Solving Two-dimensional Single-parameter Eigenvalue Problem Arnoldi Projection Algorithm

In this thesis we discuss the Two-dimensional Arnoldi Projection Method (TPM) and its applications for solving large sparse single parameter-dependent eigenvalue problems .In Chapter 1 and 2, we introduce the iterative methods for solving large parse eigenvalue problems, mainly Krylov subspace methods, and the core algorithm Arnoldi process.In Chapter 3, we introduce new developed Two-dimensional Arnoldi Process (TAP) and the algorithm to construct the TAP from standard Arnoldi process. We also give the reorthogonalization algorithm for TAP and numerical exam-ples. In Chapter 4, we give the method to construct a set of orthonormal ba-sis of the projection subspace which can be applied for solving large sparse sin-gle parameter-dependent eigenvalue problems like (A +δB)x =λCx. We give complete algorithm for such method, called Two-dimensional Arnoldi Projection Method (TPM). Moreover, we also give two different explicitly restarted strategies for TPM. After that, we use TPM to compute parameter-dependent eigenvalue problem arising from passivity check and enforcement problem and bifurcation problem in dynamic systems. We give plenty examples to analyze the properties of TPM we developed here. We also compare the two explicitly restarted strate-gies with the original algorithm. Finally, we compare TPM with known Krylov subspace methods and achieve good results.In Chapter 5, we give a note for Sherman-Morrison-Woodbury formula which is useful for computing eigenvalues. For solving eigenvalue problem as (A + UD^-1 V)x =λx by inverse iteration with shift, we show that if we use Sherman-Morrison-Woodbury (SMW) formula to solve the near singular linear systems in inverse iteration, we can also reach desired eigenpairs. When A, U, V are large sparse matrices, it takes less time than we use LU factorization to solve near sin- gular linear systems.