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Summary We describe randomized algorithms for computing the dominant eigenmodes of the generalized Hermitian eigenvalue problem Ax = λBx, with A Hermitian and B Hermitian and positive definite. The algorithms we describe only require forming operations Ax,Bx and B−1x and avoid forming square roots of B (or operations of the form, B1/2x or B−1/2x). We provide a convergence analysis and a posteriori error bounds and derive some new results that provide insight into the accuracy of the eigenvalue calculations. The error analysis shows that the randomized algorithm is most accurate when the generalized singular values of B−1A decay rapidly. A randomized algorithm for the generalized singular value decomposition is also provided. Finally, we demonstrate the performance of our algorithm on computing an approximation to the Karhunen–Loeve expansion, which involves a computationally intensive generalized Hermitian eigenvalue problem with rapidly decaying eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.

The eigenproblems of the rank-one updates of the matrices have lots of applications in scientific computation and engineering such as the symmetric tridiagonal eigenproblems by the divide-andconquer method and Web search engine. Many researchers have well studied the algorithms for computing eigenvalues of Hermitian matrices updated by a rank-one matrix [1–6]. Recently, Ding and Zhou studied a spectral perturbation theorem for rank-one updated matrices of special structure in [7] and considered two applications. Cheng, Luo, and Li considered the bounds of the smallest and largest eigenvalues for rank-one modification of Hermitian matrices [8]. Eigenvalue bounds for perturbations of Hermitian matrices have been considered by Ipsen and Nadler in [9]. In this paper, we consider the bounds of the eigenvalues for rank-one modification of Hermitian matrices. The ideas of this paper were mainly motivated by one of [9]. We study the following form

SUMMARY Strang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {eiφ}-circulant Strang-type preconditioner. Copyright © 2011 John Wiley & Sons, Ltd.

SUMMARY In this note, some errors in the article (Numer. Linear Algebra Appl. 2007; 14:217–235) are pointed out and some correct results are presented. Copyright © 2011 John Wiley & Sons, Ltd.

The performance of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for the non-Hermitian positive definite system of linear equations is largely dependent on the choice of its parameter value. In this work, an efficient scaled norm minimization (SNM) method is proposed to compute the parameter value of the HSS preconditioner. In addition, by choosing different parameters for the Hermitian and the skew-Hermitian matrices in the HSS preconditioner, a two-parameter HSS preconditioner is proposed. Moreover, an efficient and practical formula for computing the parameter values of this new preconditioner is also derived by using the SNM method. Numerical examples are illustrated to verify the performances of the HSS and the two-parameter HSS preconditioners when their parameters are computed by the SNM method.

Through a Hermitian-type (skew-Hermitian-type) singular value decomposition for pair of matrices (A, B) introduced by Zha (Linear Algebra Appl. 1996; 240:199–205), where A is Hermitian (skew-Hermitian), we show how to find a Hermitian (skew-Hermitian) matrix X such that the matrix expressions A − BX ± X*B* achieve their maximal and minimal possible ranks, respectively. For the consistent matrix equations BX ± X*B* = A, we give general solutions through the two kinds of generalized singular value decompositions. As applications to the general linear model {y, Xβ, σ2V}, we discuss the existence of a symmetric matrix G such that Gy is the weighted least-squares estimator and the best linear unbiased estimator of Xβ, respectively. Copyright © 2007 John Wiley & Sons, Ltd.