How can I find a correct value of the guess for my system of non linear equations ??

I want to solve the following system of 3 non linear equations using fsolve.The unknowns to be found are X , Y and Z.

X-241.4*[(1.919*10^-8)*Y^1.429-(5.492*10^-10)*Y^1.714]^(1/2)=0

X*(3.121*10^5)+42.29-X*Z*1117-2.482*Z=0

X+0.002222-0.001976*Y*Z^(-1/2)=0

In order to solve this system using fsolve I have to enter a value of a guess for the values of X , Y and Z.

Please is there any way of finding a correct guess for this system ????

When I run fsolve with a random guess, I receive the message

 Maximum number of function evaluations reached:
 increase options.MaxFunEvals.

Answer 1

[Walter Roberson]

You can use the first equation to isolate X in terms of Y. Substitute that X in to the second equation and then you can isolate Z in terms of Y. Substitute the X and Z in to the third equation to get an expression for Y. You can then work on just that expression in one variable.

When the floating point coefficients are converted to rational form, the expression for Y contains as a repeated sub-expression sqrt(56414126863825657569157648*Y^(1429/1000)-1614519984922625768635418*Y^(857/500)) . You can solve() this for Y and take the larger of the two solutions (the smaller is 0) and use that as an initial guess for your fsolve(). The actual solution is reasonably close by.

[kamal kiki]

Hi Walter Roberson

Can you please explain more the second part of your answer (starting from the floating point coefficients ), because I understood only the first part of your answer.

Answer 2

[Alex Sha]

it is not easy to get correct initial start values, try to use global optimization algorithm instead of local ones like fsolve, refer to solution below:

x: 29.1663547800969
y: 246736.308525207
z: 279.389144571331