In this dissertation, we consider the symmetric eigenvalue problem and the buckling eigenvalue problem. We study existing algorithms and propose stable variants for both eigenvalue problems.

We first analyze Hotelling's deflation for the symmetric eigenvalue problem $Ax=\lambda x$, where $A$ is a symmetric matrix. Hotelling's deflation is a technique to displace computed eigenvalues of $A$. It is combined with an eigensolver to compute a partial eigendecomposition of $A$. Numerical stability of Hotelling's deflation is not well understood. In this dissertation, we derive computable upper bounds on the loss of orthogonality of computed eigenvectors and on the backward error norm of computed eigenpairs. From the upper bounds, we identify crucial quantities associated with the shifts and derive sufficient conditions for the backward stability of Hotelling's deflation. Based on these results, we propose a shift selection scheme for stabilizing Hotelling's deflation.

Next we consider the buckling eigenvalue problem $Kx=\lambda K_Gx$, where the matrix $K$ is positive semi-definite, the matrix $K_G$ is indefinite, and the matrices $K$ and $K_G$ share a common nullspace. When $K$ is positive definite, the shift-invert Lanczos method is a widely accepted method for the buckling eigenvalue problem. However, in our case, there are two issues associated with the method. First, the shift-invert operator $(K - \sigma K_G)^{-1}$ does not exist or is ill-conditioned. Second, the Lanczos vectors fall rapidly into the nullspace of $K$. The inner product induced by $K$ leads to rapid growth of the Lanczos vectors in norm. The large norms introduce large round-off errors to the orthogonalization process, leading to loss of accuracy of compute solutions and even break down of the method. In this dissertation, we address these issues by generalizing the buckling spectral transformation to the singular pencil $K - \lambda K_G$ and regularizing the inner product to bound the norms of the Lanczos vectors. We propose a shift-invert Lanczos method for the buckling eigenvalue problem and develop a validation scheme using inertias.

This dissertation works towards building a fundamental theory of general $\sigma_k$ equations and general inverse $\sigma_k$ equations, showing up in many different fields. For example, PDE, differential geometry, and complex geometry. Our primary goal is to construct nice algebra tools, especially related to real algebraic geometry, so that we can generalize previous classical equations to more complicated settings. Once the framework is settled, we aim to look for a priori estimates and further obtain the solvability of these equations. To be more precise, first, we introduce a special class of multilinear polynomials and a special class of univariate polynomials which are related to the convexity of these equations. Second, we study a priori estimates of these equations provided that the convexity and a $C$-subsolution are given. Last, by collecting these equations which have a priori estimates, we obtain a special algebraic set to apply the method of continuity and further look for the solvability. As an application, we apply our theory and results to the deformed Hermitian--Yang--Mills equation, an equation discovered around the same time by Mariño--Minasian--Moore--Strominger \cite{marino2000nonlinear} and Leung--Yau--Zaslow using different points of view when studying mirror symmetry in string theory. We confirm one of the conjectures by Collins--Jacob--Yau of the deformed Hermitian--Yang--Mills equation when the complex dimension equals three or four.