a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Linear-scaling methods, or methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high-performance computers. The linear-scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas are then discussed. The applications of linear-scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear-scaling methods are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/75/3/036503/pdf

O(N) methods in electronic structure calculations.

(1992). Matrices: Methods and Applications. Journal of the Operational Research Society: Vol. 43, No. 12, pp. 1185-1185.

/pdf/matrices-methods-and-applications-3mhi2vv26h.pdf

Matrices: Methods and Applications

Beginning with an introduction to carbon-based nanomaterials, their electronic properties, and general concepts in quantum transport, this detailed primer describes the most effective theoretical and computational methods and tools for simulating the electronic structure and transport properties of graphene-based systems. Transport concepts are clearly presented through simple models, enabling comparison with analytical treatments, and multiscale quantum transport methodologies are introduced and developed in a straightforward way, demonstrating a range of methods for tackling the modelling of defects and impurities in more complex graphene-based materials. The authors also discuss the practical applications of this revolutionary nanomaterial, contemporary challenges in theory and simulation, and long-term perspectives. Containing numerous problems for solution, real-life examples of current research, and accompanied online by further exercises, solutions and computational codes, this is the perfect introductory resource for graduate students and researchers in nanoscience and nanotechnology, condensed matter physics, materials science and nanoelectronics.

/pdf/introduction-to-graphene-based-nanomaterials-from-electronic-5dvjocs8ly.pdf

Introduction to Graphene-Based Nanomaterials: From Electronic Structure to Quantum Transport

A COCR method for solving complex symmetric linear systems

A new family of k-Fibonacci numbers

A linear algebraic method named the shifted conjugate-orthogonal conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green's function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green's function, and dynamically, at each step in dynamical simulations. The method is applied to both a semiconductor and a metal.

/pdf/linear-algebraic-calculation-of-the-green-s-function-for-14csn1548m.pdf

Linear algebraic calculation of the Green’s function for large-scale electronic structure theory

An extension of the conjugate residual method to nonsymmetric linear systems

In this paper, we consider an inverse problem with the k-tridiagonal Toeplitz matrices. A theoretical result is obtained that under certain assumptions the explicit inverse of a k-tridiagonal Toeplitz matrix can be derived immediately. Two numerical examples are given to demonstrate the validity of our results.

Tomohiro Sogabe

Papers

A COCR method for solving complex symmetric linear systems

A new family of k-Fibonacci numbers

Linear algebraic calculation of the Green’s function for large-scale electronic structure theory

An extension of the conjugate residual method to nonsymmetric linear systems

Inversion of k-tridiagonal matrices with Toeplitz structure