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Encyclopedia of Parallel Computing pp 1209–1216Cite as

Multifrontal Method

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Synonyms

Linear equations solvers

Definition

The multifrontal method is a direct method for solving systems of linear equations Ax = b, when A is a sparse matrix and x and b are vectors or matrices. The multifrontal method organizes the operations that take place during the factorization of sparse matrices in such a way that the entire factorization is performed through partial factorizations of a sequence of dense and small submatrices. It is guided by a tree that represents the dependencies between those partial factorizations. In the following, the multifrontal method is formulated first for finite-element analysis and later generalized to assembled sparse matrices.

The Multifrontal Method

The multifrontal method is a direct method so that if the aim is to solve the equations

$$Ax = b,$$
(1)

where A is sparse, then this is done by performing the matrix factorization

$$PAQ = LU,$$
(2)

where P and Q are permutation matrices, L is a lower triangular matrix, and Uis an upper...

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Bibliography

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Author information

Authors and Affiliations

  1. INPT, Université de Toulouse ENSEEIHT-IRIT, 2 rue Charles-Camichel, F-31071, Toulouse cedex 7, France

    Patrick Amestoy Professor

  2. CNRS, Université de Toulouse, 2 rue Charles-Camichel, F-31071, Toulouse cedex 7, France

    Alfredo Buttari Dr.

  3. Rutherford Appleton Laboratory, Science & Technology Facilities Council, Chilton, OX11 0QX, Didcot, Oxfordshire, UK

    Iain Duff Professor

  4. LaBRI, Université de Bordeaux, 351, cours de la Libration, F-33405, Talence, France

    Abdou Guermouche Professor

  5. INRIA, ENS Lyon, 46 Allée d’Italie, F-69364, Lyon, France

    Jean-Yves L’Excellent Dr.

  6. CNRS, ENS Lyon, 46 Allée d’Italie, F-69364, Lyon, France

    Bora Uçar Dr.

Authors
  1. Patrick Amestoy Professor
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  2. Alfredo Buttari Dr.
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  3. Iain Duff Professor
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  4. Abdou Guermouche Professor
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  5. Jean-Yves L’Excellent Dr.
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  6. Bora Uçar Dr.
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Editor information

Editors and Affiliations

  1. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    David Padua

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Amestoy, P., Buttari, A., Duff, I., Guermouche, A., L’Excellent, JY., Uçar, B. (2011). Multifrontal Method. In: Padua, D. (eds) Encyclopedia of Parallel Computing. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09766-4_86

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