### 4.11 How to numerically solve a set of non-linear equations?

Numerically solve the following three equations for $$e,a,f$$ \begin {align*} eq1 &= r1 - a(e-1) \\ eq2 &= delT - \sqrt {\frac {a^3}{\mu }}(e*\sinh (f)-f) \\ eq3 &= r2 - a(e \cosh (f)-1) \end {align*}

 Mathematica ClearAll[a, e, f] delT = 5.215*60*60; r1 = 200 + 6378; r2 = 130000; mu = 3.986*10^5; eq1 = r1 - a*(e - 1) == 0; eq2 = delT - Sqrt[a^3/mu]* (e*Sinh[f] - f) == 0; eq3 = r2 - a*(e*Cosh[f]-1)==0; sol = NSolve[{eq1, eq2, eq3}, {a, e, f}, Reals]  {{a->12029.39633, e->1.546827108, f->2.721303232}} 

 Matlab clear all syms a e f; delT = 5.215*60*60; r1 = 200+6378; r2 = 130000; mu = 3.986*10^5; eq1 = r1 - a*(e-1); eq2 = delT - sqrt(a^3/mu)*(e*sinh(f)-f); eq3 = r2 - a*(e*cosh(f)-1); sol = feval(symengine,'numeric::solve',... {eq1,eq2,eq3}); vpa(sol,6)  ans = [a == 12029.4, e == 1.54683, f == 2.7213]  Another option is solve but slower sol = solve(eq1,eq2,eq3,a,e,f);  sol = a: [1x1 sym] e: [1x1 sym] f: [1x1 sym] sol.a 12029.396328714435126444927089827 sol.e 1.5468271075497087492979481526009 sol.f 2.7213032317471583123822097902877