plots:-inequal(
     {cos(x) <= y and y <= sin(x) or sin(x) <= y and y <= cos(x)},
     x= 0..4*Pi, y= -1..1, xtickmarks= "piticks", scaling= constrained, color= red
);

nprintf("%5.3f", a);

It's possible, although a bit tricky, to program this so that the 5 and the 3 are not hardcoded. I'll work on it if you need that.

Dr Venkat Subramanian: Here's my Lobatto procedure. The amount by which I increase Digits (essentially, by n^(2/3)) may be excessive, but high-degree polynomial evaluations are notoriously prone to round-off error. Nevertheless, this gives accurate results with very few integrand evaluations.

Download Lobatto.mw

Your model is a linear function of the unknown parameters a, b, and c, so you should use LinearFit. The fact that it's a nonlinear function of the independent variable t is irrelevant.

Statistics:-LinearFit([1, t, t^2], X, Y, t);

Nonetheless, NonlinearFit will still work. Unlike the other respondents, I was able to load and execute your entire worksheet (in Maple 16), including the NonlinearFit command, without error and without making any changes. I do however agree with the other respondents that you should never embed a large dataset like this in a worksheet. It should be read in from a file.

The result was

One way would be to use three colors, like this:

plot([seq([cos(t), sin(t), t= 2*Pi/3*(k-1)..2*Pi/3*k], k= 1..3)], color= [green, yellow, red]);

x=1;
map(`*`, %, 2);
map(`^`, %, 2); or map2(`^`, 2, %); (It's not clear which you meant.)
map(`^``, %, 1/3);

All the above work the same way for inequalities (even when they are not valid for inequalities---be careful!).

The most important commands for manipulating equations and inequalities are rhs and lhs, which extract the right side and the left side, respectively.

The vast, vast majority (I'd guess more than 99%) of Maple procedures are not mentioned in the help. That doesn't mean that they are in the kernel. It usually means that they were not intended for use by end users. The procedure in question, TRDpolynomial_ring, is a local of module RegularChains. That means that it was only intended to be called from other code within the module. To see its code:

kernelopts(opaquemodules= false):  #Allow access to module locals.
showstat(RegularChains:-TRDpolynomial_ring);

restart:
alias(X= RootOf(z^8+z^6+z^5+z^3+1)):
int_to_poly:= proc(n::nonnegint, x::{name, specfunc(anything, RootOf)})
local k, B2:= convert(n, base, 2);
     add(B2[k]*x^(k-1), k= 1..nops(B2))
end proc:

A:= <140, 155, 162, 64;
     218, 12,  245, 50;
     36,  251, 34, 253;
     171, 251, 184, 37>:

B:= int_to_poly~(A, X);
 

For most uses of the new Matrix B, you'll need to append mod 2 to the command. For example, to compute the determinant,

Det(B) mod 2;

You can let plot choose the number of points and their positions. This is called adaptive plotting. For example, using the "more complex" oscillator that you gave in a Reply---the one that required numpoints= 100*tmax---do

U:= eval(u[1](t), res):
V:= eval(v[1](t), res):
plot([U(t), V(t), t= 0..tmax]);

You'll get essentially the same plot as you got with odeplot using numpoints= 100*tmax. The number of points chosen by plot is 6293. The problem with this command is that it takes a little more than an hour to produce the plot. Unlike odeplot, the t-values used by plot aren't evenly spaced.

plot([seq([i, sin(Pi/12*(i-1))], i= 1..12)], style= point, symbol= diamond, symbolsize= 9);

You need a percent sign with the T, like this:

a.a^%T;

A plus sign with no percent sign also works:

a.a^+;

You asked:

Does [Length of output exceeds limit of 1000000] mean that the function couldn't be solved and I have to modify the mathematical model?

No, it doesn't mean that. It means that the solution has been found, but is just too long to display.

If the results are just too long to be displayed, how can I see the results?

You should seriously reconsider your desire to see the results. They could be hundreds of pages long. I have seen one over 4000 pages long. Once you start the display, there is no way to stop it other than by killing the Maple process. The stop button does nothing in this case.

And, practically, what would you do with such a solution?

You should use definite integrals, not indefinite integrals that you then evaluate between limits of integration.

Examples:

int(int(r^2*cos(theta), theta= Pi/4..Pi/3), r= 15..18);

or

int(r^2*cos(theta), [theta= Pi/4..Pi/3, r= 15..18]);

The revelant commands are DEtools[DEplot] or dsolve followed by plots:-odeplot. If you have parameters, I think the dsolve / odeplot route would be easiest. If you want more help, you'll need to post your system of ODEs, their initial conditions, and your desired values of the parameters.

This is a common operation---not at all a "strange question."

The ApproximateInt command, if given exact symbolic input, attempts to compute the exact value of the integral approximation. What you want is a floating-point approximation. To get such, you simply need to include a decimal point in the function or in the interval of integration:

f:= x^2/(sin(x)+x+1.;
Student:-Calculus1:-ApproximateInt(f, x= 1. .. 2., method= simpson, partition= 20);

This'll produce the decimal answer very quickly. Only one the decimal points above is needed.