Parametric Interval Linear Solver

Abstract

Abstract.IntervalComputations'LinearSystems' is a Mathematica package supporting tools for solving parametric and non-parametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results in validated interval arithmetic. The package is designed to be easy to use, versatile, to provide a necessary background for further exploration, comparisons and prototyping, and to provide some indispensable tools for solving parametric interval linear systems.This paper presents the functionality, provided by the current version of the package, and briefly discusses the underlying methodology. A new hybrid approach for sharp parametric enclosures, that combines parametric residual iteration, exact bounds, based on monotonicity properties, and refinement by interval subdivision, is outlined.

Solvers for the verified solution of parametric linear systems

“…3.1, Algorithm 1 by Rump [19] with the modification by Popova [15] (optional use of the sharp iteration matrix [C( p)]) is used in the implementation. The cost of Algorithm 1 is dominated by the computation of [C] and [z] and the computation of the approximate inverse R. The cost for the computation of R is O(n 3 ) and does not depend on the number of parameters k or the sparsity of the coefficient matrices.…”

“…If the coefficient matrices are sparse, the products in the computation of Our new implementation is based on the previous work described in [15,17]. The old solver uses an older C-XSC version and performs all computations in maximum precision using the long accumulator, which is very slow.…”

“…In the following we present a new version of our solvers for this problem using our C++ class library C-XSC [7], which is much more efficient than previous versions [15,17], both in terms of runtime and (through the optional use of sparse matrices) in terms of memory requirement. The new version also uses shared-memory parallelization via OpenMP to utilize the potential of modern multi-core processors.…”

Improved Solution Enclosures for Over- and Underdetermined Interval Linear Systems

“…Following this approach, over-/underdetermined linear system solvers with result verification were implemented in the environments of PASCAL-XSC [2] and C-XSC [1]. For problems with interval (real or complex) input data these implementations are based on solving square interval linear systems (4), (5) assuming that all the elements vary independently in their intervals. A close look at the structure of the matrices in the systems (4), (5) shows that each element of the matrix A appears twice in the augmented square matrix which means that this matrix involves dependencies.…”

“…For problems with interval (real or complex) input data these implementations are based on solving square interval linear systems (4), (5) assuming that all the elements vary independently in their intervals. A close look at the structure of the matrices in the systems (4), (5) shows that each element of the matrix A appears twice in the augmented square matrix which means that this matrix involves dependencies. As is well-known "the dependency problem" in interval analysis causes severe overestimation of the corresponding solution set and may lead to an interval iteration matrix involving singular ones.…”