Restarted Hessenberg method for solving shifted nonsymmetric linear systems

It is known that the restarted full orthogonalization method (FOM) outperforms the restarted generalized minimum residual (GMRES) method in several circumstances for solving shifted linear systems when the shifts are handled simultaneously. Many variants of them have been proposed to enhance their performance. We show that another restarted method, the restarted Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille, France, 1996] based on Hessenberg procedure, can effectively be employed, which can provide accelerating convergence rate with respect to the number of restarts. Theoretical analysis shows that the new residual of shifted restarted Hessenberg method is still collinear with each other. In these cases where the proposed algorithm needs less enough CPU time elapsed to converge than the earlier established restarted shifted FOM, weighted restarted shifted FOM, and some other popular shifted iterative solvers based on the short-term vector recurrence, as shown via extensive numerical experiments involving the recent popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference

Computing the matrix Mittag-Leffler function with applications to fractional calculus

The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision

Mittag-Leffler functions and their applications in network science

We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag–Leffler functions. This overarching theory includes as special cases well known centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag–Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modelling and computational issues, and provide guidelines on parameter 10 selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided

A Fast Monte Carlo algorithm for evaluating matrix functions with application in complex networks

We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function. This contrasts with the "classical" Monte Carlo method which only works with one entry at a time, resulting in a significant better convergence rate than the "classical" approach. To assess the applicability of our method, we compute the subgraph centrality and total communicability of several large networks. In all benchmarks analyzed so far, the performance of our method was significantly superior to the competition, being able to scale up to 64 CPU cores with a remarkable efficiency.Comment: Submitted to the Journal of Scientific Computin

Efficient variants of the CMRH method for solving a sequence of multi-shifted non-Hermitian linear systems simultaneously

Multi-shifted linear systems with non-Hermitian coefficient matrices arise in numerical solutions of time-dependent partial/fractional differential equations (PDEs/FDEs), in control theory, PageRank problems, and other research fields. We derive efficient variants of the restarted Changing Minimal Residual method based on the cost-effective Hessenberg procedure (CMRH) for this problem class. Then, we introduce a flexible variant of the algorithm that allows to use variable preconditioning at each iteration to further accelerate the convergence of shifted CMRH. We analyse the performance of the new class of methods in the numerical solution of PDEs and FDEs, also against other multi-shifted Krylov subspace methods.Comment: Techn. Rep., Univ. of Groningen, 34 pages. 11 Tables, 2 Figs. This manuscript was submitted to a journal at 20 Jun. 2016. Updated version-1: 31 pages, 10 tables, 2 figs. The manuscript was resubmitted to the journal at 9 Jun. 2018. Updated version-2: 29 pages, 10 tables, 2 figs. Make it concise. Updated version-3: 27 pages, 10 tables, 2 figs. Updated version-4: 28 pages, 10 tables, 2 fig

Nonstandard finite difference schemes for a fractional-order Brusselator system

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