Column Interchanges Research Articles

Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix

For a sparse symmetric matrix, there has been much attention given to algorithms for reducing the bandwidth. As far as we can see, little has been done for the unsymmetric matrix A, which has distinct lower and upper bandwidths l and u. When Gaussian elimination with row interchanges is applied, the lower bandwidth is unaltered, while the upper bandwidth becomes $l+u$. With column interchanges, the upper bandwidth is unaltered, while the lower bandwidth becomes $l+u$. We therefore seek to reduce $\min (l,u)+l+u$, which we call the total bandwidth. We compare applying the reverse Cuthill–McKee algorithm to $A+A^T$, to the row graph of A, and to the bipartite graph of A. We also propose an unsymmetric variant of the reverse Cuthill–McKee algorithm. In addition, we have adapted the node‐centroid and hill‐climbing ideas of Lim, Rodrigues, and Xiao to the unsymmetric case. We have found that using these to refine a Cuthill–McKee‐based ordering can give significant further bandwidth reductions. Numerical results for a range of practical problems are presented and comparisons made with the recent lexicographical method of Baumann, Fleischmann, and Mutzbauer.

Stability of the Gauss-Huard algorithm with partial pivoting

This paper considers elimination methods to solve dense linear systems, in particular a variant of Gaussian elimination due to Huard [13]. This variant reduces the system to an equivalent diagonal system just like Gauss-Jordan elimination, but does not require more floating-point operations than Gaussian elimination. To preserve stability, a pivoting strategy using column interchanges, proposed by Hoffmann [10], is incorporated in the original algorithm. An error analysis is given showing that Huard’s elimination method is as stable as Gauss-Jordan elimination with the appropriate pivoting strategy. This result is proven in a similar way as the proof of stability for Gauss-Jordan elimination given in [4]. Numerical experiments are reported which verify the theoretical error analysis of the Gauss-Huard algorithm.

The implicit LX method of the ABS class

We describe an algorithm of the ABS class, which solves a general qonsingular linear system in n 3/3 + 0(n 2) multiplications without the assumption that the coefficient matrix be regular. The method can be viewed as a variation of the implicit LU algorithm of the ABS class, whose associated factorization contains a factor which is not triangular (but can be reduced to triangular form after suitable row permutations). We describe king the Abaffan properties of the method, including in particular an efficient way of upd matrix after column interchanges. Such a problem arises in the application to the simplex algorithm, where the implicit LX algorithm provides a faster technique than the standard LU factorization for the pivoting operation if the number of equality constraints m is greater than n/2

Determinants of Latin Squares of Order 8

A latin square is an n × n array of n symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-n polynomial in n variables. Can two latin squares L, M have the same determinant, up to a renaming of the variables, apart from the obvious cases when L is obtained from M by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to beno if n ≤ 7; we show that it is yes for n = 8. The latin squaresfor which this situation occurs have interesting special characteristics.

Triangular Decomposition Methods for Solving Reducible Nonlinear Systems of Equations

This paper generalizes to the nonlinear case a standard way to solve general sparse systems of linear equations. In particular, Duff [J. Inst. Math. Appl., 19 (1977), pp. 339–342] has suggested that row and column interchanges be applied to permute the coefficient matrix of a linear system into block lower triangular forma The linear system then can be solved by using the associated block Gauss–Seidel or forward block substitution scheme. This is the approach taken in the Harwell MA28 routine. If more than one matrix with the same sparsity pattern is to be factored, then the reordering need not be redone. In extending this approach to nonlinear problems, it is necessary to assume as in the linear case that component equations can be evaluated separately from equations in other blocks. The algorithm for doing the reordering is very fast, and if the equations can be put into block lower triangular form with each block size much less than the dimension of the system, then a large savings is in order because only the diagonal blocks need to be factored. In the nonlinear variants considered here, there are additional advantages. Not only are just the diagonal blocks of the Jacobian matrix computed and factored, but the off diagonal partial derivatives need not even exist. Numerical tests and analytic results affirm the intuition that these variants are superior locally to Newton’s method. Current work is concerned with globalizing these methods and with variants especially suited to parallel implementations.

Parallel matrix inversion on a subcube-grid

In this paper we propose a new medium-grain parallel algorithm for computing a matrix inverse on a hypercube multiprocessor. The algorithm implements Gauss-Jordan inversion with column interchanges. The hypercube network is configured as a two-dimensional subcube-grid to support submatrix partitionings. For some algorithms on some types of hypercubes, submatrix partitionings are known to have communication advantages not shared by partitions limited to rows or columns We show that such advantages can be extended to Gauss-Jordan inversion on an Intel iPSC/860, the most current third-generation of hypercubes, and that there is little extra programming effort to include it in the subcube-grid library used in various other matrix computations. An actual aggregate execution rate of 200 MFLOPS (Million Floating-point Operation Per Second) is achieved when inverting a 2000 × 2000 matrix (in double-precision Fortran 77) using 64 iPSC/860 processors configured as an 8 × 8 subcube-grid.

A Scheme for Handling Rank-Deficiency in the Solution of Sparse Linear Least Squares Problems

Several schemes for computing sparse orthogonal factorizations using static data structures have been proposed recently. One novel feature of some of these schemes is that the data structures are large enough to store both the orthogonal transformations and upper triangular factor explicitly. However, in order to make use of the static storage schemes, the orthogonal factorization has to be computed without column interchanges, which is sufficient when the observation matrix has full rank. When the least squares matrix is rank-deficient, computing the minimum-norm solution to the least squares problem requires the knowledge of the rank of the matrix, which may be difficult to get if the factorization is computed without column pivoting. In this paper an algorithm is developed that makes use of the resulting factorization to solve rank-deficient least squares problems. The new algorithm has three major features. First, it uses the original triangular factor. Second, this factor is not altered. Third, the rank decision is based on the singular value decomposition and not on the diagonal elements of the triangular factor. The techniques used are similar to those employed by Björck.

Rehabilitation of the Gauss-Jordan algorithm

In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.

Rank and null space calculations using matrix decomposition without column interchanges

The most widely used stable methods for numerical determination of the rank of a matrix A are the singular value decomposition and the QR algorithm with column interchanges. Here two algorithms are presented which determine rank and nullity in a numerically stable manner without using column interchanges. One algorithm makes use of the condition estimator of Cline, Moler, Stewart, and Wilkinson and relative to alternative stable algorithms is particularly efficient for sparse matrices. The second algorithm is important in the case that one wishes to test for rank and nullity while sequentially adding columns to a matrix.

An efficient method for the solution of systems of linear equations based on planerotation transformations

An algorithm is presented based on Givens or plane-rotation transformations for solving systems of simultaneous linear equations frequently appearing in problems of the theory of elasticity for which the accuracy of the obtained results is critical in order to gain valuable information of the related problems. This algorithm is compared with Crout's method with an iterative process for convergence (N.A.G. Library routine FO4AEA) and the algorithm based on Householder transformations presented by Businger and Golub[1], after a slight modification, in order to perform one major step less and avoid column interchanges. They have been tested on the same data and the Givens transformations method proved to be the most efficient of all, both in accuracy and in run time.

Partitioning of invariant imbedding systems

The choice of partitioning the system matrix for a system of N linear ordinary differential equations may determine the ease or difficulty of obtaining a solution by the invariant imbedding method of Scott. This paper shows how the configuration and partitioning of the system matrix is reflected in the fundamental matrix. The partitioning of the fundamental matrix is the key to the ease or difficulty of obtaining a solution. If the fundamental matrix is known for a given system matrix configuration and partitioning, then the fundamental matrix associated with a new system matrix configuration may be derived by the same row and column interchanges that transformed the old system matrix into the new system matrix. The fundamental matrix for the new system matrix does not have to be recalculated anew from the Kronecker delta initial conditions.

Comparison of two pivotal strategies in sparse plane rotations

Let the rectangular matrix A be large and sparse. Assume that plane rotations are used to decompose A into QDR where Q TQ = I, D is diagonal and R is upper triangular. Both column and row interchanges.have to be used in order to preserve the sparsity of matrix A during the decomposition. If the column interchanges are fixed, then the number of non-zero elements in R does not depend on the row interchanges used. However, this does not mean that the computational work is also independent of the row interchanges. Two pivotal strategies, where the same rule is used in the choice of pivotal columns, are described and compared. It is verified (by many numerical examples) that if matrix A is not very sparse, then one of these strategies will often perform better than the other both with regard to the storage and the computing time. The accuracy and the robustness of the computations are also discussed. In the implementation described in this paper positive values of a special parameter, drop-tolerance, can optionally be used to remove all “small” elements created during the decomposition. The accuracy lost by dropping some non-zero elements is normally regained by iterative refinement. The numerical results indicate that this approach is very efficient for some matrices.

Comparison of two pivotal strategies in sparse plane rotations

Let the rectangular matrix A be large and sparse. Assume that plane rotations are used to decompose A into RDS where R^T R = I, D is a diagonal and S is upper triangular. Both column and row interchanges have to be used in order to preserve the sparsity of matrix A during the decomposition. It is proved that H the column interchanges are fixed, then the number of non-zero elements in S does not depend on the row interchanges used. However, this does not mean that the computational work is also independent of the row interchanges. Two pivotal strategies, where the same rule is used in the choice of pivotal columns, are described and compared. It is verified (by many numerical examples) that if matrix A is not very sparse, then one of these strategies will often perform better than the other both with regard to the storage and the computing time. The accuracy and the robustness of the computations are also discussed.

The Movement and Permutation of Columns in Magnetic Bubble Lattice Files

In this paper, we propose a layout of access channels in a new kind of magnetic bubble memory called a bubble lattice file. We further assume that interchange of bubble columns under certain channels is possible. We then propose an algorithm for moving a column to any other location using minimal number of column interchanges. The algorithm and the proof of its optimality involves signed-digit representation of numbers by a mixed radix system. Based on this, we next propose a simple algorithm to permute a set of columns. Analysis of this algorithm for both the worst and average case is given. The algorithm is shown to be close to optimal for both cases. Finally, based on an optimization consideration, a geometric layout of access channels is suggested for the best performance of the proposed algorithm.