Other special cases.

The technique used for polynomials can be extended to other cases, when we know how to solve some transcendental equations. In Maple, the function W(x)[2,4,3] is the solution of W(x)e^W(x)=x. Hence many combinations of powers of x with e^x and with $\ln x$ are invertible. The resulting iterators have a lower number of occurrences of x on the rhs, and are considered better. Some of them are shown in table 4.

 

Table 4: Inverting expressions with the aid of the W(x) function

equation	iterator
xaebx = F(x)	$x=\frac{a W \left( \frac{bF(x)^{1/a}}{a} \right)}{b}$
$a \ln x+bx = F(x)$	$\frac{a W(b \frac{e^{F(x)/a}}{a})} {b}$
xx=F(x)	$e^{W( \ln F(x))}$
$x \ln x = F(x)$	eW(F(x))

 

Table: Iterators generated by solving a lhs involving $\ln x$, e^x and x in terms of W(x)

66644 random equations producing 3174 iterators
method		failures
converged to a root	1.83%	converged to a non-root	.07%
failed	98.17%	diverged	.80%
average time	.128	outside domain	1.14%
time per root	6.985	too many iterations	.20%
		iterator fails	0.00%
iterators per equation	.048	no iterators	97.79%

Table 5 shows the simulation results for lhs which can be resolved with the W(x) function. The iterators generated in this way are also very successful in producing roots, just as the non-linear polynomials. In this case however, only 2% of the equations can produce such iterators, and their relative usefulness is decreased. The W(x) function accepts a branch indicator, then as before we can compute an arbitrarily large number of iterators which will normally converge to different roots.