Efficient Parallel Solvers for Large Dense Systems of Linear Interval Equations

Verified solvers for dense linear (interval-)systems require a lot of re- sources, both in terms of computing power and memory usage. Computing a verified solution of large dense linear systems (dimension n > 10000) on a single machine quickly approaches the limits of today's hardware. Therefore, an efficient parallel verified solver for distributed memory sys- tems is needed. In this work we present such a solver, implemented in C++ and using the C-XSC library for scientific computing (10, 8). The solver utilizes MPI (27) for communication between the nodes in the parallel environment and, where applicable, high performance ScaLAPACK (4) and BLAS (3) routines for fast computing times. High precision dot products (5, 18, 19, 21) are used to compute narrow enclosures of the solution of the system. We present test results on several high performance distributed mem- ory systems with different architectures, which show that our solver a- chieves good results, both in terms of numerical accuracy as well as com- puting time, and is highly portable. Furthermore, even very large systems (n ≥ 100000) can be solved given a cluster with sufficient resources.

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