Pivot Row Research Articles

A fast Faddeev array

A systolic array for the fast computation of the Faddeev algorithm is presented. Inversion of an n*n matrix on a systolic array is known to tend to 5 n inner product steps under the assumption that no data are duplicated. The proposed Faddeev array achieves matrix inversion in just 4 n steps with O(n/sup 2/) basic cells using careful duplications of some data. The array consists of two half-arrays which compute two separate but coupled triangularizations. The coupling is resolved by an on-the-fly decoupling process which duplicates pivot row data and passes them between the arrays using only nearest neighbor connections.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Symbolic Factorization for Sparse Gaussian Elimination with Partial Pivoting

Let $Ax = b$ be a large sparse nonsingular system of linear equations to be solved using Gaussian elimination with partial pivoting. The factorization obtained can be expressed in the form $A = P_1 M_1 P_2 M_2 \cdots P_{n - 1} M_{n - 1} U$, where $P_k $ is an elementary permutation matrix reflecting the row interchange that occurs at step k during the factorization, $M_k $ is a unit lower triangular matrix whose kth column contains the multipliers, and U is an upper triangular matrix.Consider the kth step of the elimination. Suppose we replace the structure of row k of the partially reduced matrix by the union of the structures of those rows which are candidates for the pivot row and then perform symbolically Gaussian elimination without partial pivoting. Assume that this is done at each step k, and let $\bar L$ and $\bar U$ denote the resulting lower and upper triangular matrices, respectively. Then the structures of $\bar L$ and $\bar U$, respectively, contain the structures of $\Sigma _{k = 1}^{n - 1} M_k $ and U. This paper describes an algorithm which determines the structures of $\bar L$ and $\bar U$, and sets up an efficient data structure for them. Since the algorithm depends only on the structure of A, the data structure can be created in advance of the actual numerical computation, which can then be performed very efficiently using the fixed storage scheme. Although the data structure is more generous than it needs to be for any specific sequence $P_1 ,P_2 , \cdots ,P_{n - 1} $, experiments indicate that the approach is competitive with conventional methods. Another important point is that the storage scheme is large enough to accommodate the $QR$ factorization of A, so it is also useful in the context of computing a sparse orthogonal decomposition of A. The algorithm is shown to execute in time bounded by $|\bar L| + |\bar U|$, where $|M|$ denotes the number of nonzeros in the matrix M.

Symbolic Givens Reduction and Row-Ordering in Large Sparse Least Squares Problems

In the solution of large sparse least squares problems by Givens factorization, a preliminary symbolic step that determines a good processing order and a data structure for the matrix factor is used. In this paper, it is shown that a processing order equivalent to sequential processing by rows can be as good as any processing order using a single pivot row in each column. A notion of local acceptability in row-ordering is introduced and shown to reduce fill globally. Row-orderings satisfying this notion are essentially equivalent to sequential processing by rows and the nature of intermediate fill produced makes implicit representation of fill possible. This forms the basis for a symbolic Givens reduction algorithm that operates in a fixed data structure.

Effect of Equilibration on Residual Size for Partial Pivoting

It is shown that column pivoting with row equilibration satisfies the same type of error bound as does row pivoting without scaling, and that row pivoting with column equilibration satisfies the same type of bound as does column pivoting without scaling. An interesting consequence for row pivoting is that column equilibration is sufficient to ensure that the norm of the residual is reasonably small.

Iterative refinement implies numerical stability for Gaussian elimination

Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect. It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense. Also, it is shown that without iterative refinement row pivoting is inferior to column pivoting in situations where the norm of the residual is important.

On the Bartels—Golub decomposition for linear programming bases

Many implementations of the simplex method require the row of the inverse of the basis matrixB corresponding to the pivot row at each iteration for updating either a pricing vector or the nonbasic reduced costs. In this note we show that if the Bartels--Golub algorithm [1, 2] or one of its variants is used to update theLU factorization ofB, then less computing is needed if one works with the factors of the updatedB than with those ofB. These results are discussed as they apply to the column selection algorithms recently proposed by Goldfarb and Reid [4, 5] and Harris [6].

A Generated Cut for Primal Integer Programming

This paper develops a new cutting plane for primal integer programming. This cut, when it exists and is vised as pivot row, has the property that the minimum of the simplex evaluators of the next tableau is strictly larger (by an integer) than the minimum for the current tableau. The paper gives a sufficient condition in the form of a linear program for the existence of such a cut, and proposes an algorithm for generating such cuts. It also presents two additional sufficient conditions. The process of cut generation is then imbedded into a convergent primal algorithm to be used in an attempt to overcome the proof-of-optimality problem experienced with the primal algorithm.

A note on a modified primal‐dual algorithm to speed convergence in solving linear programs

AbstractThe primal‐dual algorithm is modified in a two part procedure. In the first part, the pivot row is selected so that an artificial variable is always dropped. The end of the first part usually produces some basic variables with negative values. The second part consists of selecting the most negative basic variable. The equation, represented by the selected basic variable, is multiplied through by minus one and then added to all equations with negative basic variables; it is then augmented by an artificial variable. This procedure produces feasibility for all basic variables and maintains canonical form. The standard primal‐dual method is then used to complete the solution. Computational results are presented.

A Principal Pivoting Simplex Algorithm for Linear and Quadratic Programming

This paper presents a simplex algorithm for finding a nonnegative solution (or demonstrating the inconsistency) of y = a + Ax where A is positive semi-definite. Linear and quadratic programming problems are of this form. The function exhibited in the proof of finiteness does not appear in other algorithms. If a primal feasible solution is available in the linear programming case, the actual choice of pivot rows is exactly that made in the usual lexicographic simplex method.

A modified inversion procedure for product form of the inverse linear programming codes

This paper describes a new algorithm for the selection of the pivot row in matrix inversion when using the product form of the inverse. This algorithm has been developed for linear programming codes; however, it would be valuable for the inversion of any non-dense matrix. The procedures described in this paper have been thoroughly tested and have been in operation on the Esso Research and Engineering IBM 7090 computer for nine months. Substantial computer cost savings have been realized because of this procedure.