@sand15 You wrote:

• It's rather difficult to compare the results produced by Q3 with the true ones. 

I knew that the limitations of hardware-float arithmetic would kick in at some point, so I was already suspicious. To check the results, I simply summed them. The command add can be applied to any list or rtable without needing an index, as in add(L). For NumberOfDice > 22, the sums are different from 1, although they're close. For NumberOfDice > 394, the sums are undefined, which indicates that you not only overflowed the mantissas (the exact part of an hfloat), but you also overflowed the exponents.

• I suppose your observation definitely discredits Q3?

It's hard to argue for undefined as a credible answer.  :-)

A procedure to do convolutions in Maple's default exact rational arithmetic could be written in 3-5 lines. It wouldn't be as fast as hfloat, but you'd probably be surprised how fast it is.

Another alternative is to use LinearAlgebra:-Modular, doing the convolutions for a set of prime moduli (all less than 2^25) whose product exceeds the product of the number of sides of the dice, and reconstructing the exact result with LinearAlgebra:-Modular:-ChineseRemainder. You could use ArrayTools:-FlipDimension to express each entry of the convolution as a dot product of two vectors. It sounds complicated, but it may possibly be worth it because Modular is amazingly, blindingly fast when used in hfloat mode with moduli < 2^25. I think it could be done in 20-30 lines.

@mmcdara It's a good try, and I do believe that there's a good solution to this problem using convolutions. However, your procedure Q3 has several shortcomings, and one very serious error that will be difficult to fix: SignalProcessing:-Convolution uses hardware-float arithmetic, where the largest integer that can be represented exactly is (IIRC) 2^53 - 1 (I'm sure that it's some number close to that). Because of this, Q3 returns incorrect results for NumberOfDice > 22 and total nonsense for NumberOfDice > 394 (using the "default" initial die [1,2,3,4,5,6], with similar cutoffs for other initial dice).

My procedure for a dice-sum's distribution is much shorter, much more versatile, and several times faster and more memory efficient:

DiceSumDistr:= (D::list(list(integer)))-> 
   table((degree=icontent)~([op(expand(mul((add@`~`[`^`])~(_x,D))))])) /~ mul(nops~(D))
:

The dice can have any number of sides, the sides can have any integers (positive, negative, zero, or repeated), and any mixture of different dice can be used in a group. Try:

CodeTools:-Usage(DiceSumDistr([[$1..6] $ 1000])): #1000 6-sided dice
Dice:= ['['rand(-6..6)()' $ rand(2..12)()]' $ 100]; #100 randomly sized and spotted dice
CodeTools:-Usage(DiceSumDistr(Dice)):

@Preben Alsholm I'm just guessing again, but I suspect that the GUI/kernel synchronization issue might be more pronounced in document mode or with 2d Input.

@Jodelkoenig Please post a worksheet that exhibits this behavior as an attached file. You can attach files to a post by using the green uparrow on the toolbar of the MaplePrimes editor.

In the meantime, tbis might help (but I'm just guessing):

#First execution group:
restart:

#New execution group:
Threads:-Sleep(2); #Give it a 2-second break to finish the restart.

#New execution group:
interface(imaginaryunit= j);
j^2;

@Matador33 Why do you "have to" use a shooting method and 4th order Runge-Kutta? These are not the best numeric methods in Maple. Shooting methods have been superceded by a dedicated finite-element BVP solver (as used in Tom Leslie's Answer), and 4th-order Runge-Kutta has been superceded by 4th/5th-order Runge-Kutta-Fehlberg. These are the default methods for BVPs and IVPs respectively (that distinction being detected automatically in the vast majority of cases).

Once the BVP solution is obtained, it can be used to obtain the initial conditions needed to recast the the problem as an IVP. Then it can be solved for independent-variable values outside the boundaries of the original BVP.

I am confused by what you say about psi. If psi = C*sqrt(dA(z)), then psi is not a constant, and the system is not linear. Yet you also said psi=100 and that the system is (sometimes) linear.

Maple's numeric ODE solvers can generally handle isolated points of non-differentialability (such as where your dA(z) changes to constant 0), and systems can be input using piecewise to specify what happens at those points.

You'll need to define the variables before I can answer that: Which are scalar, and which are matrices or vectors? And is i just a variable, or is it sqrt(-1)?

@acer I think that your first interpretation of the Question was correct, i.e., reflected the OP's intent. My Reply was in no way intended as a criticism of your original Answer. I was just pointing out an anomalous observation that was tangential to the Question and was meant to cast doubt on the meaningfulness of ProcessClock.

@Ramakrishnan You wrote:

• I understand from your explanation that the PC [ProcessClock] timings are more accurate to nanoseconds.

I think that you misunderstood me. Perhaps they'd be more accurate if they were at all believable. From your timings above, for every case where (Delta ProcessClock) <> 0, we have (Delta time[real]) / (Delta ProcessClock) on the close order of a million (recall that "nano-" means 10^(-9)). For one thing, that degree of consistency is not believable to me^[*1], and for another thing, a million seems way too large. So, I have three problems:

• I can't figure out what ProcessClock is attempting to or supposed to report;
• I can't figure out what it's actually reporting;
• I can't figure out whether it consistently reports anything at all.

^[*1] I mean that if ProcessClock is meaningful, then it's not believable that it would be so consistent with time[real].

@radaar I didn't say that at all. If I thought that there was no way, I wouldn't have mentioned two packages for it. Let's start by looking at your program.

But, yes, there is "no way" to take a program that wasn't explicitly designed or re-designed for multiprocessing and make it run significantly faster by using multiple processors. On the other hand, multiprocessing of garbage collection in Maple is now automatic. If your program uses a significant percentage of its time for garbage collecting (as a great many Maple programs do), then it may be possible to speed things up by increasing the value of kernelopts(gcmaxthreads) and possibly also lowering the value of kernelopts(gcthreadmemorysize). 

A key feature of Maple and other computer algebra systems (CASs) that distinguish them from other mathematical software, including that $2 calculator, is that results are expressed exactly unless an approximation is forced (in this case by evalf or by using a decimal point).

How about providing us with "the" equation that you want to solve?

@acer I am not at all surprised, of course, by the difference between the OP's two timings. On the contrary, I am surprised by a similarity (which may be a coincidence): For the two timings, the ratio (delta time[real]) / (delta ProcessClock) matches to 3 significant digits, those ratios being 1.033e6 and 1.029e6 (and time[real] is only reported to three significant digits anyway). And is nanoseconds the correct unit for ProcessClock? Shouldn't that be microseconds?

I've noticed some other anomalies with ProcessClock that lead me to be skeptical of it.

@Magma The procedure Bases does it:

Bases:= f-> [seq(`if`(g::`^`, op(1,g), g), g= [`if`(f::`*`, op(f), f)])]:

Usage:

Bases(f);

where f is the polynomial, as before.

The complexity of the above code is due to the need to handle the two degenerate cases: factors that don't have an exponent and polynomials that only have a single factor. 
 

@ecterrab But Kitonum does not want to enter any Sum or sum command. I think that you may be confused because he did in fact enter such a command, even though he did not want to. He wants to enter a command akin to

convert((a+b)^n, Sum) assuming n::posint;  #or something similar

and have Maple respond with a summation formula.