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Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors
Author(s) -
Yoon Gil Ho,
Donoso Alberto,
Carlos Bellido José,
Ruiz David
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6442
Subject(s) - eigenvalues and eigenvectors , eigenvalue perturbation , jacobian matrix and determinant , mathematics , eigenvalues and eigenvectors of the second derivative , sensitivity (control systems) , defective matrix , modal matrix , basis (linear algebra) , matrix (chemical analysis) , algorithm , diagonalizable matrix , symmetric matrix , geometry , physics , materials science , quantum mechanics , electronic engineering , engineering , composite material
Summary It is well known that the sensitivity analysis of the eigenvectors corresponding to multiple eigenvalues is a difficult problem. The main difficulty is that for given multiple eigenvalues, the eigenvector derivatives can be computed for a specific eigenvector basis, the so‐called adjacent eigenvector basis. These adjacent eigenvectors depend on individual variables, which makes the eigenvector derivative calculation elaborate and expensive from a computational perspective. This research presents a method that avoids passing through adjacent eigenvectors in the calculation of the partial derivatives of any prescribed eigenvector basis. As our method fits into the adjoint sensitivity analysis , it is efficient for computing the complete Jacobian matrix because the adjoint variables are independent of each variable. Thus our method clarifies and unifies existing theories on eigenvector sensitivity analysis. Moreover, it provides a highly efficient computational method with a significant saving of the computational cost. Additional benefits of our approach are that one does not have to solve a deficient linear system and that the method is independent of the existence of repeated eigenvalue derivatives of the multiple eigenvalues. Our method covers the case of eigenvectors associated to a single eigenvalue. Some examples are provided to validate the present approach.