# Non-typical system of nonlinear equations

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KarolN on 30 Dec 2021
Commented: KarolN on 30 Dec 2021
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?

Walter Roberson on 30 Dec 2021
Edited: Walter Roberson on 30 Dec 2021
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 on 30 Dec 2021
The best answer of all submitted and works perfect! Thanks! Happy New Year!

Yusuf Suer Erdem on 30 Dec 2021
Edited: Walter Roberson on 30 Dec 2021
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 on 30 Dec 2021
This works perfect! Thanks! Happy New Year!

R2021b

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