{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,23]],"date-time":"2024-08-23T02:47:04Z","timestamp":1724381224395},"reference-count":34,"publisher":"Association for Computing Machinery (ACM)","issue":"3","funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP0559083"],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2012,4]]},"abstract":"\n We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called\n \u03d5<\/jats:italic>\n -functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the\n matlab<\/jats:sc>\n function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.\n <\/jats:p>","DOI":"10.1145\/2168773.2168781","type":"journal-article","created":{"date-parts":[[2012,5,7]],"date-time":"2012-05-07T18:47:42Z","timestamp":1336416462000},"page":"1-19","update-policy":"http:\/\/dx.doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":142,"title":["Algorithm 919"],"prefix":"10.1145","volume":"38","author":[{"given":"Jitse","family":"Niesen","sequence":"first","affiliation":[{"name":"University of Leeds"}]},{"given":"Will M.","family":"Wright","sequence":"additional","affiliation":[{"name":"La Trobe University"}]}],"member":"320","published-online":{"date-parts":[[2012,4]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/100788860"},{"key":"e_1_2_2_2_1","doi-asserted-by":"crossref","unstructured":"Butcher J. C. 2008. Numerical Methods for Ordinary Differential Equations 2nd Ed. John Wiley & Sons Ltd. Chichester. Butcher J. C. 2008. Numerical Methods for Ordinary Differential Equations 2nd Ed. John Wiley & Sons Ltd. Chichester.","DOI":"10.1002\/9780470753767"},{"key":"e_1_2_2_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2008.03.021"},{"key":"e_1_2_2_4_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100023197"},{"key":"e_1_2_2_5_1","unstructured":"Davis T. A. 2007. The University of Florida sparse matrix collection. Tech. rep. REP-2007-298 CISE Department University of Florida. Davis T. A. 2007. The University of Florida sparse matrix collection. Tech. rep. REP-2007-298 CISE Department University of Florida."},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1956-0084194-4"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/62038.62043"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01060992"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/0913071"},{"key":"e_1_2_2_10_1","volume-title":"Solving Ordinary Differential Equations. I 2nd Ed. Springer Series in Computational Mathematics","volume":"8","author":"Hairer E."},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/6.2.327"},{"key":"e_1_2_2_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/04061101X"},{"key":"e_1_2_2_13_1","unstructured":"Higham N. J. 2008. Functions of Matrices. Society for Industrial and Applied Mathematics (SIAM) Philadelphia PA. Higham N. J. 2008. Functions of Matrices . Society for Industrial and Applied Mathematics (SIAM) Philadelphia PA."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995280572"},{"key":"e_1_2_2_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827595295337"},{"key":"e_1_2_2_16_1","doi-asserted-by":"crossref","unstructured":"Horn R. A. and Johnson C. R. 1991. Topics in Matrix Analysis. Cambridge University Press Cambridge UK. Horn R. A. and Johnson C. R. 1991. Topics in Matrix Analysis . Cambridge University Press Cambridge UK.","DOI":"10.1017\/CBO9780511840371"},{"key":"e_1_2_2_17_1","volume-title":"Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics","volume":"33","author":"Hundsdorfer W."},{"key":"e_1_2_2_18_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.2790085"},{"key":"e_1_2_2_19_1","first-page":"303","article-title":"ADI finite difference schemes for option pricing in the Heston model with correlation","volume":"7","author":"In \u2019t Hout K. J.","year":"2010","journal-title":"Int. J. Numer. Anal. Model."},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2008.03.016"},{"key":"e_1_2_2_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2006.04.032"},{"key":"e_1_2_2_22_1","doi-asserted-by":"publisher","DOI":"10.1145\/1499096.1499101"},{"key":"e_1_2_2_23_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-010-0273-5"},{"key":"e_1_2_2_24_1","unstructured":"Minchev B. and Wright W. M . 2005. A review of exponential integrators for semilinear problems. Tech. rep. 2\/05 Department of Mathematical Sciences NTNU Norway. http:\/\/www.math.ntnu.no\/preprint\/. Minchev B. and Wright W. M . 2005. A review of exponential integrators for semilinear problems. Tech. rep. 2\/05 Department of Mathematical Sciences NTNU Norway. http:\/\/www.math.ntnu.no\/preprint\/."},{"key":"e_1_2_2_25_1","doi-asserted-by":"publisher","DOI":"10.1137\/S00361445024180"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.532"},{"key":"e_1_2_2_27_1","doi-asserted-by":"crossref","unstructured":"Niesen J. and Wright W. M. 2011. A Krylov subspace method for option pricing. http:\/\/ssrn.com\/abstract=1799124. Niesen J. and Wright W. M. 2011. A Krylov subspace method for option pricing. http:\/\/ssrn.com\/abstract=1799124.","DOI":"10.2139\/ssrn.1799124"},{"key":"e_1_2_2_28_1","doi-asserted-by":"publisher","DOI":"10.1137\/0729014"},{"key":"e_1_2_2_29_1","first-page":"1","article-title":"Evaluating matrix functions for exponential integrators via Carath\u00e9odory-Fej\u00e9r approximation and contour integrals","volume":"29","author":"Schmelzer T.","year":"2007","journal-title":"Electron. Trans. Numer. Anal."},{"key":"e_1_2_2_30_1","doi-asserted-by":"publisher","DOI":"10.1145\/285861.285868"},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2008.03.035"},{"key":"e_1_2_2_32_1","unstructured":"Sofroniou M. and Spaletta G. 2007. Efficient computation of Krylov approximations to the matrix exponential and related functions. Personal communication. Sofroniou M. and Spaletta G. 2007. Efficient computation of Krylov approximations to the matrix exponential and related functions. Personal communication."},{"key":"e_1_2_2_33_1","unstructured":"Tavella D. and Randall C. 2000. Pricing Financial Instruments. John Wiley & Sons Ltd. New York. Tavella D. and Randall C. 2000. Pricing Financial Instruments . John Wiley & Sons Ltd. New York."},{"key":"e_1_2_2_34_1","doi-asserted-by":"crossref","unstructured":"Trefethen L. N. and Bau III D. 1997. Numerical Linear Algebra. SIAM Philadelphia PA. Trefethen L. N. and Bau III D. 1997. Numerical Linear Algebra . SIAM Philadelphia PA.","DOI":"10.1137\/1.9780898719574"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2168773.2168781","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,30]],"date-time":"2022-12-30T06:54:31Z","timestamp":1672383271000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2168773.2168781"}},"subtitle":["A Krylov Subspace Algorithm for Evaluating the \u03d5-Functions Appearing in Exponential Integrators"],"short-title":[],"issued":{"date-parts":[[2012,4]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["10.1145\/2168773.2168781"],"URL":"http:\/\/dx.doi.org\/10.1145\/2168773.2168781","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,4]]},"assertion":[{"value":"2009-07-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-03-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-04-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}