Highly efficient general method for sensitivity analysis of eigenvectors with repeated eigenvalues without passing through adjacent eigenvectors

Abstract

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.

A Rellich's result revisited and sensitivity of solutions of parametrized linear systems

“…Combining Equations ( 8) and ( 9) we can obtain a way to compute the derivatives inspired in Nelson's method (that was originally developed for eigenvector sensitivity, see [4,5,22]). Note that the general solution of Equation (8) must be of the form…”

“…[3] is a classical reference on the subject. Sensitivity of eigenvalues and eigenvectors of linear problems is a matter of great practical interest in many engineering contexts, as for instance and just for citing one among many, structural design [4,5]. Of course, the problems we study could be seen as particular cases of eigenvector sensitivity analysis, although our questions are even more elementary and our aim here is to check how far we can get pushing forward the elementary ideas in the proof of Theorem 1.…”

“…Finally, Section 4 is devoted to the exposition of two sensitivities computation algorithms for the multi-parameter case and for (n − 1)-deficient systems. We provide a direct method, inspired in Nelson's Method for simple eigenvector derivative calculation [20], and an adjoint method (more efficient for large values of N) inspired by [4,5].…”

Rayleigh quotient with bolzano booster for faster convergence of dominant eigenvalues

“…Sensitivity is a way of knowing whether the influence of a scalar outside the Bolzano algorithm will affect the result of an eigenvector [24], [25], [26], [27]. In this research, sensitivity was carried out on the Bolzano method, Rayleigh quotient, and the results of the combination of both algorithms.…”

A Rule-Based Decision Support Method Combining Variable Precision Rough Set and Stochastic Multi-Objective Acceptability Analysis for Multi-Attribute Decision-Making

“…Depending on the information provided, the methods of determining attribute weights are divided into subjective methods and objective methods [2]. In subjective methods, attribute weights are derived from value judgments about attributes given by decision makers (DMs), including eigenvectorbased method [3][4][5][6], weighted averaging or least square method [7][8][9][10], AHP or ANP-based method [11,12], and D-TOPSIS method [13]. And the objective methods determine attribute weights according to objective information (e.g., the information in the decision matrix), such as principal component analysis method [14], entropy method [15,16], and multi-objective programming model-based method [17,18].…”