Solving Linear Systems with Polynomial Parameter Dependency with Application to the Verified Solution of Problems in Structural Mechanics
2013, Garloff, Jürgen, Popova, Evgenija D., Smith, Andrew P.
We give a short survey on methods for the enclosure of the solution set of a system of linear equations where the coeffficients of the matrix and the right hand side depend on parameters varying within given intervals. Then we present a hybrid method for finding such an enclosure in the case that the dependency is polynomial or rational. A general-purpose parametric fixed-point iteration is combined with efficient tools for range enclosure based on the Bernstein expansion of multivariate polynomials. We discuss applications of the general-purpose parametric method to linear systems obtained by standard finite element analysis of mechanical structures and illustrate the efficiency of the new parametric solver.
Bounding the Range of a Rational Functiom over a Box
2012, Narkawicz, Anthony, Garloff, Jürgen, Smith, Andrew P., Munoz, César A.
A simple method is presented by which tight bounds on the range of a multivariate rational function over a box can be computed. The approach relies on the expansion of the numerator and denominator polynomials into Bernstein polynomials.
Bounds on the Range of Multivariate Rational Functions
2012, Garloff, Jürgen, Schabert, Antek, Smith, Andrew P.
By utilising the expansion of a polynomial into Bernstein polynomials, we construct bounds for the range of a multivariate rational function over a box.