The topic of this paper is a convergence analysis of preconditioned inverse iteration (PINVIT). A sharp estimate for the eigenvalue approximations is derived; the eigenvector approximations are controlled by an upper bound for the residual vector. The analysis is mainly based on extremal properties of various quantities which define the geometry of PINVIT.

This paper presents an example of an algebraic eigenvalue problem which can be used to motivate the study of numerical techniques for solving such problems. The problem consists of finding the axis and angle of rotation from a 3 x 3 rotation matrix and is referred to as the axis-angle problem. The problem is used to demonstrate the inverse power method for finding eigenvectors. The axis-angle p...

Technische Universit at Dresden Herausgeber: Der Rektor A Generalized Inverse Iteration for Computing Simple Eigenvalues of Nonsymmetric Matrices Hubert Schwetlick and Ralf L osche IOKOMO-07-97 December 1997 Preprint-Reihe IOKOMO der DFG-Forschergruppe Identi kation und Optimierung komplexer Modelle auf der Basis analytischer Sensitivitatsberechnungen an der Technischen Universitat Dresden ...

Power Method We now describe the power method for computing the dominant eigenpair. Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation is known. Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision. The inverse power method is then invoked to refine the numerical values and...

We discuss variants of the Jacobi–Davidson method for solving the generalized complex-symmetric eigenvalue problem. The Jacobi–Davidson algorithm can be considered as an accelerated inexact Rayleigh quotient iteration. We show that it is appropriate to replace the Euclidean inner product xy in C by the bilinear form x y. The Rayleigh quotient based on this bilinear form leads to an asymptotical...

Newton's Method constitutes a nested iteration scheme with the Newton step as the outer iteration and a linear solver of the Jacobian system as the inner iteration. We examine the interaction between these two schemes and derive solution techniques for the linear system from the properties of the outer Newton iteration. Contrary to inexact Newton methods, our techniques do not rely on relaxed t...

This thesis deals with the solution of a nonlinear inverse problem arising in digital image registration. In image registration one seeks to compute a transformation between two images such that they become more similar in some sense. In the first part, we define the problem as the minimization of a regularized nonlinear least-squares functional, which measures the image difference and smoothne...

Abstract. Augmented Lagrangian method (ALM) has been popularly used for solving constrained optimization problems. Its convergence and local convergence speed have been extensively studied. However, its global convergence rate is still open for problems with nonlinear inequality constraints. In this paper, we work on general constrained convex programs. For these problems, we establish the glob...

K.V. Fernando developed an efficient approach for computation of an eigenvector of a tridiagonal matrix corresponding to an approximate eigenvalue. We supplement Fernando’s method with deflation procedures by Givens rotations. These deflations can be used in the Lanczos process and instead of the inverse iteration.

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inv...