A bisection method for measuring the distance of a quadratic matrix polynomial to the quadratic matrix polynomials that are singular on the unit circle

Computing the distance to continuous-time instability of quadratic matrix polynomials

“…Remark 2 Item 2 of Theorem 4 is in line with a result in [15], where it is shown that a condition of the form s < dp − 2η, where η is the norm of the backward error of the generalized Schur form of (34), is necessary to numerically guarantee that the pencil (34) has no eigenvalues on the unit circle. In other words, if the pencil (34) has an eigenvalue on the unit circle, then dp ≤ s + 2η.…”

“…Remark 3 In view of Theorem 4 and ( 11), ( 15), ( 18), it appears that only the case p = 2 makes it possible to compute a lower bound with a guaranteed accuracy using the algorithm developed in [15]. The cases p = 1 and p = ∞ are limited by the constraints | Im λ| > 1, | Im λ| ≤ 1 and Im λ > 0, Im λ ≤ 0, (see (11) Algorithm 2 is tested below on the following three quadratic matrix polynomials hospital, pdde stability, and sign2 taken from the NLEVP collection [1], see also [22].…”

“…The scaling (20) is not used. Step 6 is based on the QZ algorithm and the Laub trick (see [15] for more details).…”

“…In [5] a Newton-based procedure is implemented to accelerate local convergence of an approximation to the distance. A theory of complex and real stability radii of matrix polynomials is developed in [18,6] and closely related issues are addressed in [8,14,15,22].…”

Modified bisection method for solving nonlinear equations

“…Also fixed point iteration method is root finding method of ( ) = 0 using the form of ( ) = , in which a sequence is generated so that it converges to a root. The bisection method [3,4,5,6,7] is one of root finding method. Cut half of the interval and use the Intermediate Value Theorem [8] to select the small sub-interval, where the root is present.…”