@Markiyan Hirnyk The information about continuity is not displayed if you include the phrase assuming n::posint.

@Christopher2222 wrote:

One could instead create a proc that would circumvent the issue when StandardDeviation command is called, yes?

Yes, Statistics:-StandardDeviation could be overloaded so that when weights are passed in, it is diverted to separate user-supplied code; and when weights are not passed in, it uses the current code. Like this:

restart:
sav_std_dev:= eval(Statistics:-StandardDeviation):
unprotect(Statistics:-StandardDeviation):
Statistics:-StandardDeviation:= overload([
     proc(A::{rtable,list})
          option overload;
          local p, weights, sd;
          if not membertype(identical(':-weights')={Vector,list}, [args], p) then
               error "invalid input:"  # Kicks to next proc in overload
          end if;
          weights:= rhs([args][p]);
          print('weights' = weights);
          sd:= Statistics:-StandardDeviation(A);
          # Do whatever you want here to compute with the weights.
     end proc,

     sav_std_dev
]):
    
protect(Statistics:-StandardDeviation):

Please explicitly show in a worksheet the code that works and the code that doesn't work. Please state explicitly why you say that it hasn't worked.

@Carl Love To analyze where f(a,c,n) < 0, the 3d animation should be viewed in conjunction with an animation of the level curves f(a,c,n) = 0. This can be done with implicitplot.

plots:-animate(plots:-implicitplot, [f(a,c,n) = 0, c= 0..2, n= 0..4, gridrefine= 3], a= 0..2);

@Christopher2222 Yet it seems that the weights are used somehow in the computation. Changing the weights often changes the last digit or two of the standard deviation.

@Markiyan Hirnyk You're right; I missed that the OP's primary interest was the sign. The animation can be improved with the view option.

plots:-animate(plot3d, [f(a,c,n), c= 0..2, n= 0..4, view= [0..2, 0..4, -300..0]], a= 0..2);

@Carl Love Incorporating Joe Riel's %m hack into a convert procedure, we get

`convert/decimalfraction`:=
     (x::float)-> sscanf(sprintf("#%m%m", op(1,x), 10^(-op(2,x))), "%m")[]:

@Preben Alsholm Just to be absolutely clear, Preben's comment was merely meant to point out a flaw in the OP's final argument. It was not meant to deny the fact that there is a bug in product at play here.

@Joe Riel Obviously the "#" is the key to this hack. Can you describe other hacks using the %m technique?

@love maths Here's an example. I couldn't upload an executed example (some bug in MaplePrimes), so you'll have to execute this yourself.

Eigen:= proc(A::'Matrix'(2,2))
local
     a:= A[1,1], b:= A[2,1], c:= A[1,2], d:= A[2,2],
     s:= sqrt((a-d)^2+4*b*c)/2,
     e1:= (a+d)/2 + s, e2:= (a+d)/2 - s,
     ev1:= < c, (d-a)/2+s >, ev2:= < c, (d-a)/2-s >
;
     if s=0 then [] else [e1, e2, ev1, ev2] end if
end proc:

A:= LinearAlgebra:-RandomMatrix(2,2) + I*LinearAlgebra:-RandomMatrix(2,2);
Sol:= Eigen(A);
A.Sol[3] - Sol[1].Sol[3];
simplify(%);

@sarra There is no mistake. You can ignore the higher-order terms. The expression is still O(h^4) (as h-> 0). The purpose of big-O notation is to let you ignore higher-order terms.

@Preben Alsholm Thanks. I corrected the Answer.

@YasH See ?emptysymbol . Using `` as a function name has the same effect as using f or any other undefined symbol, except that `` does not print as anything.

I just answered that two days ago, here.

Is your primary objective to get the answers, or to do it by the Newton-Raphson method?