In two previous papers by Neymeyr: A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient, LAA 322: (1-3), 61-85, 2001, and A geometric theory for preconditioned inverse iteration II: Convergence estimates, LAA 322: (1-3), 87-104, 2001, a sharp, but cumbersome, convergence rate estimate was proved for a simple preconditioned eigensolver, which computes the s...

We study the solutions of Hermitian positive definite Toeplitz systems Tnx = b by the preconditioned conjugate gradient method. For preconditioner An the convergence rate is known to be governed by the distribution of the eigenvalues of the preconditioned matrix A−1 n Tn . New properties of the circulant preconditioners introduced by Strang, R. Chan, T. Chan, Szegö/Grenander and Tyrtyshnikov ar...

We consider the solution of «-by-« Toeplitz systems T„x = b by preconditioned conjugate gradient methods. The preconditioner Cn is the T. Chan circulant preconditioner, which is defined to be the circulant matrix that minimizes \\B„ T„\\f over all circulant matrices B„ . For Toeplitz matrices generated by positive In -periodic continuous functions, we have shown earlier that the spectrum of the...

In this paper, we analyze the convergence of the preconditioned GMRESmethod for the first order finite element discretizations of the Helmholtz equation in media with losses. We consider a Laplace preconditioner, an inexact Laplace preconditioner and a two-level preconditioner. Our analysis is based on bounding the field of values of the preconditioned system matrix in the complex plane. The an...

It is often observed in practice that matrix sequences {An}n generated by discretization methods applied to linear differential equations, possess a Spectral Symbol, that is a measurable function describing the asymptotic distribution of the eigenvalues of An. Sequences composed by Hermitian matrices own real spectral symbols, that can be derived through the axioms of Generalized Locally Toepli...

The implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is employed to discretize fractional diffusion equations. The resulting systems are Toeplitz-like and then the fast Fourier transform can be used to reduce the computational cost of the matrix-vector multiplication. The preconditioned conjugate gradient normal residual method with a circula...

We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by Lvu := −∆u + a(x, y)ux + b(x, y)uy + d(x, y)u, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator Lcu is const...

In this paper we present the convergence analysis of the inexact infeasible path-following (IIPF) interior point algorithm. In this algorithm the preconditioned conjugate gradient method is used to solve the reduced KKT system (the augmented system). The augmented system is preconditioned by using a block triangular matrix. The KKT system is solved approximately. Therefore, it becomes necessary...

We construct, analyze and implement SSOR-like preconditioners for non-Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew-Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR-like iteration methods as well as the cor...