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

Three widely-used nodal resequencing algorithms were tested and compared for their ability to reduce matrix profile and root-mean-square (rms) wavefront, the latter being the most critical parameter in determining matrix decomposition time in the NASTRAN finite element computer program. The three algorithms are Cuthill–McKee (CM), Gibbs–Poole–Stockmeyer (GPS), and Levy. Results are presented for a diversified collection of 30 test problems ranging in size from 59 to 2680 nodes. It is concluded that GPS is exceptionally fast, and, for the conditions under which the test was made, the algorithm best able to reduce profile and rms wavefront consistently well. An extensive bibliography of resequencing algorithms is included.

A comparasion of three resequencing algorithms for the reduction of matrix profile and wavefront

Indexing schemes of main interest are the bit map, address map, row-column, and the threaded list Major variations of the indexing techniques above mentioned are noted, as well as the particular indexing scheme inherent in diagonal or band matrices. The concluding section of the paper compares the types of methods, discusses their suitabihty for different types of processing, and makes suggestions eoneernlng the adaptability and flexibility of the maj or exmting methods of indexing algorithms for application to user problems

A Survey of Indexing Techniques for Sparse Matrices

A comparison in the context of sparse matrices is made between the Product Form of the Inverse PFI (a form of Gauss-Jordan elimination) and the Elimination Form of the Inverse EFI (a form of Gaussian elimination). The precise relation of the elements of these two forms of the inverse is given in terms of the nontrivial elements of the three matrices L, U, U-1 associated with the triangular factorization of the coefficient matrix A; i.e., A = L. U, where L is lower triangular and U is unit upper triangular. It is shown that the zero- nonzero structure of the PFI always has more nonzeros than the EFI. It is proved that Gaussian elimination is a minimal algorithm with respect to preserving sparseness if the diagonal elements of the matrix A are nonzero. However, Gaussian elimination is not nec- essarily minimal if A has some zero diagonal elements. The same statements hold for the PFI as well. A probabilistic study of fill-in and computing times for the PFI and EF sparse ma- trix algorithms is presented. This study suggests quantitatively how rapidly sparse matrices fill up for increasing densities, and emphasizes the necessity for reordering to minimize fill-in. I. Introduction. A sparse matrix is a matrix with very few nonzero elements. In many applications, a rough rule seems to be that there are O(N) nonzero entries; typically, say 2 to 10 nonzero entries per row. If the dimension N of the matrix is not large, then there is no compelling reason to treat sparse matrices differently from full matrices. It is when N becomes large and one attempts computations with the sparse matrix that it becomes necessary to take advantage of the zeros. The reason for this is obvious: there is a storage requirement of the order of N2 and arithmetic operations count of the order of N3 for many matrix algorithms using the full matrix. On the other hand, by storing only nonzero quantities and using logical operations to decide when an arithmetic operation is necessary, the storage requirement and arithmetic operations count can be reduced by a factor of N in many instances. Of course, this not only becomes a sizable savings of computer time, but also dictates whether or not some problems can be attempted. Computations with sparse matrices are not new. Iterative techniques for these matrices, especially those related to the solution of partial differential equations have been extensively developed (1). Sparse matrix methods for solving linear equa- tions by direct methods have been used for a long time in linear programming and there is a large body of literature, computational experience, programs, and artfulness, which has been built up in this area (2)-(15). In most linear programming codes, the product form of the inverse (PFI) is the method used to solve linear equations (16)-(19), although there are exceptions (20)-(21). Methods for scaling, pivoting for accuracy and sparseness, structuring data, and handling input-output have been extensively developed (22)-(72). However, there do not seem to exist rigorous results

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Some results on sparse matrices

The optimal (requiring the minimum number of multiplications) ordering of a sparse symmetric system of linear algebraic equations to be used with Gaussian elimination is first developed as a graph problem which is then treated using the functional equation techniques of dynamic programming. A simple algorithm is proposed as an alternative to the more lengthy procedures of dynamic programming and this algorithm is shown to be effective for systems whose graphs are \"grids\". Introduction. The motivation for this work is the fact that there exists a large class of physical problems which give rise to sparse symmetric linear systems, for which the computational effort required to obtain a solution by elimination is highly dependent upon the ordering of the equations. Here a system of n linear algebraic equations A'x' = b' (1) is called symmetric if the coefficient matrix is symmetric and sparse if A' has a large number of zero elements. In many problems dealing with structures, networks, finite difference formulations, etc., this is precisely the case. Certainly if there are no zero elements, there is no such thing as an optimum procedure in the sense in which the term is used here. The origins of this work can be traced back to Ivron [1] in the work which he calls \"Diakoptics\" and more recently to the work of Branin [2] and Roth [3]. It has been pointed out (see also [4], [5]) that to solve these sparse systems by first computing the inverse system matrix can be highly inefficient, and that Gaussian elimination, which is in fact a special case of one of Kron's techniques, is apparently the most efficient procedure, excluding special cases such as, e.g., systems which are highly symmetric (systems in which there is much repetition of elements or groups of elements). There are now digital computer programs available for the automatic analysis of many physical systems. Since the computational effort and therefore the cost is sensitive to the procedure used, it is important to proceed efficiently. In the following, Gaussian elimination is first developed as a graph problem; its functional equation is then treated using the dynamic programming techniques of Bellman [6]; and finally a simple algorithm is discussed which is a computationally attractive alternative to the dynamic programming procedures and which can be easily included in computer programs for automatic analysis. Gaussian elimination. Given a system in the form of Eq. (1), the question considered here is how to find the solution matrix x' using elimination, so that the computing time, and therefore computing cost, is as small as possible. Following von Neumann, the number of multiplications required will be counted as a measure of the computing time. *Received April 11, 1966; revised manuscript received July 21, 1967. 126 W. R, SPILLERS AND NORRIS HICKERSON [Vol. XXVI, No. 3 Consider the distinct phases, elimination and backsubstitution. Generally, a typical step in the elimination phase consists of using, e.g., the zth row of A' to remove all the nonzero terms except a'tl in the jth column by taking linear combinations of rows. Here, a restrictive form is used in which the zth row is used only to delete terms in the ith column. This implies that the system is well conditioned; it also, however, makes the problem tractable. In the backsubstitution phase, known components of the solution matrix x' are used to compute, e.g., the unknown ith component. In this restricted form, the numerical procedure is completely specified for a given system once the equations have been ordered. Further, if only the number of multiplications are to be counted, a reduced matrix A whose elements are defined to be a,, = 1 if a'a 9^ 0 ^

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W. R. Spillers

Papers

Optimal elimination for sparse symmetric systems as a graph problem.