# Systems of nonlinear equations

Consider the nonlinear system of equations $$(1) \quad qk^2a_1^2E^2+wna_0a_1AE+pnka_0^2a_1E+rn^2a_0^2A^2-rn^2a_0^3A^2+qk^2a_0a_1ABE-qk^2na_0^2E^2=0,$$ $$(2) \quad nwa_1^2AE+nwa_0a_1AB+2pnka_0a_1^2AE+2pnka_0^2a_1AB+2qk^2a_1^2ABE-qk^2na_1^2ABE-2qk^2na_0a_1AmE+qk^2na_0a_1AB^2+2rn^2a_0a_1A^2-3rn^2a_0^2a_1A^2=0,$$ $$(3) \quad rn^2a_1^2A^2+nwa_1^2AB-nwa_0a_1Am+pnka_1^2AE+2pnka_0a_1^2AB-pnka_0a_1^2Am-2qk^2a_1^2m+qk^2a_1^2B^2-3qk^2na_0a_1AmB+qk^2na_1^2AB^2-qk^2na_1^2B^2+2qk^2na_1mE-2qk^2na_1AmE=0,$$ $$(4) \quad - nwa_1^2Am-rn^2a_1^3A^3+pnka_1^3AB-pnka_0a_1^2Am-2qk^2a_1^2mB+2qk^2na_1^2mB-3qk^2na_1^2AmE+2qk^2na_0a_1Am^2=0,$$ $$(5) \quad qk^2-pnka_1A+2qk^2nAm-qk^2nm=0;$$ My question for this nonlinear system is:

Using Matlab symbolic computation solve for $a_0$,$a_1$ and $w$ in terms of the rest variables $(q,k,p,n,A,B,E,r,m)$.

Here my tag is under matlab; nonlinear system; symbolic computation:::: if any please correct me

• Welcome to SciComp! As posed, this question (1) looks like a homework problem, (2) doesn't state what the broader scientific or computational problem is, and (3) doesn't show any attempt that you, the original poster, have made to solve the problem. Please address these issues. – Geoff Oxberry Apr 13 '14 at 6:37
• It is not a home work problem. I tried to solve by hand but i could not handle it. And this kind of system of equation can be solved by matlab: – Rida Tassew Apr 13 '14 at 14:35
• @RidaTassew What have you tried? What difficulty are you having? Have you tried just using solve()? If your question is primarily about how to use MATLAB's symbolic capabilities I think this not the right site to get help. – Doug Lipinski Apr 13 '14 at 14:48
• Actually i am new for matlab what i tried is just to solve by hand but i couldn't do that: and a friend of mine tried to solve by matlab but he is this much at matlab – Rida Tassew Apr 13 '14 at 15:06
• By itself, MATLAB is not a computer algebra system; it does not do symbolic manipulation. For that, you either need the MATLAB Symbolic Math Toolbox, or you need other software (for instance, you could do symbolic solves in Maple, or Mathematica). If you know Python, I recommend using SymPy or Sage, since these are free alternatives with a symbolic calculation capability. – Geoff Oxberry Apr 13 '14 at 21:58

I think it is unlikely that these equations have a solution. This is already apparent by the fact that you have 5 equations for three variables, but one can look at it in more detail:

• Take equation 5. It is linear in $a_1$ and does no involve the other variables. You can solve it for $a_1$ and plug the value into the other equations.

• If you do this, equation 4 becomes linear in the remaining variables, $a_0$ and $w$. Solve for one of them in terms of the other and plug the result into the first three equations. Let's say you solve for $w$ in terms of $a_0$.

• This makes equation 3 a quadratic equation in $a_0$ because it contains a term proportional to $a_0w$. Being a quadratic equation, it has two solutions that are easily computed.

• Equation 2 is also quadratic equation in $a_0$. You can again compute its two solutions. Only those values that simultaneously solve equations 2 and 3 can possibly be solutions of the joint system.

• Equation 1 is cubic in $a_0$. If any of the joint solutions of equations 2 and 3 also solve this equation (easily verified by plugging it in and seeing whether left and right hand sides of the equation are equal) then you have an answer.

This is a polynomial system of equations. You can try to find a closed form solution by computing a Groebner basis using a lexicographic monomial ordering. Doing so should give you similar information to that outlined by Wolfgang Bangerth in his answer.

• M.Bellow Riv: may be it can be solved using what you have said but the problem is that i never used before the software you are talking about. Furthere more i didn't take any numerical course. Dear brother thank you for your comment and answer. – Rida Tassew Apr 14 '14 at 21:58