In this thesis we study the quadratic eigenvalue problem (QEP), that is for given \(M, C, K \in \mathbb{C}^{n \times n}\), task is to compute an eigenvalue \( \lambda \in \mathbb{C}\) and an eigenvector \(x \in \mathbb{C}^n\), \(x \neq 0\) \((M \lambda^2 + C\lambda + K)x = 0\). This QEP has \(2n\) eigenvaluess, some of them can even be infinite. We present two different numerical methods for solving QEP; quadeig method and SOAR (Second Order ARnoldi) method. Quadeig method is a complete (direct) method, meaning that it computes all \(2n\) eigenvalues, and we recommend applying this method when the dimension of the initial QEP is not too big. One of the improvements that quadeig offers is scaling of the given matrices M,C and K. The way that quadeig method deals with infinite eigenvalues and zero eigenvalues improves the accuracy of the approximations for the finite eigenvalues. Our numerical experiments compare quadeig method with MATLAB’s built-in function polyeig that solves the same QEP problem. Also, we investigate the SOAR method which belongs to the class of iterative methods. Unlike the complete methods, iterative methods are applied when we are interested only in a subset of the spectrum (only few eigenvalues). Those iterative methods are effective way of dealing with the problem of large dimension. Applying a complete method to such big problems would cause great memory cost and immense time consumption. SOAR method, that we suggest in this thesis for effectively solving QEP, is a generalization of the Arnoldi method for the standard eigenvalue problem and the basic idea is to find suitable subspace that is very close to the subspace that is spanned by the wanted eigenvectors. One of the good subspaces which is also the basis for the SOAR method is a second order Krylov subspace. The main problem of iterative methods is the convergence question. Enhancing the convergence is still one of the main topics in many researches and improvements are being made. All the numerical experiments in this thesis are made in MATLAB.