Euler-Richardson method preconditioned by weakly stochastic matrix algebras: a potential contribution to Pagerank computation
Electronic Journal of Linear Algebra, vol. 32, pp. 254–272, 2017
Let $S$ be a column stochastic matrix with at least one full row. Then $S$ describes a Pagerank-like random walk since the computation of the Perron vector $x$ of $S$ can be tackled by solving a suitable M-matrix linear system $Mx = y$, where $M = I − \tau A$, $A$ is a column stochastic matrix and $\tau$ is a positive …
Adaptive matrix algebras in unconstrained minimization
Linear Algebra and its Applications, vol. 471, pp. 544–568, 2015
In this paper we study adaptive $L(k)QN$ methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix …
Optimal Rank Matrix Algebras Preconditioners
Linear Algebra and its Applications, vol. 438, pp. 405–427, 2013
When a linear system $Ax = y$ is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner $P$. The use of such preconditioner changes the spectrum of the matrix defining the system and could result into a great acceleration of the convergence rate. …
A preconditioning approach to the PageRank computation problem
Linear Algebra and its Applications, vol. 435, pp. 2222–2246, 2011
Some spectral properties of the transition matrix of an oriented graph indicate the preconditioning of Euler-Richardson (ER) iterative scheme as a good way to compute efficiently the vertexrank vector associated with such graph. We choose the preconditioner from an algebra $\mathcal U$ of matrices, thereby obtaining an …