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Result verification for eigenvectors and eigenvalues. (English) Zbl 0813.65077
Herzberger, Jürgen, Topics in validated computations. Proceedings of the IMACS-GAMM international workshop, Oldenburg, Germany, 30 August - 3 September 1993. Amsterdam: Elsevier. Stud. Comput. Math. 5, 209-276 (1994).
We quote substantially from the introduction:
The paper is addressed to the matrix eigenproblem for $$n \times n$$ matrices $$A$$ and to related topics such as the generalized eigenproblem, the singular value problem and the inverse eigenvalue problem. Mainly, the considerations are restricted to the case of real eigenvalues $$\lambda^*$$ and real eigenvectors $$x^*$$. The following problems are considered:
1. Find an interval $$[\lambda]$$ which contains at least one (or exactly one) real eigenvalue $$\lambda^*$$ of $$A$$. In particular, find $$[\lambda]$$ such that the bounds $$\underline\lambda$$, $$\overline \lambda$$ of $$[\lambda]$$ differ only in the last few digits. (In this case their common leading digits coincide with those of $$\lambda^*$$. This aspect is particularly important if one wants to control rounding errors when approximating $$\lambda^*$$ on a computer.)
2. Find interval enclosures $$[\lambda]_ i$$ simultaneously for all real eigenvalues $$\lambda_ i$$ of $$A$$.
3. Find an interval vector $$[x]$$ which contains at least one (or exactly one) eigenvector $$x^*$$ associated with an eigenvalue $$\lambda^*$$. (Again tight bounds $$\underline x$$, $$\overline x$$ are required in order to gurantee digits of $$x^*.)$$
4. Find the interval quantities in 1.–3. when $$A$$ is replaced by an interval matrix $$[A] \in M_{nn} (I(\mathbb{R}))$$. In this case $$[\lambda]$$ means an interval which contains for each $$A \in [A]$$ at least one (or exactly one) eigenvalue of $$A$$; furthermore, $$[x]$$ means an interval vector with the following property: for any $$\lambda^* \in [\lambda]$$ and any $$A \in [A]$$ for which $$\lambda^*$$ is an eigenvalue, the vector $$[x]$$ contains at least one (or exactly one) eigenvector of $$A$$ associated with $$\lambda^*$$.
In detail, the arrangement of the contents is as follows. Sections 2-4 are concerned with preliminaries (notations and auxiliary results), quadratic systems, and some (classical) existence theorems for eigenvalues $$\lambda^*$$ and corresponding eigenvectors $$x^*$$ (yielding also estimates for $$\lambda^*$$ and $$x^*)$$. In Sec. 5, a method for general matrices $$A$$ with simple, double or nearly double eigenvalues is presented, whereas, in Sec. 6, for the case of symmetric $$A$$ also multiple eigenvalues and clusters of eigenvalues are admitted.
In Sec. 7, the generalized eigenproblem is considered, and, in Sec. 8, a method is derived in order to verify and enclose singular values and singular vectors. In Sec. 9, an inverse eigenvalue problem is described for which the methods of Secs. 5-6 apply. Sec. 10 terminates with a brief sketch on additional topics in result verification for eigenpairs and with biographical remarks. The paper includes 9 examples and a list of 144 references.
For the entire collection see [Zbl 0803.00016].

##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 65G30 Interval and finite arithmetic
##### Software:
EISPACK; C-XSC 2.0; LAPACK