Disguised and new quasi-Newton methods for nonlinear eigenvalue problems

In this paper we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type $M(\lambda)v=0$, where $M:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh's theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier's residual inverse iteration and Ruhe's method of successive linear problems

Jacobi-Davidson Methods and Preconditioning with Applications in Pole-zero Analysis

this paper, Sect. 2 introduces the pole-zero problem. Section 3 gives an overview of conventional methods used in pole-zero analysis, and describes the Jacobi-Davidson style methods as an alternative. In Sect. 4, the methods will be compared by numerical results, concluding with some future research topic

Limited Memory Block Preconditioners for Fast Solution of Fractional Partial Differential Equations

An innovative block structured with sparse blocks multi iterative preconditioner for linear multistep formulas used in boundary value form is proposed here to accelerate GMRES, FGMRES and BiCGstab(l). The preconditioner is based on block (Formula presented.)-circulant matrices and a short-memory approximation of the underlying Jacobian matrix of the fractional partial differential equations. Convergence results, numerical tests and comparisons with other techniques confirm the effectiveness of the approach