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

This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.

A survey of sparse matrix research

An efficient heuristic algorithm for solving a cluster problem associated with the tearing of an undirected graph is presented via the concept of a contour tableau. The required computation time is shown to be bounded by \theta (nb) , where n and b are the number of nodes and branches of the input graph, respectively. Experimental results show that our algorithm is highly efficient and yields near optimal solutions.

http://ieeexplore.ieee.org/iel5/31/23464/01084298.pdf

An efficient heuristic cluster algorithm for tearing large-scale networks

SUMMARY The selection of parts and machines poses an important problem in the design and planning phases of cellular manufacturing and flexible manufacturing systems. In most real-life situations, this grouping invariably leads to 'bottleneck’ parts and machines. This paper discusses a method of identifying the minimal number of bottle-neck cells (machines or parts) which, when dealt with through either duplication of machines or subcontracting of parts, will result in perfect part-machine groupings with no overlap. The polynomially bounded algorithms used in the analysis are oriented towards finding minimal cut-nodes in either partition of the bipartite part-machine graph. †Research of Vannelli was partially supported by Natural Sciences and Engineering Research Council of Canada under grant U0381.

A method for finding minimal bottle-neck cells for grouping part-machine families†

A methodology is developed for optimal design and/or operation of a water distribution system that is to operate under one or several loading conditions. Decision variables may be design variables, such as pipe diameters, or control variables, such as heads and flows. The objective function may include the initial cost of the design, the cost of operation, the physical measures of performance, and the penalties for violating constraints. Constraints may be imposed on the decision variables and on the performance of the system under each loading. Flow solutions are obtained by a modified Newton-Raphson method employing sparse matrix techniques. Optimization is obtained by a combination of the generalized reduced gradient and penalty methods. Implementation in a computer program and its use on a test problem in both batch and time-sharing modes are described, and it is concluded that the method is computationally feasible. The many different ways in which it can be used to analyze, design, and operate water distribution systems are outlined.

Optimal Design and Operation of Water Distribution Systems

This is a concise critical survey of the theory and practice relating to the ordered Gaussian elimination on sparse systems. A new method of renumbering by clusters is developed, and its properties described. By establishing a correspondence between matrix patterns and directed graphs, a sequential binary partition is used to decompose the nodes of a graph into clusters. By appropriate ordering of the nodes within each cluster and by selecting clusters, one at a time, both optimal ordering and a useful form of matrix banding are achieved. Some results pertaining to the compatibility between optimal ordering for sparsity and the usual pivoting for numerical accuracy are included.

/pdf/sparsity-directed-decomposition-for-gaussian-elimination-on-2lvye1ri3n.pdf

John W. Walker

Papers

Sparsity-Directed Decomposition for Gaussian Elimination on Matrices