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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,
xf(x)/f'(x)
66641 random equations 
method 
failures 
converged to a root 
34.36% 
converged to a nonroot 
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 (2point method)
66641 random equations 
method 
failures 
converged to a root 
37.40% 
converged to a nonroot 
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


e^{x0.001} 

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 x_{0}=1.23
equation 
iterator 
result 
e^{x}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 xx 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
19980708