@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?

I can confirm all three of your observations on Windows 8.1/64 using Maple 17.02/64:

1. That there is a memory leak; indeed it is some of the fastest memory consumption that I've ever seen. The memory usage is not reported on the Maple status line.
2. That restart does not release the memory, even when followed by gc();
3. That the difference of the two integrals that should be zero is not even close.

Since your code executes without any explicit error message, you'll need to explain why you think that there is a problem.

@mkharban Pi is capitalized in Maple. Also, after you execute restart, you need to re-execute the other lines also. It looks like you did not re-execute the lines defining the ranges X and Y.

And please read your email, at whatever email address you used to sign up for MaplePrimes.