Linear Approx to Nonlinear - Sara.sagews

var("M,S")
Mprime = 24*M-2*M^2-3*M*S
Sprime = 15*S-S^2-3*M*S
J = jacobian([Mprime,Sprime],[M,S])
show(J)

(M, S)

$\displaystyle \left(\begin{array}{rr} -4 \, M - 3 \, S + 24 & -3 \, M \\ -3 \, S & -3 \, M - 2 \, S + 15 \end{array}\right)$

J36 = J.subs(M=3,S=6)
show(J36)

$\displaystyle \left(\begin{array}{rr} -6 & -9 \\ -18 & -6 \end{array}\right)$

J36.eigenvalues()

[-9*sqrt(2) - 6, 9*sqrt(2) - 6]

for eval in J36.eigenvalues():
    show(n(eval))

$\displaystyle -18.7279220613579$

$\displaystyle 6.72792206135786$