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Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues

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Abstract

An improved modal truncation method with arbitrarily high order accuracy is developed for calculating the second- and third-order eigenvalue derivatives and the first- and second-order eigenvector derivatives of an asymmetric and non-defective matrix with repeated eigenvalues. If the different eigenvalues λ 1, λ 2, …, λ r of the matrix satisfy |λ 1| ⩽ … ⩽ |λ r | and |λ s | < |λ s+1| (s ⩽ r−1), then associated with any eigenvalue λ i (i ⩽ s), the errors of the eigenvalue and eigenvector derivatives obtained by the qth-order approximate method are proportional to |λ i /λ s+1 | q+1, where the approximate method only uses the eigenpairs corresponding to λ 1, λ 2, …, λ s . A numerical example shows the validity of the approximate method. The numerical example also shows that in order to get the approximate solutions with the same order accuracy, a higher order method should be used for higher order eigenvalue and eigenvector derivatives.

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, P. R. China

    Zhen-yu Zhang  (张振宇)

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  1. Zhen-yu Zhang  (张振宇)
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Correspondence to Zhen-yu Zhang  (张振宇).

Additional information

Project supported by the National Natural Science Foundation of China (No. 11101149) and the Basic Academic Discipline Program of Shanghai University of Finance and Economics (No. 2013950575)

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Zhang, Zy. Improved modal truncation method for eigensensitivity analysis of asymmetric matrix with repeated eigenvalues. Appl. Math. Mech.-Engl. Ed. 35, 437–452 (2014). https://doi.org/10.1007/s10483-014-1803-6

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  • DOI: https://doi.org/10.1007/s10483-014-1803-6

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Chinese Library Classification

2010 Mathematics Subject Classification

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