What do you mean by "a label coordinate"?

Are the circles in the same plane with one inside the other? On first thought, that seems an easier problem than two arbitrary circles. 

@Axel Vogt

I'm just writing this to clarify Axel's advice: If you want to avoid this anomaly, use 1/10 instead of 0.1. The floating-point anomaly that you point out is just a standard example, not an bug.

@Kitonum I agree with Markiyan. What relation is there between your workaround plot and the matter discussed in the paper which the OP referenced?

@Kitonum Here's another improvement. This avoids the once-per-rotation stutter caused by 0 and 2*Pi being actually the same frame. We start the rotation a little higher than 0 based on the frame count. This improvement can be applied to any continuous-play rotating animation:

Frames:= 54:
plots:-animate(
     plots:-tubeplot,
     [[cos(t+phi), sin(t+phi), t], t= 0..6*Pi, radius= 1/3, numpoints=200],
     phi= (2*Pi)/Frames..2*Pi, frames= Frames,
     orientation= [40,70], shading= z, axes= box, style= patchnogrid,
     scaling= constrained
);

@GPY Right click on the animation. Select the Animation context menu. Check off the option Continuous. This can also be done in the animation toolbar. Use the second pull-down to the right of the slider to choose between Continuous and Single Cycle. All animations start as single cycle. There is unfortunately no way to control this programatically, i.e., with the plot/animation command.

@Kitonum 

In the standard American high-school curriculum, it is usual to indicate a jump discontinuity with a hollow circle on one  piece of the graph and a solid circle on the other piece at the jump point. So, in the case of floor, each segment would have a solid circle on the left end and a hollow circle on the right. Is this possible using option discont? Of course, I can figure out how to explicitly plot the hollow circles; what I want is a way using discont or some such.

@acer The plot can be improved with option discont as in

plot(cos(x)/(x^2+2*x), x= -10..10, discont);

The command smartplot seems to ignore the option: neither results nor an error message appears.

Were you trying to approximate by using a context menu or by a directly typed evalf command?

@GuruYerram The link that you gave works, but there is no file found there. Possibly it has been removed. Even if I shorten it to mkhebcha.math.science.cmu.ac.th/206455, it is not found. Can you just take a photo of the page with your phone and post it?

You'll need to post the complete code, that is, the definition for G1, G2, and Loss.

@GPY Can you post a photo, a link to a photo, or a direct link to a webpage or PDF that has the formula from Cheney and Kincaid? Meanwhile, I'm about to post some code showing order of convergence for Halley's method. The code can be easily modified to show the order of convergence for any iterative root-finding algorithm.

The pseudocode that you give appears as nonsense to me. I can't see any relation to Muller's method. See the Wikipedia article "Muller's method".

The iterator that you give does not converge to the root at 0 for any starting value other than 0. It does converge linearly (and a very slow linearly at that) to the root at 1 for any positive starting value that I've tried. Are you sure about that iterator? I've looked through the entire Wikipedia category "Root-finding algorithms" and I can't find it. The only method that I know of with cubic convergence is Halley's method:

x[n+1]:= x[n] - 2*f(x[n])*D(f)(x[n])/(2*D(f)(x[n])^2 - f(x[n])*(D@@2)(f)(x[n]))

This has a denominator similar to yours, but there's no square root and there's the 2 coefficient. There are a few methods that use square roots, but AFAIK none has cubic convergence.

Your function has no simple roots. It only has the multiple root at 1.