Approximate method for eigensensitivity analysis of a defective matrix. (English) Zbl 1210.65087
Summary: Based on the exact modal expansion method, an arbitrary high-order approximate method is developed for calculating the second-order eigenvalue derivatives and the first-order eigenvector derivatives of a defective matrix. The numerical example shows the validity of the method. If the different eigenvalues \(\mu (1),\dots ,\mu (q)\) of the matrix are arranged so that |\(\mu (1)|\leq \cdots \leq |\mu (q)\)| and satisfy the condition that |\(\mu (q_{1})|<|\mu (q_{1}+1)\)| for some \(q_{1}<q\), and if the approximate method only uses the left and right principal eigenvectors associated with \(\mu (1),\dots ,\mu (q_{1})\), then associated with \(\mu (h)(h\leq q_{1})\) the errors of the eigenvalue and eigenvector derivatives by the \(p\)th-order approximate method are nearly proportional to |\(\mu (h)/\mu (q_{1}+1)|^{p+1}\).
MSC:
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
Keywords:
eigensensitivity analysis; eigenvalue and eigenvector derivatives; defective matrix; modal expansion method; numerical exampleReferences:
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