The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The techniques of the calculus of variation and of optimization proved to be successful for several optimal shape design problems however these remain expensive both in the qualification of the engineers required to understand the method and in computing time. However it seems difficult to do without such techniques for 3-dimensional optimization problems. The field is well studied from the mathematical point of view but still in its beginnings from the industrial implementation side.

Optimal Shape Design for Elliptic Systems

The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Computational methods of linear algebra

Introduction. Basic Equations. Approximations to Derivatives. Ordinary Differential Equations. The Explicit Method. Boundary Conditions. Unequal Intervals. The Commonly Used Implicit Methods. Other Methods. Adsorption. Effects Due to Uncompensated Resistance and Capacitance. Two-Dimensional Systems. Convection. Performance. Programming. Simulation Packages. Some Mathematical Proofs. Useful Procedures. Example Programs.

Digital Simulation in Electrochemistry

Encompassing all the major topics students will encounter in courses on the subject, the authors teach both the underlying mathematical foundations and how these ideas are implemented in practice. They illustrate all the concepts with both worked examples and plenty of exercises, and, in addition, provide software so that students can try out numerical methods and so hone their skills in interpreting the results. As a result, this will make an ideal textbook for all those coming to the subject for the first time. Authors' note: A problem recently found with the software is due to a bug in Formula One, the third party commercial software package that was used for the development of the interface. It occurs when the date, currency, etc. format is set to a non-United States version. Please try setting your computer date/currency option to the United States option . The new version of Formula One, when ready, will be posted on Www.

Linear Programming 1: Introduction

Some properties of totally positive matrices

Publisher Summary This chapter presents numerical methods for functional differential equation. The survey of numerical methods is divided into three parts: (1) initial-value problems for functional ordinary differential equations, (2) boundary-value problems for functional ordinary differential equations, and (3) functional partial differential equations. In discussing the part 1 of the surveys, the author considered several cases, namely: (1) ordinary differential equations, (2) retarded and neutral differential equations, (3) Volterra integro-differential equations, and (4) functional differential equations of retarded type. For each special case of (1) several aspects of numerical methods are considered, such as Euler's method, one-step methods: general theory, Runge–Kutta methods, block methods, Splines and Hermte methods, multistep methods, Chebyshev series and the tau method, nonsmooth solutions, extrapolation and deferred correction, efficiency, highly stable methods, and special methods. It was pointed out by Bailey and Shampine that in the case of ordinary differential equations, the shooting method and the method of finite differences complement one another in the sense that the shooting method gives good results when the finite-difference method gives poor results and vice versa. Under suitable hypotheses, Thompson shows that the Lax–Richtmyer theory for difference approximations to initial-value problems for partial differential equations.

Numerical methods for functional differential equations

LetA = (a ij ) be a real n x n matrix such that |a ij | < 1. It has been conjectured by WILKINSON that if the process of Gaussian elimination with complete pivoting is applied to A then all the pivots are less than or equal to n in absolute value. This conjecture is proved forn=4.

Pivot size in gaussian elimination

Two algorithms for solving free boundary problems in two dimensions are described. The algorithms use the method of finite differences and are automated versions of methods due to Southwell. The algorithms have been implemented as a general program FREEBOUN, and the numerical results that were obtained using this program are discussed.

On the Approximate Solution of Free Boundary Problems Using Finite Differences

It is shown that the backward difference multistep method

$$\mathop \Sigma \limits_{m = 1}^q \frac{1}{m}\nabla ^m y_p = hf_p$$

for the numerical integration ofy′(x)=f(x, y) is stable in the sense of Dahlquist iff 1≦q≦6.

Colin W. Cryer

Papers

Some properties of totally positive matrices

Numerical methods for functional differential equations

Pivot size in gaussian elimination

On the Approximate Solution of Free Boundary Problems Using Finite Differences

On the instability of high order backward-difference multistep methods