32 x-x -1 converged to too many iterations converged to 3.443...+3.812...i

Next: Extension to systems of Up: Partial inverse heuristic for Previous: Inverting functions
Gaston Gonnet
1998-07-08