Exhaustive search with 4 variables

Dear Friends., I have two non-linear equations(Highly non linear simultaneous equations with 4th degree) with 5 unknowns such as x, y, a1,a2 and a3. x and y variables must be real and exact solution. however a1, a2 and a3 can be any arbitrary values , satisfying the following constraints. a1+a2+a3=10; a1+a2<=10; a2>0 a3>0 Can any body please suggest me how to solve this problem? If needed I can provide those two equations. Please help me. this is research problem. Thanks in advance.

Answer 1

[John D'Errico]

You can't. Well, you can, in a sense. But you won't have some nice, clean "solution" that can be written on paper.

You have 5 unknowns, in the 5 variables {x,y,a1,a2,a3}. You have 3 equalities (two nonlinear, one linear), and a few inequalities that really do not bound things very much anyway. What matters is that if any solutions do exist, there will generally be infinitely many such solutions, that will live on some curved surface in that 5-dimensional space of your unknowns.

Since your equations are of a high enough total degree that almost certainly no analytical solution exists, the only description of that solution locus will be the set of equations you already have.

You can decide to solve it parametrically, choosing perhaps values of a2 and a3 such that both are positive. That will allow you to choose a1 to satisfy the linear constraint exactly. Then, given numeric values for a1,a2,a3, solve for x and y using a tool like fsolve or vpasolve. Note that zero, one, or more solutions may well exist, although fsolve will generate only one solution for any starting values (x0,y0). (Generally it would be a finite number of solutions.) Since you are restricting the solutions to real variables, it is even more difficult to predict how many solutions exist.

[Dr. Hareesha N G]

Dear John D'Errico, Thank you very much for your kind reply and useful suggestions. I completely agree with the solution you have provided in the third paragraph. By using exhaustive search method for { a1, a2 and a3 } and substituting these variables in the eqn-1 and eqn-2 we can solve them, Since, we get two non linear equations in two unknowns {x, y}. I am using sylvester's dialytic elimination method to solve these equations. But, my problem is how to search these variables {a1, a2 and a3 } satisfying the constraints as mentioned earlier. I have a research paper where this approach is used. if required i'll send that paper. If possible, kindly send me your mail ID to this mail ID: [e-mail] to discuss more about this topic. I will be very grateful to you if you can help me solve this issue.

The equations are as below.

Eqn-1: 4*a2^2*x^3*y^4 + 8*a2^2*x^3*y^2 + 4*a2^2*x^3 - 4*a2^2*x*y^4 - 8*a2^2*x*y^2 - 4*a2^2*x + 4*a2*a3*x^4*y^3 + 4*a2*a3*x^4*y + 4*a2*a3*x^3*y^4 - 4*a2*a3*x^3 + 4*a2*a3*x*y^4 - 4*a2*a3*x - 4*a2*a3*y^3 - 4*a2*a3*y - 2*a1*a2*x^3*y^4 - 4*a1*a2*x^3*y^2 - 2*a1*a2*x^3 - 2*a1*a2*x*y^4 - 4*a1*a2*x*y^2 - 2*a1*a2*x + 6*a3^2*x^4*y^3 - 6*a3^2*x^4*y + 12*a3^2*x^2*y^3 - 12*a3^2*x^2*y + 6*a3^2*y^3 - 6*a3^2*y - 2*a1*a3*x^4*y^3 - 2*a1*a3*x^4*y - 4*a1*a3*x^2*y^3 - 4*a1*a3*x^2*y - 2*a1*a3*y^3 - 2*a1*a3*y

Eqn-2: - a1^2*x^4*y^4 - 2*a1^2*x^4*y^2 - a1^2*x^4- 2*a1^2*x^2*y^4 - 4*a1^2*x^2*y^2 - 2*a1^2*x^2- a1^2*y^4 - 2*a1^2*y^2 - a1^2+ 2*a1*a2*x^4*y^4 + 4*a1*a2*x^4*y^2 + 2*a1*a2*x^4- 2*a1*a2*y^4 - 4*a1*a2*y^2 - 2*a1*a2+ 2*a1*a3*x^4*y^4- 2*a1*a3*x^4+ 4*a1*a3*x^2*y^4 - 4*a1*a3*x^2+ 2*a1*a3*y^4 - 2*a1*a3 - 2*a2^2*x^4*y^4- 4*a2^2*x^4*y^2- 2*a2^2*x^4+ 12*a2^2*x^2*y^4 + 24*a2^2*x^2*y^2 + 12*a2^2*x^2 - 2*a2^2*y^4 - 4*a2^2*y^2 - 2*a2^2- 4*a2*a3*x^4*y^4+ 4*a2*a3*x^4+ 16*a2*a3*x^3*y^3 + 16*a2*a3*x^3*y + 16*a2*a3*x*y^3 + 16*a2*a3*x*y+ 4*a2*a3*y^4 - 4*a2*a3- 3*a3^2*x^4*y^4+ 18*a3^2*x^4*y^2- 3*a3^2*x^4 - 6*a3^2*x^2*y^4 + 36*a3^2*x^2*y^2 - 6*a3^2*x^2- 3*a3^2*y^4 + 18*a3^2*y^2 - 3*a3^2

Thanks in advance. Hareesha N G

[Alex Sha]

There are multi-solutions like:

1:

a1: 5.46730225513182
a2: 0.851618737586692
a3: 3.68107900728144
x: 0.000246447680731675
y: 2.16879039641196

2:

a1: 5.85785865095287
a2: 4.14212848033602
a3: 1.286871370499E-5
x: 2.41422385831
y: 0.564810456396699

3:

a1: 6.32500394092923
a2: 0.0208116349433613
a3: 3.65418565174466
x: 0.417501331245315
y: 1.94129445225702