I made this new thread because I cannot make replies in the original thread.

Now we turn to an example of the usage of the Dragilev method with Maple. Let us consider the curve
from
http://www.mapleprimes.com/questions/143454-How-To-Produce-Such-Animation .

Its points can be found by the Dragilev method as follows.

This is an effective method of solving systems of  N nonlinear and nonalgebraic equations in N+1 real-valued variables:
F(x)=0, where F=(f1,f2,..., fN) and x=(x1,x2,...,xN+1). (1)               
                                                                  
All the functions fj(x) are assumed to be continuously differentiable in some domain D in R^(N+1).
In general, such systems have an infinite set of the solutions which form a space curve in R^(N+1).
 The Optimization and DirectSearch solvers have difficulties with it.
The idea of the Dragilev method consists in the following. Let us assume that the space curve (1) can be parametrized through the  parameter t
which can be taken as the  length of the arc (for example, see http://en.wikipedia.org/wiki/Curve ).
That presentation provides to avoid the difficulties associated with self-intersections and singularities.
We will obtain the coordinates of the points of the space curve under consideration as the solution
of the Cauchy problem for certain system of ODEs. We put the coordinates of a known point of (1) as the initial conditions.
The system of ODEs is obtained by the differentiation  with respect to t:

We treat (2) as a linear system in the derivatives diff(xj(t),t),j=1..N+1. In general, its solutions form an infinite set.
We choose the  solution of (2) as follows. We put diff(xN+1)= -Determinant(Matrix(n, (i,j)->diff(fi(x),xj))).
This is one of the key points of the method under consideration.
Next, we find diff(xj,t), j=1..N. At last, we numerically solve that system of ODEs, obtaining an array of points.
 The general case of a system of N nonlinear  equations in N variables
G(y)=0, where y=(y1, y2, ..., yN), G=(g1,g2,..., gN) (3),
is reduced to the previous one by homotopy. We choose a point y0= (y01, ..., y0N) and consider the system
G(y)- v*G(y0)=0 (4)
of N equations in N+1 variables (v is taken as  a dummy variable). Changing v from 1 to 0, we arrive to (3).
We don't consider any mathematical justification of the above pattern here as that does not deal with Maple.
I would like to point out the sites

http://forum.exponenta.ru/viewtopic.php?t=3892 and
http://forum.exponenta.ru/viewtopic.php?t=11284 , where the Dragilev method is discussed. Unfortunately, in Russian.
The googling of "Dragilev" and "Draghilev" does not produce many info.
A short biographical sketch: Dr. Anatoliy Vladimirovich Dragilev (1923 - 1997) was a professor at  Rostov State University in Russia.
See his articles here: http://www.mathnet.ru/php/person.phtml?&personid=32359&option_lang=eng and
http://www.zentralblatt-math.org/zmath/en/search/?q=au%3A%22dragilev%2C%20a*%20v*%22 .
 

I am very grateful to Professor Alexey B. Ivanov, who is an enthusiast of the Dragilev method, for the very useful discussions.

The new option added to MaplePrimes about two hours ago----to show Answers by date----only shows the Answers in reverse chronologocal order. That's ridiculous! Even more so with the comments still being in forward chronological order. The reason that people have requested date order so vehemently is to preserve the continuity of conversation. With the reverse chronological order, the continuity is worse than before.

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