LinearOperator – a generic, high-level expression syntax for linear algebra

We introduce an expression syntax for the evaluation of matrix-matrix, matrix-vector and vector-vector operations. The implementation is similar to the well-known general concept of expression templates as used, for example, in the C++ linear-algebra libraries Eigen and Blaze. The novelty of the approach that is discussed here lies in the use of new C++11 features like lambda expressions and std::function objects that avoid the majority of the implementational complexity that usually comes with a pure template solution. A concrete implementation of the expression syntax has been developed within the framework of the finite-element library deal.II, but it is fairly generic: the LinearOperator implementation only requires a minimal vector and matrix interface, that all of deal.II's concrete vector and matrix types adhere to. This makes the interface fully transparent with respect to the concrete implementation, in particular to the storage strategy (full matrix, sparse structure), and memory strategy (local, shared, distributed). The paper concludes with a number of performance comparisons and examples that demonstrate that the framework results in efficient, short and concise code. The performance comparisons show that the overhead introduced by std::function objects is negligible for moderately sized matrices, even when compared to native expression-template implementations.