Alternative methods for representing the inverse of linear programming basis matrices

Methods for representing the inverse of Linear Programming (LP) basis matrices are closely related to techniques for solving a system of sparse unsymmetric linear equations by direct methods. It is now well accepted that for these problems the static process of reordering the matrix in the lower block triangular (LBT) form constitutes the initial step. We introduce a combined static and dynamic factorisation of a basis matrix and derive its inverse which we call the partial elimination form of the inverse (PEFI). This factorization takes advantage of the LBT structure and produces a sparser representation of the inverse than the elimination form of the inverse (EFI). In this we make use of the original columns (of the constraint matrix) which are in the basis. To represent the factored inverse it is, however, necessary to introduce special data structures which are used in the forward and the backward transformations (the two major algorithmic steps) of the simplex method. These correspond to solving a system of equations and solving a system of equations with the transposed matrix respectively. In this paper we compare the nonzero build up of PEFI with that of EFI. We have also investigated alternative methods for updating the basis inverse in the PEFI representation. The results of our experimental investigation are presented in this pape

Multigrid Methods in Lattice Field Computations

The multigrid methodology is reviewed. By integrating numerical processes at all scales of a problem, it seeks to perform various computational tasks at a cost that rises as slowly as possible as a function of $n$, the number of degrees of freedom in the problem. Current and potential benefits for lattice field computations are outlined. They include: $O(n)$ solution of Dirac equations; just $O(1)$ operations in updating the solution (upon any local change of data, including the gauge field); similar efficiency in gauge fixing and updating; $O(1)$ operations in updating the inverse matrix and in calculating the change in the logarithm of its determinant; $O(n)$ operations per producing each independent configuration in statistical simulations (eliminating CSD), and, more important, effectively just $O(1)$ operations per each independent measurement (eliminating the volume factor as well). These potential capabilities have been demonstrated on simple model problems. Extensions to real life are explored.Comment: 4

An orthogonally based pivoting transformation of matrices and some applications

In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, andPruned a [Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming, John Wiley, New York, 1999] andits multiple applications. The meaning of each sequential tableau appearing during the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors andits complement. This transformation, which is basedon the orthogonality concept, allows us to solve many problems of linear algebra, such as calculating the inverse and the determinant of a matrix, updating the inverse or the determinant of a matrix after changing a row (column), determining the rank of a matrix, determining whether or not a set of vectors is linearly independent, obtaining the intersection of two linear subspaces, solving systems of linear equations, etc. When the process is appliedto inverting a matrix andcalculating its determinant, not only is the inverse of the final matrix obtained, but also the inverses and the determinants of all its block main diagonal matrices, all without extra computations

An efficient basis update for asymptotic linear programming

AbstractFor a linear program in which the constraint coefficients vary linearly with the time parameter, we showed in a previous paper that a basic feasible solution can be evaluated using O((k + 1)m3) arithmetic operations, where m is the number of constraints and k is the index of the basis matrix pair. Here we show, in the special case when k = 1 for all basis matrix pairs, and when one of the matrices in each pair has nearly full rank, how the (possibly singular) matrix factorization can be updated with only O(m2) operations, using rank-one update techniques. This makes the arithmetic complexity of updating the basis in asymptotic linear programming comparable to that of updating the inverse in ordinary linear programming, in this case. Moreover, we show that the result holds, in particular, when computing a Blackwell optimal policy for Markov decision chains in the unichain case or when all policies have only a small number of recurrent subchains

Determination of the source parameter in a heat equation with a non-local boundary condition

AbstractIn this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods