Non-typical system of nonlinear equations

I tried to solve this system of equations, using fsolve and various newton-raphson functions, but to no avail:

x*y-z^2-1 = 0
x*y*z+y^2-x^2-2 = 0
e^x+z-e^y-3 = 0

How to bite this?

Answer 1

[Walter Roberson]

There two real-valued solutions and two complex-valued solutions.

syms x y z
eq1=x.*y-z.^2-1 == 0;
eq2=x.*y.*z+y.^2-x.^2-2 == 0;
eq3=exp(x)+z-exp(y)-3 == 0;
eqns = [eq1, eq2, eq3];
string(eqns)

ans = 1×3 string array
    "x*y - z^2 - 1 == 0"    "y^2 - x^2 + x*y*z - 2 == 0"    "z + exp(x) - exp(y) - 3 == 0"

solx = solve(eqns(1),x)

solx = 

eqns2 = subs(eqns(2:end), x, solx)

eqns2 = 

soly = solve(eqns2(1), y, 'maxdegree', 4)

soly = 

eqns3 = subs(eqns2(2:end), y, soly)

eqns3 = 

solz = arrayfun(@vpasolve, eqns3)

solz = 

backz = solz

backz = 

backy = simplify(arrayfun(@(Y,Z) subs(Y, z, Z), soly, backz))

backy = 

backx = simplify(arrayfun(@(Y,Z) subs(solx, {y, z}, {Y, Z}), backy, backz))

backx = 

X = double(backx(:));
Y = double(backy(:));
Z = double(backz(:));

sols = table(X, Y, Z)

sols = 4×3 table
                      X                                       Y                                      Z                 
    _____________________________________    ___________________________________    ___________________________________

       1.77767191801074+0i                    1.42396059788849+0i                    1.2374711177317+0i                
      0.312665572783089-1.92640000714264i     -3.6173710991782+6.60644699784i       3.50141439437029+1.29006613162679i 
      -6.00007674738141+0i                   -1.82891828362435+0i                   3.15810862169672+0i                
    -0.0235291143948645-1.64861787491676i    0.226753337733853+4.16490693352808i    2.42290841496534-0.097367728502186i

[KarolN]

The best answer of all submitted and works perfect! Thanks! Happy New Year!

Answer 2

[Yusuf Suer Erdem]

Try these codes below please;

syms x y z
eq1=x.*y-z.^2-1 == 0;
eq2=x.*y.*z+y.^2-x.^2-2 == 0;
eq3=exp(x)+z-exp(y)-3 == 0;
solve(eq1,eq2,eq3,x,y,z)

Warning: Unable to solve symbolically. Returning a numeric solution using vpasolve.

ans = struct with fields:
    x: 1.7776719180107405499159549245637
    y: 1.4239605978884890880290410180788
    z: 1.2374711177317033717819050711785

[KarolN]

This works perfect! Thanks! Happy New Year!