using "solve" for a system of nonlinear equations

I have a set of 6 nonlinear equations as follows (a1,a2,a3,b1,b2,b3 are unknowns). I know that one of the answers of this system must be as following: a1=3, a2=1, a3=3, b1=0.2, b2=0.3, b3=0.4 (it exactly satisfies my equations). When I use "solve" it gives me 4 different set of answers but surprisingly none of them is the above mentioned answer that I am anticipating. In fact "solve" is neglecting at least one of the answers of this system. How can I adjust or use "solve" to give me that answer?

syms a1 a2 a3 b1 b2 b3

X1=0.16;
X2=0.9466667;
X3=2.97;
Z1=2.97;
Z2=0.9866667;
Z3=0.38;

[a1, a2, a3, b1, b2, b3]=solve(X1==b1-b2*b3/a1,X2==a2-b3^2/a1,X3==a3-b2^2/a1,Z1==a1-b2^2/a3,Z2==a2-b1^2/a3,Z3==b3-b1*b2/a3); 

a1=eval(a1)
a2=eval(a2)
a3=eval(a3)
b1=eval(b1)
b2=eval(b2)
b3=eval(b3)

2- Do you know any other command or user defined function to use for solving system of nonlinear equations and gives me all the answers ? (does not require initial guess like fsolve)

Answer 1

[Matt J]

2- Do you know any other command or user defined function to use for solving system of nonlinear equations and gives me all the answers ? (does not require initial guess like fsolve)

No, that's not possible for general nonlinear equations and numerical solvers. The set of solutions can be infinite for example, as in the system

      x-y=0
     x^2-y^2=0

When I use "solve" it gives me 4 different set of answers but surprisingly none of them is the above mentioned answer that I am anticipating.

What answers does it give you?

[Sayed]

Dear Matt J, Thanks for your answer. Here is my answer set for "solve", but does not include 3, 1, 3, 0.2, 0.3, 0.4

a1 =
  -0.640000000000000
  -0.640000000000000
   6.014182570014500
  12.345817429985500
a2 =
   0.946666666666667
   0.946666666666667
   1.112940302087347
   1.027665758518714
a3 =
  -0.640000000000000
  -0.640000000000000
   6.014182570014500
  12.345817429985500
b1 =
   0.160000000000000
   0.160000000000000
   0.871454357496375
  -0.711454357496375
b2 =
   1.520000000000000
   1.520000000000000
   4.278816396215563
 -10.758816396215565
b3 =
     0
     0
     1
     1

[Matt J]

Yeah. Well, you clearly have lots of alternative solutions to the system of equations at hand. Yet, there is something that makes you prefer [3, 1, 3, 0.2, 0.3, 0.4] over these other solutions. If there is something that distinguishes this solution from the others, you should express that distinguishing property in equation form and add that equation to your system. That will make the solution you want more uniquely identifiable by the solver.

[Sayed]

Dear Matt J, Thanks for all your helpful comments. Actually this answer is my only desired one for this system of equations, but I'm still working on finding a criteria to distinguish the desired answer between the answers set. However, you mentioned that "you clearly have lots of alternative solutions". I need to find all the existing answers for the above system. Would you please let me know that how I should use solve (or any other command) to give me the rest of answers? (doesn't matter if it can't give all of them simultaneously) I appreciate your response in advance.

[Matt J]

I need to find all the existing answers for the above system.

Do you know how many there are? Are you sure there are not infinite solutions?

[Sayed]

No, I am not sure whether the answers are really infinite or not, but it is something important to know. Is there any way to find that or at least find some other answers (with solve or ...)?

[Matt J]

If you have the Global Optimization Toolbox, you could try MultiStart

http://www.mathworks.com/help/gads/example-finding-global-or-multiple-local-minima.html#bsds4eg

This would involve basically rewriting the objective function as a least squares problem. MultiStart then manages the running of a least squares minimization algorithm from many starting points and can find multiple solutions.