The Study Of Numerical Algorithms For Solving Large Sparse Quadratic Eigenvalue Problem

The large-scale quadratic eigenvalue problem (QEP) arises in many scientificand engineering applications, such as structural mechanics dynamic analysis,simulation telecommunications, signal processing, and recent modeling ofmicroelectronic mechanical problems, etc. Typically, these problems are large, sparseand have better structure. Thus, the traditional linearization methods are not effectiveat all.Taking into account these problems, Su et al. developed a second-order Arnoldimethod, i.e. SOAR. This method can not only take advantage of Rayleigh-Ritzorthogonal projection technology directly solving quadratic eigenvalue problems butalso maintaining the structure of matrix of the original problem.In this dissertation, we studied the numerical algorithms for quadratic eigenvalueproblem. Inspired by the ideas of SOAR, we select two new and appropriateprojection subspaces to study the large-scale sparse quadratic eigenvalue problem andpresent the theoretical analysis of new methods. We get two variants of SOARalgorithm（i.e., VSOAR I and VSOAR II） to solve quadratic eigenvalue problems.Both in theory and numerical experiments, we can see the new algorithms willperformance better than original one. This dissertation includes four chapters, whichis organized as follows:Firstly, the research background, research status and related preliminaries ofsolving quadratic eigenvalue problem (QEP) of Krylov subspace methods are given.Furthermore, the main contents of this paper are briefed.In the second chapter, the variant of second-order Arnoldi method I for thesolution of the quadratic eigenvalue problem is given. Then we discuss theconvergence of the algorithm and numerical experiments.In the third chapter, the variant of second-order Arnoldi method II for thesolution of the quadratic eigenvalue problem is given. Then the convergence of thealgorithm and numerical experiments are presented. Finally, the research work of this dissertation is summarized and the futureresearch directions based on this work are discussed.