Francisco Jubete

Orthogonal sets and polar methods in linear algebra : applications to matrix calculations, systems of equations, inequalities, and linear programming

LINEAR SPACES AND SYSTEMS OF EQUATIONS. Basic Concepts. Orthogonal Sets. Matrix Calculations Using Orthogonal Sets. More Applications of Orthogonal Sets. Orthogonal Sets and Systems of Linear Equations. CONES AND SYSTEMS OF INEQUALITIES. Polyhedral Convex Cones. Polytopes and Polyhedra. Cones and Systems of Inequalities. LINEAR PROGRAMMING. An Introduction to Linear Programming. The Exterior Point Method. APPLICATIONS. Applications. Appendices. References. Index.

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, and Pruneda [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] and its 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 and its complement. This transformation, which is based on 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 applied to inverting a matrix and calculating 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.

Obtaining simultaneous solutions of linear subsystems of inequalities and duals

Abstract Given a set S of linear relations (equations and/or inequalities) among n variables, the problem of solving the systems resulting after selecting any subsets of S is dealt with. An algorithm that obtains all the necessary information to solve this problem, even if the operator in each linear relation is chosen, at wish, as an equality or inequality ⩽, is given. In addition, this algorithm simultaneously obtains the orthogonal set (dual cone) of a linear space (cone) generated by any subset of a given set of vectors (including sign selection), and allows simplifying the representation of the resulting linear spaces and cones to their minimal representations. The proposed methods are illustrated with several examples.

The Γ-algorithm and some applications

In this paper the power of the Γ-algorithm for obtaining the dual of a given cone and some of its multiple applications is discussed. The meaning of each sequential tableau appearing during the process is interpreted. It is shown that each tableau contains the generators of the dual cone of a given cone and that the algorithm updates the dual cone when new generators are incorporated. This algorithm, which is based on the duality concept, allows one to solve many problems in linear algebra, such as determining whether or not a vector belongs to a cone, obtaining the minimal representations of a cone in terms of a linear space and an acute cone, obtaining the intersection of two cones, discussing the compatibility of linear systems of inequalities, solving systems of linear inequalities, etc. The applications are illustrated with examples.

UPDATING INVERSES IN MATRIX ANALYSIS OF STRUCTURES

We deal with the problem of updating inverses. Several methods which allow calculating the inverse of a matrix when one or several rows (columns) are changed, or one or several rows and the same number of columns are added or removed are given. They are based on a method for calculating inverses given by Jubete and Castillo, which uses the concept of orthogonal sets, and lead to a considerable saving in computational power. The methods are ideal for being used in the design process of structures, where stiffness matrices are sequentially modified by simply changing rows (columns), and adding or removing rows and columns, as the result of modifying the geometric or structural characteristics of its pieces, the structures degrees of freedom, and/or the boundary conditions. Some examples of simple structures are given to illustrate the methodology. Finally, a discussion about its practical application and some conclusions and recommendations are given.

An expert system for coherent assessment of probabilities in multigraph models

In this paper, we present a method that allows a coherent assessment of probabilistic Bayesian networks over sets of nodes or variables that share a common subset. We motivate and illustrate the models with a diseases-symptoms knowledge base defined by multiple Bayesian networks. Since the parameter values for the joint probability distribution of diseases and symptoms must satisfy some compatibility conditions, the probability assessment becomes complicated and can be monitored by an expert system, in order to maintain coherence in the knowledge base. The bases of this expert system are presented. Once the human expert selects one parameter in the model to be assessed, the expert system calculates and provides the human expert with a range of feasible values of the parameter. Any value of the parameter within this range is coherent with the probability axioms. Then, the human expert can specify either a single value or an interval for that parameter in the feasible range. Accordingly, the expert system then updates the feasible ranges for all other parameters. The process can be repeated until all the parameters have been specified. At the end, the expert system gives a final set of values or intervals for all parameters. We apply the proposed system to the dependent-symptoms and the independent-symptoms probabilistic models. We give two numerical examples to illustrate the method.

A complete description of cones and polytopes including hypervolumes of all facets of a polytope

In this paper methods and algorithms for identifying the main elements (edges and facets of any dimension) of a cone and a polytope, and calculating the corresponding hypervolumes are presented. The cones and polytopes are supposed to be given as the non-negative linear combination and the convex hull generated by a, not necessarily minimal, set of vectors (points), respectively, and they can be degenerated (of a dimension smaller that that of the proper space in which they are contained). First a minimum set of generators (edges and vertices) are obtained by eliminating the redundant vectors. In the case of cones, the linear space basis and the minimal cone generators are obtained. Second the set of all facets of any dimension are identified. Finally, an algorithm for obtaining the associated hypervolumes of any dimension, i.e. the length of its edges, the areas of its faces of dimension two, and the hypervolumes of its facets of any dimension, is introduced. The proposed formula leads to a recursion that gives the hypervolumes of dimension n as a function of other hypervolumes of dimension n − 1. Examples are used to illustrate the proposed methods and algorithms.