Solvers for the verified solution of parametric linear systems

Abstract

We present a newly developed version of our solvers for the verified solution of dense parametric linear systems, i.e. linear systems whose system matrix and right-hand side depend affine-linearly on parameters that vary inside prescribed intervals. The solvers use our C++ class library for reliable computing, C-XSC. The C-XSC library provides many features, especially easy to handle data types for dense and sparse matrices and vectors and the ability to compute dot products and dot product expressions in arbitrary precision. The new solvers can use either sparse or dense matrices as the coefficient matrices for the parameters. The use of sparse coefficient matrices can result in huge improvements in both performance and memory consumption. BLAS and LAPACK routines are used where applicable, and OpenMP is used for the parallelization on multi-core and multi-processor systems. The solvers also provide the ability to compute not only an outer but also a componentwise inner enclosure of the solution set of the system and to choose between two versions of the algorithm, one being very fast and one giving sharp results and extending the range of solvable systems. We give some examples for parametric linear systems (also from real world examples such as worst-case tolerance analysis of linear electric circuits), give performance measurements of our solvers and also demonstrate that they scale very well when using multiple cores or processors.