Solving system of 9 non-linear equations with 9 variables

I am trying to solve a system of 9 equations with 9 unknown variables. But I could not find the solution. How can I resolve this? Thanks!

Case11phi11 = sym('Case11phi11','real');Case11phi21 = sym('Case11phi21','real');Case11phi31 = sym('Case11phi31','real');Case11phi41 = sym('Case11phi41','real');Case11phi51 = sym('Case11phi51','real');Case11phi61 = sym('Case11phi61','real');Case11phi71 = sym('Case11phi71','real');Case11phi81 = sym('Case11phi81','real');Case11phi91 = sym('Case11phi91','real');
Case11Equation1=0.15353635291833572086082838106454*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.15353635291833572086082838106454*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi11*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation2=0.27666294814139843879130234163141*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.27666294814139843879130234163141*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi21*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation3=0.34499305343666716537323839398319 *((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.34499305343666716537323839398319 *(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi31*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation4=-0.062661935670297878162282668473888*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))-0.062661935670297878162282668473888*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi41*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation5=-0.1129129064828849881524578433522 *((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))-0.1129129064828849881524578433522 *(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi51*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation6=-0.1408000913806153399899016724867*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))-0.1408000913806153399899016724867*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi61*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation7=0.1712854132141495204893606176184*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.1712854132141495204893606176184*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi71*(((2.5898^2)+1)/12)*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation8=0.3086456496635007095471901313621 *((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.3086456496635007095471901313621 *(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi81*(((2.5898^2)+1)/12)*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11Equation9=0.38487482990651228555131625015817*((Case11phi11^2)+(Case11phi21^2)+(Case11phi31^2)+(Case11phi41^2)+(Case11phi51^2)+(Case11phi61^2))+0.38487482990651228555131625015817*(((2.5898^2)+1)/12)*((Case11phi71^2)+(Case11phi81^2)+(Case11phi91^2))-Case11phi91*(((2.5898^2)+1)/12)*((Case11phi11+Case11phi21+Case11phi31)+0.3*(Case11phi41+Case11phi51+Case11phi61))
Case11sol=solve(Case11Equation1,Case11Equation2,Case11Equation3,Case11Equation4,Case11Equation5,Case11Equation6,Case11Equation7,Case11Equation8,Case11Equation9)
Case11phi=[Case11sol.Case11phi11 Case11sol.Case11phi21 Case11sol.Case11phi31 Case11sol.Case11phi41 Case11sol.Case11phi51 Case11sol.Case11phi61 Case11sol.Case11phi71 Case11sol.Case11phi81 Case11sol.Case11phi91]'
Warning: Explicit solution could not be found. 
> In solve at 179 
 
Case11sol =
 
[ empty sym ]
 
Comma separated list expansion has cell syntax for an array that is not a cell.

Error in sym/subsref (line 1578)
            [inds{k},refs{k}] = privformat(inds{k});

[Mihret Melese]

Anyone please!

[Walter Roberson]

You will can solve the first three equations in the first three variables by specifying that you want MaxDegree 4 in solve(). The 4 results will be long, but you can substitute each one of them in turn together with the equations up to the 4th, 5th, and 6th equations. Mixing in the 7th equation gives you two sets of solutions, but you can show that for one of the sets, when the inputs are all positive like you have asserted, that the two quadratics have complex roots and so should probably be ruled out (though I did not check whether those complex roots are raised to powers such that the terms would be real-valued.)

I am still working on the form for the 9th variable.

I think what is happening is that MATLAB is deciding that the solutions cannot be positive. I do not know at the moment if it is mistaken on that.

WIth the assumptions you have in place, MATLAB can solve for the first 3 in the first 3 unknowns, but says there is no solution for the first 4 in the first 4 unknowns.

[Mihret Melese]

Thanks Walter for the reply. The equations above have a real valued solutions. The roots are the ones listed below. You may ask why I need to solve this if the values are known already. But I have the same problem i.e. other 9 equations with 9 unknowns. Solving this can be seen as a validation to proceed with the other unsolved problems. The equations are derived from the matrix analysis of the underneath equations for first mode. The variables Case11phi11- Case11phi91are elements of the first eigen vector (first mode). The problem is finding Phi 1 knowing the first mode modal mass distribution (S1). Please let me know your finding. Thanks.

mass =

[ M, 0, 0, 0, 0, 0, 0, 0, 0]

[ 0, M, 0, 0, 0, 0, 0, 0, 0]

[ 0, 0, M, 0, 0, 0, 0, 0, 0]

[ 0, 0, 0, M, 0, 0, 0, 0, 0]

[ 0, 0, 0, 0, M, 0, 0, 0, 0]

[ 0, 0, 0, 0, 0, M, 0, 0, 0]

[ 0, 0, 0, 0, 0, 0, (M*(Lx^2 + Ly^2))/12, 0, 0]

[ 0, 0, 0, 0, 0, 0, 0, (M*(Lx^2 + Ly^2))/12, 0]

[ 0, 0, 0, 0, 0, 0, 0, 0, (M*(Lx^2 + Ly^2))/12]

Case11phi11=0.256212

Case11phi21=0.461678

Case11phi31=0.575703

Case11phi41=-0.10457

Case11phi51=-0.18842

Case11phi61=-0.23496

Case11phi71=0.445042

Case11phi81=0.801938

Case11phi91=1

[Mihret Melese]

I have edited the previous assumed positive value to real value.

[Mihret Melese]

Walter Roberson please let me know your finding! Thanks.

[Walter Roberson]

My system ran out of memory trying to find explicit solutions.

Note that solve() is only suitable for the case where you need closed form indefinitely-precise solutions, especially if you need all of them. Otherwise, you should use vpasolve()