A simple method for error bounds of eigenvalues of symmetric matrices. (English) Zbl 0981.65042
If \(\rho_1\leq\rho_2\leq\cdots\leq\rho_n\) are the eigenvalues of a symmetric matrix \(A\) and \(\widetilde{\rho}\) is an approximate eigenvalue, then, given \(\delta_1,\delta_2\geq 0\), it is shown how to compute an interval \([\widetilde{\rho}-\varepsilon(\delta_1),\widetilde{\rho}+\varepsilon(\delta_2))\) that contains the eigenvalues \(\rho_k,\ldots,\rho_{k+r}\), where \(k-1\) and \(k+r\) are the number of negative eigenvalues of \(Y_1=A-(\widetilde{\rho}-\delta_1)I\) and \(Y_2=A-(\widetilde{\rho}+\delta_2)I\), respectively. The \(\varepsilon(\delta_i)\) are computed from the numerical error in the Cholesky decompositions of \(Y_i\). This technique can also be applied to the generalized eigenvalue problem \(Ax=\rho Bx\) if \(A\) and \(B\) are positive definite.
Reviewer: Adhemar Bultheel (Leuven)
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 65G30 | Interval and finite arithmetic |
Keywords:
error bounds; symmetric matrix; interval arithmetic; Cholesky decompositions; generalized eigenvalue problemReferences:
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