SUMMARY

The following table shows various methods for finding roots in increasing order of relative efficiency.  All methods were started at x0=1.23. 

Performance of various iteration formulas
tested on 66644 random equations
method iterators success
  exist ratio
Inverse quadratic interp 100.00% 32.45%
Newton 100.00% 34.36%
Secant 100.00% 37.40%
W(x) iterators 4.00% 1.83%
Euler (Direct quadratic interp) 100.00% 47.14%
Halley 100.00% 48.58%
Secant (3 point) 100.00% 48.69%
Ostrowski 100.00% 51.73%
Rational polynomial iterators 21.60% 11.64%
Homotopy 100.00% 58.10%
Polynomial iterators 69.96% 44.64%
Homotopy (Euler direction) 100.00% 65.68%
Simple iterators 96.33% 69.50%
Spiral 100.00% 74.39%
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