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Alexey Ivanov

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8 years, 258 days
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For example:

nn := nextprime(10^100); zz := 1;
for ii from 0 to 100000 do zz := `mod`(zz^2+1, nn); if `or`(ii > 99997, zz = 66388502) then print("ii=", ii, "zz=", zz) end if end do:

Blue - the denominator sin(x+(1/3)*Pi-theta) = 0.
theta, I think, has a period of Pi, and x has a period of Pi / 3. The solution is obtained by Draghilev method. This numerical solution of ordinary differential equations with initial conditions theta (0) = 0, x (0) = Pi / 3.


This way you get all the real solutions for any real value of any parameter for the polynomial

equations N * N.


The Draghilev method to find all the points of zero and Pi/2 slope of an implicit function f(x1,x2)=0 while driving along the section of the line connected.

Example solution of equation and searching of all the points of zero and Pi/2 slope (with animation):



The Draghilev method for F(x,y,z)=0. Many points with good precision (variants with h and h1). 


restart; with(RootFinding):

 f := Z^6-3*Z^4+3*Z^2+Z-1;

 rhs(op(1, Isolate(f, Z)));

 rhs(op(2, Isolate(f, Z)));

restart; with(RootFinding):

 Z := x+I*y;

 f := Z^6-3*Z^4+3*Z^2+Z-1;

 Isolate([evalc(Re(f)), evalc(Im(f))], [x, y]);

It seems that the system is not consistent in any combination of 12 equations…


 g := (3*x^2-3*y^2)/(x-y);

 x := `5`; y := `2`;


       At the urging of Markiyan Alexeyevich I show the text. Because I can't find out the proper code at once, here is a very close substitute. The Draghilev method is used in my animation.

Only for real roots...



eq1 := x^4+(2*(3*y+1))*x^2+(5*y^2+4*y+11)*x-y^2+10*y+2;

eq2 := y^3+(x-2)*y+x^2+x+2;

 Isolate([eq1, eq2], [x, y]);

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