Eigenvalue and eigenvector computations are extremely important and have various applications in engineering, physics and other scientific disciplines. The aim of this study is partly to examine the existing methods used to solve the generalized eigenvalue problem GEP and the Quadratic eigenvalue problem QEP with definite Hermitian matrices. Furthermore, the research investigates new algorithms and computer programs that reduce the cost of eigenpair (eigenvalue, eigenvector) computations and improve the speed of eigenpair solutions.