## 9 Conclusions  
A novel methodology was developed and demonstrated to accurately calculate higher-order sensitivities of the eigenvalues and eigenvectors for eigenfrequency problems using the hypercomplex automatic differentiation (HYPAD) method. The method integrates HYPAD with exact semi-analytical expressions developed from differentiating the equations of the generalized eigenvalue problem, and to compute arbitrary-order sensitivities of the structural matrices. Sensitivities with respect to the material and geometrical parameters were obtained and verified for the analysis of the free vibration of a discrete cantilever beam with a known analytical solution. Two types of problems were analyzed: (i) a model with distinct eigenvalues, and (ii) a model with repeated eigenvalues, with high accuracy in both cases. The advantages of this approach are that both truncation and subtractive errors are eliminated from the sensitivity analysis, and a single general expression is used for the computation of the sensitivities of both eigenvalues and eigenvectors up to arbitrary-order. This results in accurate and efficient sensitivity results. In this case, the error of the sensitivities is dependent upon the error of the numerical method used to solve the generalized eigenvalue problem, which can be further reduced with mesh refinements. An additional advantage of the presented methodology is given by the possibility of evaluating higher-order sensitivities for only a specific number of modes, not altogether as happens with other approaches. This is beneficial; since, in large systems, finding the spectrum of eigenvalues and eigenvectors is known to be expensive. In addition, the HYPAD method can calculate all higher-order sensitivities with respect to all the input model parameters using only one function evaluation, which makes it easy to implement in commercial practice. Finally, regardless of the hypercomplex nature of the operations used to calculate the sensitivities, the method presented here uses real only numbers when solving the generalized eigenvalue problem. This is crucial since no commercial hypercomplex eigensolver is currently available. In the future, if hypercomplex solvers become available, that could impose the need to reformulate the methodology. Multiple applications of the higher-order sensitivities are envisioned to perform a sensitivity analysis in structural systems. As examples, the sensitivities can be used to improve the performance of topology optimization models, finite element updating algorithms, and other algorithms where accurate derivatives are crucial to correctly guide the processes. Further research efforts will be focused on implementing the present methodology within commercial structural analysis software to include the calculation of sensitivities in damped systems. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13127125/s1](https://www.mdpi.com/article/10.3390/app13127125/s1)]([https://www.mdpi.com/article/10.3390/app13127125/s1](https://www.mdpi.com/article/10.3390/app13127125/s1)), Supplementary Material A: Analytical Demonstration of Higher-Order Sensitivities Obtained with ZTSE. Supplementary Material B: Construction of Hyperdual Arrays for Higher-Order Sensitivities. Supplementary Material C: Cantilever Beam Problem: Semi-Analytical demonstration with a single element. Supplementary Material D: Other Sensitivities for Numerical Verification Problem. Supplementary Material E: Fortran Code: Arbitrary-order derivatives for eigenfrequency problems. Supplementary Material F: Finite Differences Step Size Refinement. J.C.V.-G.: Methodology, conceptualization, formal analysis, software, writing--original draft, writing--review and editing. J.D.N. : Formal analysis, methodology, software and conceptualization, writing--review and editing. A.M.: Conceptualization, supervision, funding acquisition, writing--review and editing. M.A. : Conceptualization, software, data analysis, writing--review. H.R.M.