The solution of linear systems of equations with many right hand sides is mostly seen as a trivial extension of solving a linear system and the algorithmic developments mostly focus on the efficient computation of the LU decomposition. This is, however, not regarding the case where many right hand sides increase the runtime influence of the forward/backward substitution. In this contribution we present a GPU accelerated Gauss-Jordan-elimination based all-at-once solution scheme which focuses on minimizing the runtime and the energy consumption by switching the forward/backward substitution in favor of a more suitable operation. We obtain a multi-GPU aware algorithm which is up to 2.5 times faster than the current state-of-the-art LU decomposition based solution process of MAGMA and saves 48 % required energy.

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For a given matrix, we are interested in computing GR decompositions A = GR, where G is an isometry with respect to given scalar products. The orthogonal QR decomposition is the representative for the Euclidian scalar product. For a signature matrix, a respective factorization is given as the hyperbolic QR decomposition. Considering a skew-symmetric matrix leads to the symplectic QR decomposition. The standard approach for computing GR decompositions is based on the successive elimination of subdiagonal matrix entries. For the hyperbolic and symplectic case, this approach does in general not lead to a satisfying numerical accuracy. An alternative approach computes the QR decomposition via a Cholesky factorization, but also has bad stability. It is improved by repeating the procedure a second time. In the same way, the hyperbolic and the symplectic QR decomposition are related to the LDL T and a skew-symmetric Cholesky-like factorization. We show that methods exploiting this connection can provide better numerical stability than elimination-based approaches.