Substitute all non-polynomials.

Another possible way of constructing an iterator is to inspect the equation top-down and force the evaluation of any subexpression which is a special function, a power to a non-integer exponent, or anything else that is not algebraic. If after this is done, we are left with an expression containing x, this must be a rational polynomial expression. The solution of this is equivalent to the solution of a polynomial and we can proceed as with the previous iterators. For example

$$\frac{x - \sin x}{1-x^2/2} = \frac{\tan x - x}{1+x^2/2}$$

for x₀=1.23 would be handled by evaluating the $\sin x$and $\tan x$, obtaining

$$\frac{x - 0.94248...}{1-x^2/2} = \frac{2.8198... - x}{1+x^2/2}$$

which has the roots -3.3331..., 1.2024.... From these possible values for x₁ we select, as before, the closest to x₀, x₁=1.2024.... More formally, using Maple's RootOf notation, the iterator becomes

$$x_{i+1} = {\tt RootOf} \left ( \frac{z - \sin x_i}{1-z^2/2} = \frac{\tan x_i - z}{1+z^2/2}, z \right )$$

 

Table 3: Rational polynomial iterators, obtained by solving all the transcendental subexpressions first

66644 random equations producing 14398 iterators
method		failures
converged to a root	11.64%	converged to a non-root	.39%
failed	88.36%	diverged	6.22%
average time	.672	outside domain	3.51%
time per root	5.768	too many iterations	1.15%
		iterator fails	0.00%
iterators per equation	.216	no iterators	88.73%

Table 3 shows the simulation results for this heuristic.