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I have 3 equations with 3 variables and I want a solution for the 3 of them. Moreover, I'm not really concerned for solutions outside of the range between 0 and 1. I tried using Solve to do this, but it's taking way too long. Any suggestions? Each of the 3 equations needs to be solved for 0. I need to do this several times (say, 20-30) so it would be optimal to have a way to do this that it wouldn't take the whole day and without having mathematica crashing in the middle of it. 

Solve[{(-0.8702809089282518` + 1.2424405081409913` Y2^0.5` + 
  1.7541599570585076` Y2 + 1.2424405081409913` Y2^1.` - 
  1.2424405081409913` Y2^1.5` - 1.1056986938629452` Y2^2 - 
  1.2424405081409913` Y2^2.` + 0.3397298264448023` Y2^2.5` + 
  0.2218196457326892` Y2^3 + 0.3397298264448023` Y2^3.` + 
  Y1^2 (-0.016997674002504914` + 0.02773304705671855` Y2^0.5` + 
     0.016997674002504914` Y2 + 0.02773304705671855` Y2^1.`) + 
  Y0^2 (-0.10623546251565573` + 0.17333154410449098` Y2^0.5` + 
     0.10623546251565573` Y2 + 0.17333154410449098` Y2^1.`) + 
  Y1 (0.2447665056360708` - 0.35498300232599744` Y2^0.5` - 
     0.3705492932546072` Y2 - 0.35498300232599744` Y2^1.` + 
     0.16085167292896763` Y2^1.5` + 0.1257827876185364` Y2^2 + 
     0.16085167292896763` Y2^2.`) + 
  Y0 (0.6119162640901772` - 0.8874575058149936` Y2^0.5` - 
     0.926373233136518` Y2 - 0.8874575058149936` Y2^1.` + 
     0.402129182322419` Y2^1.5` + 0.314456969046341` Y2^2 + 
     0.402129182322419` Y2^2.` + 
     Y1 (-0.08498837001252461` + 0.13866523528359276` Y2^0.5` + 
        0.08498837001252461` Y2 + 
        0.13866523528359276` Y2^1.`)))/((1.` + 
    Y2^0.5`) (1.7777777777777777` - 0.5555555555555556` Y0 - 
    0.2222222222222222` Y1 - 1.` Y2)^2 Y2^0.5` (-8.` + 5.` Y0 + 
    2.` Y1 + 1.` Y2)) == 0, 
  1/(3 + Y0^0.5` + 2 Y1^0.5`)^2 ((
  0.03668478260869564` (1 + Y0^0.5`) (3 + Y0^0.5` + 
     2 Y1^0.5`) (3 - Y0 - 2 Y1))/(3 - 1.4375` Y0 - 1.375` Y1 - 
    0.1875` Y2)^2 + (
  0.07336956521739128` (1 + Y1^0.5`) (3 + Y0^0.5` + 
     2 Y1^0.5`) (3 - Y0 - 2 Y1))/(3 - 1.4375` Y0 - 1.375` Y1 - 
    0.1875` Y2)^2 + (
  0.16843100189035914` (1.` + Y0^0.5`) (3.` + Y0^0.5` + 
     2.` Y1^0.5`) (-1.9393939393939392` + 1.` Y0 + 
     0.7878787878787878` Y1 + 
     0.15151515151515152` Y2))/((-2.0869565217391304` + 1.` Y0 + 
     0.9565217391304348` Y1 + 0.13043478260869565` Y2) (-1.6` + 
     1.` Y0 + 0.4` Y1 + 0.2` Y2)) - (
  0.08421550094517957` (1.` + Y0^0.5`) (-3.` + Y0 + 
     2.` Y1) (-1.9393939393939392` + 1.` Y0 + 
     0.7878787878787878` Y1 + 0.15151515151515152` Y2))/(
  Y1^0.5` (-2.0869565217391304` + 1.` Y0 + 
     0.9565217391304348` Y1 + 0.13043478260869565` Y2) (-1.6` + 
     1.` Y0 + 0.4` Y1 + 0.2` Y2)) + (
  0.21521739130434786` (3.` + Y0^0.5` + 2.` Y1^0.5`) (-3.` + Y0 + 
     2.` Y1) (-1.9393939393939392` + 1.` Y0 + 
     0.7878787878787878` Y1 + 0.15151515151515152` Y2))/(
  Y1^0.5` (-2.0869565217391304` + 1.` Y0 + 
     0.9565217391304348` Y1 + 0.13043478260869565` Y2) (-1.6` + 
     1.` Y0 + 0.4` Y1 + 0.2` Y2)) + (
  0.693675889328063` (1.` + Y1^0.5`) (3.` + Y0^0.5` + 
     2.` Y1^0.5`) (-2.4615384615384612` + 
     1.2692307692307692` Y0 + 1.` Y1 + 
     0.1923076923076923` Y2))/((-2.1818181818181817` + 
     1.0454545454545454` Y0 + 1.` Y1 + 
     0.13636363636363635` Y2) (-4.` + 2.5` Y0 + 1.` Y1 + 
     0.5` Y2)) - (
  0.3468379446640315` (1.` + Y1^0.5`) (-3.` + Y0 + 
     2.` Y1) (-2.4615384615384612` + 1.2692307692307692` Y0 + 
     1.` Y1 + 0.1923076923076923` Y2))/(
  Y1^0.5` (-2.1818181818181817` + 1.0454545454545454` Y0 + 
     1.` Y1 + 0.13636363636363635` Y2) (-4.` + 2.5` Y0 + 1.` Y1 + 
     0.5` Y2))) == 
 0, -((0.007561436672967865` (1.` + Y0^0.5`) (-3.` + Y0 + 
    2.` Y1))/((3.` + Y0^0.5` + 
    2.` Y1^0.5`) (2.0869565217391304` - 1.` Y0 - 
    0.9565217391304348` Y1 - 0.13043478260869565` Y2)^2)) - (
 0.01512287334593573` (1.` + Y1^0.5`) (-3.` + Y0 + 2.` Y1))/((3.` +
    Y0^0.5` + 2.` Y1^0.5`) (2.0869565217391304` - 1.` Y0 - 
   0.9565217391304348` Y1 - 0.13043478260869565` Y2)^2) + (
 0.031558185404339245` - 
 0.031558185404339245` Y0)/(1.2307692307692308` - 1.` Y0 - 
  0.15384615384615385` Y1 - 
  0.07692307692307693` Y2)^2 + (-0.3282051282051282` + 
 0.2512820512820513` Y0 + 0.05128205128205128` Y1 + 
 0.02564102564102564` Y2)/((-1.2307692307692308` + 1.` Y0 + 
   0.15384615384615385` Y1 + 0.07692307692307693` Y2) (-1.6` + 
   1.` Y0 + 0.4` Y1 + 0.2` Y2)) + (
0.2512820512820513` (-1.` + Y0) (-1.306122448979592` + 1.` Y0 + 
   0.20408163265306123` Y1 + 0.10204081632653061` Y2))/((1.` + 
   Y0^0.5`) Y0^0.5` (-1.2307692307692308` + 1.` Y0 + 
   0.15384615384615385` Y1 + 0.07692307692307693` Y2) (-1.6` + 
   1.` Y0 + 0.4` Y1 + 0.2` Y2)) + (
0.07173913043478261` (1.` + Y0^0.5`) (-1.9393939393939394` + 
   1.` Y0 + 0.787878787878788` Y1 + 
   0.15151515151515155` Y2))/((3.` + Y0^0.5` + 
   2.` Y1^0.5`) (-2.0869565217391304` + 1.` Y0 + 
   0.9565217391304348` Y1 + 0.13043478260869565` Y2) (-1.6` + 
   1.` Y0 + 0.4` Y1 + 0.2` Y2)) + (
0.14347826086956522` (1.` + Y1^0.5`) (-1.9393939393939394` + 
   1.` Y0 + 0.787878787878788` Y1 + 
   0.15151515151515155` Y2))/((3.` + Y0^0.5` + 
   2.` Y1^0.5`) (-2.0869565217391304` + 1.` Y0 + 
   0.9565217391304348` Y1 + 0.13043478260869565` Y2) (-1.6` + 
   1.` Y0 + 0.4` Y1 + 0.2` Y2)) - (
0.035869565217391305` (1.` + Y0^0.5`) (-3.` + Y0 + 
   2.` Y1) (-1.9393939393939394` + 1.` Y0 + 
   0.787878787878788` Y1 + 0.15151515151515155` Y2))/(
Y0^0.5` (3.` + Y0^0.5` + 2.` Y1^0.5`)^2 (-2.0869565217391304` + 
   1.` Y0 + 0.9565217391304348` Y1 + 
   0.13043478260869565` Y2) (-1.6` + 1.` Y0 + 0.4` Y1 + 
   0.2` Y2)) - (
0.07173913043478261` (1.` + Y1^0.5`) (-3.` + Y0 + 
   2.` Y1) (-1.9393939393939394` + 1.` Y0 + 
   0.787878787878788` Y1 + 0.15151515151515155` Y2))/(
Y0^0.5` (3.` + Y0^0.5` + 2.` Y1^0.5`)^2 (-2.0869565217391304` + 
   1.` Y0 + 0.9565217391304348` Y1 + 
   0.13043478260869565` Y2) (-1.6` + 1.` Y0 + 0.4` Y1 + 
   0.2` Y2)) + (
0.10760869565217392` (-3.` + Y0 + 2.` Y1) (-1.9393939393939394` + 
   1.` Y0 + 0.787878787878788` Y1 + 0.15151515151515155` Y2))/(
Y0^0.5` (3.` + Y0^0.5` + 2.` Y1^0.5`) (-2.0869565217391304` + 
   1.` Y0 + 0.9565217391304348` Y1 + 
   0.13043478260869565` Y2) (-1.6` + 1.` Y0 + 0.4` Y1 + 
   0.2` Y2)) == 0, 0 <= Y2 <= 1, 0 <= Y1 <= 1, 0 <= Y0 <= 1}, {Y2,
Y1, Y0}]
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  • 2
    $\begingroup$ Drop 0 <= Y2 <= 1, 0 <= Y1 <= 1, 0 <= Y0 <= 1}, use FindRoot instead of Solve, and replace {Y2, Y1, Y0} with {{Y2, 0.3}, {Y1, 0.2}, {Y0, 0.1}}. You should get as the answer {Y2 -> 0.169988, Y1 -> 0.209275, Y0 -> 0.290306} (which satisfies your constraints). $\endgroup$
    – JimB
    Nov 15, 2018 at 16:55
  • 1
    $\begingroup$ The search starting values for FindRoot are not critical in this instance. Look at SeedRandom[0]; Table[ FindRoot[eqns, {{Y2, RandomReal[]}, {Y1, RandomReal[]}, {Y0, RandomReal[]}}], {10}] $\endgroup$
    – Bob Hanlon
    Nov 15, 2018 at 19:39

2 Answers 2

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Call your expression e0. I removed the inequalities. Then we can make this into a system of bona fide polynomial equations in a few steps.

First rationalize to make the exponents into rationals. This also happenstance makes all coefficients into rationals.

e1 = MapAll[Expand, Rationalize[e0]];

There are a few variables in radicals. We replace them by squares (works fine since we only want positive solutions anyway).

e2 = Together[PowerExpand[e1 /. {Y0 -> y0^2, Y1 -> y1^2, Y2 -> y2^2}]];

We have denominators. We'll take a look at their square-less factors.

dene2 = Denominator[e2];
dens = Rest[FactorList[Apply[PolynomialLCM, dene2]]][[All, 1]]

(* Out[222]= {y0, y1, 
 3 + y0 + 2 y1, y2, -8 + 5 y0^2 + 2 y1^2 + y2^2, -16 + 13 y0^2 + 
  2 y1^2 + y2^2, -48 + 23 y0^2 + 22 y1^2 + 3 y2^2, -16 + 5 y0^2 + 
  2 y1^2 + 9 y2^2} *)

These are not allowed to vanish. Four our purposes (I checked this) it suffices just to make sure the individual variables do not vanish. We enforce that by adding a polynomial y0*y1*y2*yrecip - 1 (with a new variable yrecip that we really don't care about).

We now have a system that NSolve can handle in reasonable time.

AbsoluteTiming[
 sols = NSolve[Join[Numerator[e2], {y0*y1*y2*yrecip - 1}]];]

(* Out[228]= {5.081856, Null} *)

We gather the solutions of interest. First remove complex values, then remove those that fall outside the range of interest, and last take squares to account for the earlier change of variables.

solvals = {y0, y1, y2} /. sols;
realvals = Select[solvals, FreeQ[#, Complex] &];
Select[realvals, AllTrue[#, 0 < # < 1 &] &]^2

(* Out[248]= {{0.290306173455, 0.209275354184, 0.169987697235}} *)
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You can use NMinimize instaed of FindRoot[]:

If you set the equations inside of FindRoot (" first argument") by eq the solution is easily evaluated by

NMinimize[{1, eq}, {Y0, Y1, Y2}]
(*{1., {Y0 -> 0.290306, Y1 -> 0.209275, Y2 -> 0.169988}}*)

There is no need to define starting values ...

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