Kurt Ewald wrote:

Dihedralgroup(9) acts on the set 1..9 and has order 9.

Actually, DihedralGroup(n) has order 2*n.

What is SmallGroup(18,1) in somewhat standard terminology? Apparently, it is not CyclicGroup(18). I am not familiar with this GAP/"Small group" system of classification.

@Axel Vogt A brilliantly simple solution. Just make Delta local.

@xmmood You wrote:

It would be great if Maple could allow us to declare 'I' not to be the imagainary part.

Maple does allow that. Just pick another symbol to play the role of I. Let's say you pick J. Then do

interface(imaginaryunit= J);

@Markiyan Hirnyk The value 2663.73573252378 is also known as the mean absolute deviation (MAD).

If one uses Statistics:-NonlinearFit instead of ExponentialFit, then the returned residual sum of squares is the same as with DirectSearch:-DataFit(..., fitmethod= lms).

It is easier and apparently more accurate (at least in this case) to use NonlinearFit rather than ExponentialFit. Then the residual sum of squares can be obtained directly.

Statistics:-NonlinearFit(a*exp(b*x), X, Y, x, output= residualsumofsquares);

@mapleq2013 Using Maple 18, your code works for me with a final kernel (mserver) memory usage of 161.6 MB (as reported by Windows Task Manager) and the intermittent usage never going above 280 MB. If you are using earlier Maple, perhaps this shows that there is more effective garbage collection in Maple 18.

@Andriy The difference between your two example functions is that in the first example the operation k^2 is valid whether or not k has a numeric value, but in the second example fsolve requires that the number of variables which are "just a name" be the same as the number of equations.

@Markiyan Hirnyk Warning: This numeric integration takes an extreme amount of memory for me. I had to kill it from Windows Task Manager.

@Andriy "Just a name" means that k is a variable to which no numeric value has been assigned.

@mapleq2013 You wrote:

Could you please briefly explain why this is correct, if there is a reasoning at all?

Because you are extracting a procedure rather than an expression from dsolve's output.

I asked because I have never defined functions this way, I always do as the following example:

source(t):=sin(0.1* t)

I recommend that you never define a function that way. Maple will let you get away with that sometimes (especially if you use 2D input). The correct way to define a procedure (function) is

source:= t-> sin(0.1*t);

In the case of the dsolve output, you are not defining a procedure; rather, you are extracting a procedure that has already been defined, hence the t-> part is not needed.

There are two main syntaxes for plotting a procedure. The way that always works is

plot(f, a..b);

A syntax that works only if f is prepared to take symbolic input (as is source above) is

plot(f(t), t= a..b);

The procedures returned by dsolve are also prepared to take symbolic input.

I recommend that you never use the syntax

plot(f(t), a..b);

although, as I said, Maple will let you get away with it sometimes.

I've looked over the partial fractions for k= 0 through 7, and I am curious what pattern you saw (you said that you guessed the general decomposition). I can see that one fraction will always be 1/(k+1)/(x-1)^2. The other denominators are (x-1) and 2+sum(x^j, j= 1..k-1), but I can't guess their numerators.

@Kitonum You can make a factor-of-9 time improvement by including option remember in your procedure P. Without it plot3d calls dsolve repeatedly with the same value of k.

Please post your code.

If I correctly understand your situation, you should store the value computed at each iteration in a Vector. Then subtract from the true value outside the loop to get a value for the error at each iteration. I'll post an example in an Answer below.

@acer Thanks, Acer. I updated the Answer accordingly.

But I don't understand how your equality tester works. Specifically, I don't understand how member works. Groups are implemented as modules. According to ?member when M is a module, member(x, M) checks whether x is an export of M. But an element of a group is not an export of the module that represents that group! So how does it work?

@rodrigog So, if you don't want the purple shades, does that mean that you do want to go through orange, yellow, and green? To do that, and in reverse order, do

plots:-pointplot(Pts, color= [seq(COLOR(HUE, (N-k)*.65/N), k= 1..N)]);

If you don't want to go through orange, yellow, and green, then you need to tell me what you expect the color in the middle to be.