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For the sake of portability, simplicity, flexibility and efficiency the GMRES solvers have been implemented in Fortran 77 using the reverse communication mechanism for the matrix-vector product, the preconditioning and the dot product computations. For distributed memory computation, several orthogonalization procedures have been implemented to reduce the cost of the dot product calculation, which is a well-known bottleneck of efficiency for the Krylov methods. Either implicit or explicit calculation of the residual at restart are possible depending on the actual cost of the matrix-vector product. Finally the implemented stopping criterion is based on a normwise backward error.<\/jats:p>","DOI":"10.1145\/1067967.1067970","type":"journal-article","created":{"date-parts":[[2005,8,3]],"date-time":"2005-08-03T08:30:55Z","timestamp":1123057855000},"page":"228-238","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":51,"title":["Algorithm 842"],"prefix":"10.1145","volume":"31","author":[{"given":"Val\u00e9rie","family":"Frayss\u00e9","sequence":"first","affiliation":[{"name":"Kvasar Technology LLC, Boston, MA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Luc","family":"Giraud","sequence":"additional","affiliation":[{"name":"CERFACS, Toulouse Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serge","family":"Gratton","sequence":"additional","affiliation":[{"name":"CERFACS, Toulouse Cedex, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Julien","family":"Langou","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, Knoxville, TN"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,6]]},"reference":[{"key":"e_1_2_2_1_1","volume-title":"Tech. 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SIAM , Philadelphia, PA . Barrett, R., Berry, M., Chan, T. F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., and der Vorst, H. V. 1994. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Second ed. SIAM, Philadelphia, PA."},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/567806.567809"},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","unstructured":"Bj\u00f6rck \u00c5. 1994. Numerics of Gram-Schmidt orthogonalization. Linear Algebra Appl. 197--198 297--316.  Bj\u00f6rck \u00c5. 1994. Numerics of Gram-Schmidt orthogonalization. Linear Algebra Appl. 197--198 297--316.","DOI":"10.1016\/0024-3795(94)90493-6"},{"key":"e_1_2_2_5_1","volume-title":"Numerical Methods for Least Squares Problems","author":"Bj\u00f6rck","unstructured":"Bj\u00f6rck , \u00c5. 1996. Numerical Methods for Least Squares Problems . SIAM , Philadelphia . Bj\u00f6rck, \u00c5. 1996. Numerical Methods for Least Squares Problems. 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