To see why this computation converges for x = 1.4, contemplate the figure to the right.
The stairstep line shows a graphical approach to calculating y, with the y coordinates of the successive stair tops occurring at 1.4, 1.4**1.4, and 1.4**1.4**1.4. One easily sees that this process will converge. In fact, one can easily confirm that it also converges if one starts at 2.5 instead of 1.4; that is, passing through 1.4**2.5, 1.4**1.4**2.5, et cetera. So there is a range of starting values for which the computation converges.
The two curves intersect a second time, beyond the upper-right corner of the plot. That intersection corresponds a point on the y > e part of the x = y**(1/y) curve, and is unstable in the sense that the calculation only "converges" if one starts at the exact point of convergence.
The following graph shows why the calculation cannot converge for x too large: the curves don't touch, so the stairstep path will squeeze through the gap and escape to infinity. Convergence will end at the value w for which the curve y = w**x is tangent to y = x.
For small values of x and finite numbers of exponentiations,
one finds that the result diverges into two branches, depending
on whether the number of exponentiations is even or odd. In
the following figure, curves are labelled with the number of
exponentiations.
What's happening here? To the right of the cutoff point,
convergence proceeds to a point along the spiral shown to
the left.
To the left of the cutoff point, the calculation loops
endlessly in a spiral that doesn't converge to a point.
In fact, if one restarts the calculation from a starting
point closer to the intersection of the curves (red),
one loops on a spiral that grows outward.
Convergence can only be achieved for values v
such that the slope of the curve y = v**x is greater than -1
at the point of intersection with y = x. That condition defines
the lower end of the curve.
2006-01-04
My email address:
