## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1-24 |

Number of pages | 24 |

Journal | Computers and Mathematics with Applications |

Volume | 72 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2016 |

### Bibliographical note

Funding Information:This work was partially supported by the project OpenViewSHIP, “Sviluppo di un ecosistema computazionale per la progettazione idrodinamica del sistema elica-carena”, supported by Regione FVG—PAR FSC 2007–2013, Fondo per lo Sviluppo e la Coesione and by the project “TRIM—Tecnologia e Ricerca Industriale per la Mobilità Marina”, CTN01-00176-163601 , supported by MIUR, the Italian Ministry of Instruction, University and Research .

## Keywords

- Expression templates
- High performance computing
- Linear algebra