@Mathematix I don't know. Those don't look like difference equations or differential equations. There's no independent variable and no derivatives. Also, it is obvious that the system does not satisfy its own initial conditions.

I haven't looked at your worksheet yet, but generally you want to raise epsilon and decrease the precision to get rid of holes in a plot. Four significant digits should be enough for any plot, smaller differences not being perceivable. The higher precision can lead to holes for complicated integrands because there may be some points for which the integral returns unevaluated because it cannot achieve the requested precision.

You would need to have an A and a B to use the procedure. What do A and B represent? Obviously, they are n x n matrices of nonnegative integers. Are they zero-one? Where did you find procedure rOrt?

By powers of Y_1*Y_2 do you mean (Y_1*Y_2)^2, (Y_1*Y_2)^3, etc., but not Y_1^3*Y_2^2? And should the latter be considered as Y_1*(Y_1*Y_2)^2, i.e., as a "power" of Y_1*Y_2 with a coefficient of Y_1?

The code that you gave above does not parse. The first statement in the first for loop has two := assignments, which Maple does not allow. Correct the syntax errors, and then I'll have a look at the optimize issue. It is not possible that the code that you have above gave the warning messages that you describe because that code never would've gotten past the parser.

@acer Do you have any idea what went wrong with the method (and the method) that I used?

@acer Do you have any idea what went wrong with the method (and the method) that I used?

When the technique is applied to a definite integral, a spurious imaginary part is introduced, although the real part is correct. I guess that it is jumping over branch cuts.

restart:
J:= Int(sin(x)/cos(x)/sqrt(1+sin(x)^3), x= 0..Pi/4):
FWIW, note that the integrand is analytic in an open disk centered at the origin and 
containing both endpoints of integration.
IntegrationTools:-Change(J, t= sin(x));
              /0                                    
             |                       t              
             |            ----------------------- dt
             |                     (1/2)            
            /       (1/2) /  3    \      / 2    \   
             - 1/2 2      \-t  + 1/      \t  - 1/   
value(%):
evalf(%);
            0.322446488989322 + 0.555360367269405 I
evalf(J);
                       0.322446488991541

When the technique is applied to a definite integral, a spurious imaginary part is introduced, although the real part is correct. I guess that it is jumping over branch cuts.

restart:
J:= Int(sin(x)/cos(x)/sqrt(1+sin(x)^3), x= 0..Pi/4):
FWIW, note that the integrand is analytic in an open disk centered at the origin and 
containing both endpoints of integration.
IntegrationTools:-Change(J, t= sin(x));
              /0                                    
             |                       t              
             |            ----------------------- dt
             |                     (1/2)            
            /       (1/2) /  3    \      / 2    \   
             - 1/2 2      \-t  + 1/      \t  - 1/   
value(%):
evalf(%);
            0.322446488989322 + 0.555360367269405 I
evalf(J);
                       0.322446488991541

Do all of yangtheary's problems come from a list of problems posed by Robert Israel? Is yangtheary Robert Israel in disguise?

Do all of yangtheary's problems come from a list of problems posed by Robert Israel? Is yangtheary Robert Israel in disguise?

@Bendesarts I don't know about exporting it,

@Bendesarts I don't know about exporting it,

@abbeykabir Thank you for the kind words.

Sorry, I have no idea what "error analysis" means with respect to this problem.

I noticed something about the way you set the values of the x[i] that seemed strange. This might be the source of your problem. You have

x[i]:= x[i-1] + i*h

I think that should be either (both the below are equivalent)

x[i]:= x[0] + i*h

or

x[i]:= x[i-1] + h

You are essentially doing both of the above, with the result that your x[i] are not evenly spaced (and that seems strange to me).

You are right about seq, but not every for loop can be replaced by seq.

First check out what I said about the x[i]'s, then I'll show you how to replace your for loop.

@abbeykabir Thank you for the kind words.

Sorry, I have no idea what "error analysis" means with respect to this problem.

I noticed something about the way you set the values of the x[i] that seemed strange. This might be the source of your problem. You have

x[i]:= x[i-1] + i*h

I think that should be either (both the below are equivalent)

x[i]:= x[0] + i*h

or

x[i]:= x[i-1] + h

You are essentially doing both of the above, with the result that your x[i] are not evenly spaced (and that seems strange to me).

You are right about seq, but not every for loop can be replaced by seq.

First check out what I said about the x[i]'s, then I'll show you how to replace your for loop.