We discuss high-order adjoint based degenerate perturbation theory of semisimple eigenvalues resulting from nonlinear eigenvalue problems, which are typical of acoustic problems and vibrational analysis. The theory is discussed for a general matrix operator $\mathsf{L}$ with no further properties assumed. The unperturbed operator is thus in general non-normal, which requires the use of adjoint methods. The main challenge is to handle arbitrary order eigenvalue split for eigenvalue problems of arbitrary multiplicity: this is aided by a tree-structure formalism, which can be straightforwardly implemented numerically. We highlight the existence of a cross-branch solvability condition that needs to be satisfied when the perturbation causes the eigenvalues to split. After deriving the equations that govern the eigenproblem expansion order by order, we apply the method to three different test cases – which span eigenproblems of academic and industrial interest arising in structural mechanics and thermoacoustic stability – having different degrees of degeneracy and levels of complexity. We validate our algorithm by comparing the eigenvalues and eigenvectors reconstructed from perturbation theory up to 20th-order against exact solutions obtained using nonlinear eigenvalue solvers, and we discuss the advantages and limits of the method.

我们讨论由非线性特征值问题产生的半简单特征值的基于高阶伴随的简并扰动理论，这是声学问题和振动分析的典型特征。讨论了有关一般矩阵算子的理论$\mathsf{大号}$假设没有其他属性。因此，不受干扰的运算符通常是非正常的，这需要使用伴随方法。主要挑战是处理任意多重特征值问题的任意阶特征值拆分：这可以通过树结构形式主义来辅助，该树构形式主义可以通过数字直接实现。我们着重指出了当扰动导致特征值分裂时需要满足的跨分支可解性条件的存在。在按顺序得出控制本征问题扩展阶数的方程式之后，我们将该方法应用于三个不同的测试案例，这些案例涵盖了结构力学和热声稳定性方面引起的学术和工业兴趣的本征问题，并且具有不同程度的简并度和复杂度。