Result verification for the real quadratic eigenvalue problem (Q2910961)

Given \(A_0,A_1,A_2\in \mathbb{R}^{n\times n}\), \(\text{det}(A_2)\neq 0\), the author considers the real quadratic eigenvalue problem \(P(\lambda)x= 0\), \(A\in\mathbb{R}\), \(x\in\mathbb{R}^n\setminus\{0\}\), where \(P(\lambda)= A_2\lambda^2+ A_1\lambda+ A_0\). Solutions \((x^*,\lambda^*)\) of this problem are called eigenpairs. In the paper \(\lambda^*\) is restricted to be a simple real eigenvalue, i.e., a simple real zero of \(\text{det}(P(\lambda))\).NEWLINENEWLINE In order to verify and to enclose \((x^*,\lambda^*)\), an existence and uniqueness result on the solution of cubic systems is stated and proved first. This result is applied to the quadratic eigenvalue problem by looking for zeros of an appropriate nonlinear function \(f:\mathbb{R}^{b+1}\to \mathbb{R}^{n+1}\). Based on a sufficiently good approximation of \((x^*,\lambda^*)\) one is able to construct an interval vector \([x]\) and an interval \([\lambda]\) which together contain a unique eigenpair if some given component of the eigenvector is normalized by one. Starting with these interval quantities, an interval iterative method is presented such that all the interval iterates contain the eigenpair \((x^*,\lambda^*)\) and finally contract to it -- at least in exact arithmetic.NEWLINENEWLINE Using an appropriate interval software like INTLAB, the method can also be run on a computer taking into account rounding errors. The efficiency of the method is demonstrated by numerical examples.