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Alexey Ivanov

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7 years, 137 days
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Many choices
on the same basis.
PAR.mw

restart:
a := [x+1, x+2, x+3, x+4];
a := convert(a, set);
a :=minus(a, {x+2});
a := convert(a, list);

restart;
a := [x+1, x+2, x+3, x+4];
a := subs(x+2 = NULL, a);
nops(a);
op(2, a);

restart: with(plots):
t := [1, 2, 3];
f(t[1]) := [3, 4]; f(t[2]) := [11, 12]; f(t[3]) := [41, 1];
pointplot([f(t[1]), f(t[2]), f(t[3])], color = RGB(7, .3, 4), style = line, symbol = solidcircle, thickness = 5);
pointplot3d([1, op(f(t[1])), 2, op(f(t[2])), 3, op(f(t[3]))], color = RGB(7, .3, 4), style = line, symbol = solidcircle, thickness = 5);

combine(-ln(x)+ln(y), symbolic);

restart;
f := x1^2/(x2^3*x3^2);
op(1, op(2, f))^sign(op(2, op(2, f)));
op(1, op(3, f))^sign(op(2, op(3, f)));
f := algsubs(1/x2 = x2b, f);
f := algsubs(1/x3 = x3b, f);

restart;

(diff(f(x), x))/(diff(ln(x), x));

 

for example:

restart;

f := sin(x);

(diff(f, x))/(diff(ln(x), x));

Skeptik18(_for_d1.mw

 For a start point “a” = 0.5 any number of solutions. It depends on the "smax".

   1, [(0.8013209420000008)], 2.70183810879842667*10^-8                                                            

   2, [(1.0938038328000006)], 1.75716205141895898*10^-7                                                        

      3, [(1.5165511908)], 3.19181676505797540*10^-8                                                        

   4, [(1.9061358998000002)], 4.58627091859398206*10^-9                                                           

   5, [(2.1833650214000007)], 4.36816931514982798*10^-8                                                            

   6, [(2.5877177300000005)], 1.71876049503971729*10^-7                                                            

    7, [(2.937538889999998)], 9.29630750157173224*10^-9                                                           

    8, [(3.219756611999996)], 2.01128799837135830*10^-7                                                            

   9, [(3.6180527559999964)], 4.55144026911824540*10^-8                                                           

   10, [(3.953152181999997)], 3.39347172584325562*10^-8                                                            

   11, [(4.239272189999994)], 2.57192325658905930*10^-7                                                           

   12, [(4.635025125999992)], 3.01111789280383846*10^-7                                                         

   13, [(4.962516473999992)], 9.20600748965938465 10^-8                                                            

   14, [(5.2514410039999975)], 2.19850920579744980*10^-7                                                           

   15, [(5.645899321999999)], 1.82685308214303177*10^-7                                                         

   16, [5.968760335999992)], 2.30153605063065925*10^-8                                                          

   17, [6.2597571959999945)], 2.33606770816408017*10^-7                                                           

   18, [(6.653469225999989)], 1.16942534766906194*10^-7                                                           

    19, [(6.97322109999999)], 7.71696275769784279*10^-8                                                          

   20, [(7.265801703999997)], 4.86139439814792240*10^-8                                                           

   21, [(7.659044969999998)], 4.08131656026711200*10^-7                                                          

   22, [(7.976567196000005)], 6.48611420128730742*10^-8                                                    

   23, [(8.270394148000008)], 1.66075897922723926*10^-7                                                           

   24, [(8.663323925999999)], 2.60858493916771295*10^-7                                                      

      25, [(8.979170028)], 7.82206077687419565*10^-8                                                           

   26, [(9.274002000000005)], 1.76298078358172460*10^-8                                                           

   27, [(9.666711878000001)], 5.96099682503847818*10^-7                                                        

      28,[(9.981252484)], 2.84982932807764656*10^-8                                                            

   29, [(10.276911416000011)], 2.49167541710448860*10^-7 

...

The Draghilev method. Read, for example: http://www.mapleprimes.com/posts/145360-The-Dragilev-Method-1-Some-Mathematical

 

restart;
Digits := 30; 2^29.403243784;

For example:

restart:
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.

METHOD(n-1)2d.mw

MaplePrime.mw

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):

x1^3+x2^3-0.1e-1*sin(1.00001*x1+x2)=0;

D__LIST0.mw

 

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

diffanimationAA.mw 

 

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)));

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