Bounds for eigenvalues of matrix polynomials
Let \(P(\lambda)= \lambda^m A_m+ \lambda^{m-1} A_{m-1}+\cdots+ A_0\) be a polynomial in \(\lambda\) with matrix coefficients. A great variety of upper and lower bounds are derived here for \(|\lambda|\), where \(\lambda\) is an eigenvalue of \(P\). The authors concentrate on bounds that are of practical use, and many of these generalize known inequalities. Some are based on 1, \(\infty\) and 2-norms of block companion matrices \(C_L\) and \(C_U\) whose blocks are constructed from the \(A_j\). Two of the bounds generalize earlier ones of Cauchy, Montel, Carmichael and Mason, and Kojima. Others involve the singular values of \(C_U\) and \(C_L\) and the characteristic polynomial of \(P\). The authors did numerical experiments to test many of their bounds. One involved a polynomial of degree 9 with \(5\times 5\) coefficient matrices having random entries from the normal \((0,1)\) distribution. The upper bounds were quite sharp, the lower bounds more variable. Two other experiments involved polynomials of degree 2 with much larger coefficient matrices, one arising from a differential equation and the other from a structural dynamics model. Some of the upper bounds for the dominant eigenvalue were quite good, but one was a severe overestimate. The authors remark that identifying bounds suited to particular classes of problems would be valuable.
scientific article; zbMATH DE number 1864788
scientific article