@gmzsvsclk Please define what you mean by "the points (t,y)"? Which points? I gave you a method for finding those points on a graph that Maple could be  made to draw. Obviously, you didn't want that collection of points, so which points do you want? You have also been given recipes for finding arbitrary points by solving explicitly for y. So, which points do you want?

For a single variable, Newton's method takes x[n+1] as the x-intercept on the line tangent to the graph of f(x) at x[n]. The Secant method replaces this tangent line with a secant line drawn through x[n-1] and x[n].

In the multivariate case, the Jacobian matrix in Newton's method can be replaced with alternatives, such as in Broyden's method. I'm not sure that such methods follow from a simple geometry that generalizes a secant line. In fact, Newton's method in the multivariate case is derived via Taylor expansions of each function, and not through any obvious geometry. If it can be so based, I've never seen it, and would appreciate a pointer to it.

@Kitonum 

The Roots command will succeed if Digits is set to at least 17. As several responses indicate, the calculation is delicate and subject to roundoff. I guess I missed that the first time I looked. Thanks to all who provided a sharper analysis.

@amcmaple 

Delete the number=1 option, and don't specify a derivative condition as an initial condition for a first-order ODE.

With the DEtools package installed, the following syntax works:

phaseportrait(X(t) = diff(X(t), t), X(t), t = -1 .. 2.5,[[X(0)=1],[X(0)=-1]])

but not phaseportrait(X(t) = diff(X(t), t), X(t), t = -1 .. 2.5,[[X(0)=1],[D(X)(0)=-1]])

The dsolve command can obtain the solution with the derivative IC, which can be graphed separately, then joined to the plaseportrait via the display command in the plots package.

p1:=DEtools:-phaseportrait(X(t) = diff(X(t), t), X(t), t = -1 .. 2.5,[[X(0)=1]]):
p2:=plot(rhs(dsolve({X(t)=diff(X(t),t),D(X)(0)=3/4},X(t))),t=-1..2.5):
plots:-display(p1,p2)

@brian bovril 

Nice use of Maple to find a general pattern. I have to admit that I scratched around a bit before I came to the same conclusion. But I did end up using rsolve along the way.

I don't understand your statement that the loop should finish when the computed T is the same as the initial T. If it's the initial T that you want, why do any calculations? You already have it.

It would make more sense to terminate the loop when two successive computations of T are essentially the same.

Please clarify.

@Joe Riel 

The SpaceCurve command in either of the VectorCalculus packages will graph a curve defined parametrically by either a 2- or 3-dimensional vector. No need to switch graphing commands for different dimensions.

Have you followed the thread most recently appearing here:

https://www.mapleprimes.com/maplesoftblog/208901-A-Surprising-NextNumber-Puzzle

@alfarunner 

The Physics:-Vectors package has some neat functionality for vectors and differentiation. Unfortunately, it has none for visualization or integration. The Student VectorCalculus package has wonderful functionality for integration and visualization.

The Physics:-Vectors package is the only place in Maple where one can manipulate "unprojected" vectors, that is, vectors that do not have to be given components. So, vector identities can be manipulated without having to specify components for the vectors.

The Student VectorCalculus package is far more lenient with respect to setting coordinates. The VectorCalculus package demands at every turn a declaration of the coordinate system and the coordinate-variable names. The Student package, recognizing only Cartesian, polar, cylindrical, and spherical coordinates, all with default coordinate-variable names, is much more user-friendly for someone not working in esoteric systems.

@Markiyan Hirnyk 

Interesting. I have the source code for Maple V Release 4 on a set of 10 floppy disks, the original media used for the distribution of this version. My Windows 10 laptop does not have a floppy-disk reader, nor does my Windows7 desktop. How did you manage to install MVR4 on a machine capable of running Windows 10?

@mahmood1800 

Minimize in the Optimization package computes numerically. It is not able to work with the c's as parameters, and that accounts for the zero solution. Hence, the LagrangeMultipliers command, which can work symbolically.

@mahmood1800 Originally, the constraint was an equation. Now, you show that not to be the case. I'm therefore confused as to what your original problem actually was.

@Robert Israel 

Thanks for the gfun usage and the pointer to OEIS.

Note that the gfun package did not find my "interpolation-completion" of the first sequence, not did it find the next strobogrammatic number with the generating function it provided for the second.

@exality Thanks for getting to the cause of the problem. When I coded the template I never thought of potential conflicts with other packages. Will have to consult with the professional Maple developers at Maplesoft for advice. For now, it seems as if the unwith suggestion is the best alternative for you. Changing the code in the task template itself is a much more involved process.

Maple 12 gives the same null response. In Maple 2017, I can eliminate x1, x2, and x3 first, then in that result, eliminate x8. (The first result contains no x4.) The second result looks like the result of the example you posted of what you expected.