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 $x^{1.001}-x \ln x =0$

$( x \ln x)^{1/1.001}$	$\frac{x^{1.001}}{\ln x}$	ex0.001	$(\ln x)^{1000}$
1.23	1.23	1.23	1.23
.254976...	5.94285...	2.71884...	$.100017... \times 10^{-683}$
-.348812...+.00109473...i	3.34053...	2.72100...	$.186958... \times 10^{3198}$
.365255...-1.09527...i	2.77297...	2.72101...	$.989096... \times 10^{3867}$
-1.31325...-.616913...i	2.72159...	converged	$.388442... \times 10^{3950}$
-2.14588...+3.31938...i	2.72101...		$.585638... \times 10^{3959}$
-10.0449...-.0715578...i	2.72101...		$.596622... \times 10^{3960}$
-23.2443...+31.2573...i	converged		$.769617... \times 10^{3960}$
diverged			$.791411... \times 10^{3960}$
			$.793839... \times 10^{3960}$
			$.794105... \times 10^{3960}$
			$.794135... \times 10^{3960}$
			converged

 

Table: Partial inverse iterators for solving $\frac{\sin^{-1} x - \tan x}{x^4} = 0$

$\sin \tan x$	$\tan^{-1} \sin^{-1} x$
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 $x^2+5+ \ln ( \vert x - \pi \vert ) = 0$

$\pi+e^{-5-x^2}$	$\pi-e^{-5-x^2}$	$\sqrt{-5-\ln \vert x-\pi\vert}$	$-\sqrt{-5-\ln \vert x-\pi\vert}$
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
$\ln (6x)$	converged to 2.83314...
$\frac{e^x}{6}$	converged to .20448...
-W(-1/6)	is 0.20448...
$\frac{e^x-1}{5}\;\;\dagger$	converged to .20448...
$\sin z^2\ln (1+z)-\cos\sqrt {2}z$
$\sqrt {\sin^{-1}({\frac {\cos\sqrt {2}z}{\ln (1+z)}})}$	diverged
$-\sqrt {\sin^{-1}({\frac {\cos\sqrt {2}z}{\ln (1+z)}})}$	too many iterations
${e^{{\frac {\cos\sqrt {2}z}{\sin z^2}}}}-1$	too many iterations
$\frac{ \cos^{-1}(\sin z^2 \ln (1+z))} {\sqrt {2}}$	converged to 0.83102...
$\ln (x+\sqrt {\pi }+{x}^{2})-\ln (x-\pi )=4$
$\pi +{e^{-4+\ln (x+\sqrt {\pi }+{x}^{2})}}$	converged to 3.45613...
$-1/2+1/2\sqrt {1-4\sqrt {\pi }+4(x-\pi)e^4}$	converged to 50.14201...
$-1/2-1/2\sqrt {1-4\sqrt {\pi }+4(x-\pi)e^4}$	diverged
$\frac{ \pi +{e^{-4+\ln (x+\sqrt {\pi }+{x}^{2})}} - e^{-4}x}{1+e^{-4}} \;\; \dagger$	converged to 3.45613...
$e^{4+ \ln (x-\pi)} - \sqrt{\pi} - x^2$	diverged
$\sqrt{ e^{4+ \ln (x-\pi)} - \sqrt{\pi} - x}$	converged to 50.14201...
$x^x\left (1+\ln x \right )+32=0$
$\left (-\frac{32}{1+\ln x}\right )^{1/x}$	falls outside valid domain
e-32 x-x -1	converged to $0.46588... \times 10^{-14}$
$\frac{\ln \left (- \frac{32}{1+\ln (x)} \right )} { \ln x}$	too many iterations
$e^{W( \ln ( -\frac{32}{1+\ln x} ) )}$	converged to 3.443...+3.812...i