This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in (Linear Algebra Appl 435:641–658, 2011). A Arnoldi (t-Arnoldi) process is applied to reduce a large-scale Tikhonov regularization problem of small size. The data may be represented laterally oriented matrix or tensor, operator tensor. discrepancy ...

Our main aim is the accurate computation of a large number of specified eigenvalues and eigenvectors of Mathieu’s system as a multiparameter eigenvalue problem (MEP). The reduced wave equation, for small deflections, is solved directly without approximations introduced by the classical Mathieu functions. We show how for moderate values of the cut-off collocation parameter the QR algorithm and t...

We present simultaneous reduction algorithms for two (nonsymmetric) matrices A and B to upper Hessenberg and lower Hessenberg forms, respectively. One is through the simultaneous similarity reduction and the other is through a Lanczos– Arnoldi-type iteration. The algorithm that uses the Lanczos–Arnoldi-type iteration can be considered as a generalization of both the nonsymmetric Lanczos algorit...

The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large scale systems. The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This...

The aim of this paper is to examine a numerical method for the computation of approximate solution of the continuous-time algebraic Riccati equation using Krylov subspace matrix. First of all, Global Arnoldi process is initiated to construct an orthonormal basis. In addition, Krylov subspace matrix is employed as projection method because it is one of the frequently referred method in the liter...

We consider the two-sided Arnoldi method and propose a two-sided Krylov–Schurtype restarting method. We discuss the restart for standard Rayleigh–Ritz extraction as well as harmonic Rayleigh–Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer–Fike theorem and Saad’s theorem....

The PageRank model computes the stationary distribution of a Markov random walk on linking structure network, and it uses values within to represent importance or centrality each node. This is first proposed by Google for ranking web pages, then widely applied as measure networks arising in various fields such chemistry, bioinformatics, neuroscience social networks. For example, can node centra...

In this paper an interesting property of the restarted FOM algorithm for solving large nonsymmetric linear systems is presented and studied. By establishing a relationship between the convergence of its residual vectors and the convergence of Ritz values in the Arnoldi procedure, it is shown that some important information of previous FOM(m) cycles may be saved by the iteration approximates at ...