The ranking of genes plays an important role in biomedical research. The GeneRank method of Morrison et al. [11] ranks genes based on the results of microarray experiments combined with gene expression information, for example from gene annotations. The algorithm is a variant of the well known PageRank iteration, and can be formulated as the solution of a large, sparse linear system. Here we sh...

We present an approach for the acceleration of the restarted Arnoldi iteration for the computation of a number of eigenvalues of the standard eigenproblem Ax = λx. This study applies the Chebyshev polynomial to the restarted Arnoldi iteration and proves that it computes necessary eigenvalues with far less complexity than the QR method. We also discuss the dependence of the convergence rate of t...

We introduce a new filter or sum acceleration method which is the complementary error function with a logarithmic argument. It was inspired by the large order asymptotics of the Euler and Vandeven accelerations, which we show are both proportional to the erfc function also. We also show the relationship between Vandeven’s filter, the Erfc-Log filter and the “lagged-Euler” method. The theory for...

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem ...

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes which elements mimic as well as possible the Chebyshev-Loba...

It is an important fact that general families of Chebyshev and L-splines can be locally represented, i.e. there exists a basis of B-splines which spans the entire space. We develop a special technique to calculate with 4 order Chebyshev splines of minimum deficiency on nonuniform meshes, which leads to a numerically stable algorithm, at least in case one special Hermite interpolant can be const...

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with high-order filter polynomials obtained from a regularized Chebyshev expansion of a window function. After a short discussion of the conceptual foun...

A Chebyshev-based representation of the state vector is proposed for designing optimal control trajectories of unconstrained, linear, dynamic systems with quadratic performance indices. By approximating each state variable by a finite-term, shifted Chebyshev series, the linear quadratic (LQ) optimal control problem can be cast as a quadratic programming (QP) problem. In solving this QP problem,...

In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are des...