The advent of gene expression microarray technology enables the simultaneous measurement of expression levels for thousands or tens of thousands of genes in a single experiment (Schena, et al., 1995). Analysis of gene expression microarray data presents unprecedented opportunities and challenges for data mining in areas such as gene clustering (Eisen, et al., 1998; Tamayo, et al., 1999), sample...

The Lanczos algorithm for matrix tridiagonalisation suffers from strong numerical instability in finite precision arithmetic when applied to evaluate matrix eigenvalues. The mechanism by which this instability arises is well documented in the literature. A recent application of the Lanczos algorithm proposed by Bai, Fahey and Golub allows quadrature evaluation of inner products of the form ψ · ...

Recently, several proposals for the generalization of Young’s SOR method to the saddle point problem or the augmented system has been presented. One of the most practical versions is the SOR-like method given by Golub et al., [(2001). SOR-like methods for augmented systems. BIT, 41, 71–85.], where the convergence and the determination of its optimum parameters were given. In this article, a ful...

Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different problems from different applicative areas. ...

This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–ve...

A discretization method attributed to Kahan is used to approximate the May–Leonard (M–L) competition model for three species. The local dynamics of this discrete-time M–L model are analyzed. This model differs from the discrete M–L models being studied previously. This discrete model shows dynamical consistency with the continuous M–L model. Numerically, we showed that the discrete M–L model ha...

Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d > 0 be the distance between the spectra of A and C . We prove that under these assumptions the sharp value of the constant c in the condition ‖B‖ < cd implying the solvability of the operator Riccati equation XA−CX+XBX = B∗ is equal to √ 2. We also prove an extension of the Davis-Kahan...