Consider a second order iterative scheme u(k 2) Au(k l) Bu(k) Define a quadratic eigenvalue to be a complex number that satisfies det(2I A B) 0 Prove that the system is asymptotically stable if and only if all its quadratic eigenvalues satisfy 1 Look at the equivalent first order system and use Exercise 1 9 23b

Question: Consider a second order iterative scheme u(k+2) = Au(k+l) + Bu(k). Define a quadratic eigenvalue to be a complex number that satisfies det(2I - A

Consider a second order iterative scheme u(k+2) = Au(k+l) + Bu(k). Define a quadratic eigenvalue to be a complex number that satisfies
det(λ2I - λA - B) = 0.
Prove that the system is asymptotically stable if and only if all its quadratic eigenvalues satisfy |λ| < 1. Look at the equivalent first order system and use Exercise 1.9.23b.

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