{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T10:34:26Z","timestamp":1773311666284,"version":"3.50.1"},"reference-count":25,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2020,7,13]],"date-time":"2020-07-13T00:00:00Z","timestamp":1594598400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/100000005","name":"U.S. Department of Defense","doi-asserted-by":"publisher","award":["W911NF- 15-1-0456"],"award-info":[{"award-number":["W911NF- 15-1-0456"]}],"id":[{"id":"10.13039\/100000005","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007637","name":"Departamento Administrativo de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n","doi-asserted-by":"crossref","award":["6172"],"award-info":[{"award-number":["6172"]}],"id":[{"id":"10.13039\/100007637","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2020,9,30]]},"abstract":"<jats:p>Multicomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. The similarities between multicomplex and multidual algebras allowed us to create a unified library to use either one for sensitivity analysis. This library can be used to compute arbitrary order derivates of functions of a single variable or multiple variables. The storage of matrix representations of multicomplex and multidual numbers is avoided using a combination of one-dimensional resizable arrays and an indexation method based on binary bitwise operations. To provide high computational efficiency and low memory usage, the multiplication of hypercomplex numbers up to sixth order is carried out using a hard-coded algorithm. For higher hypercomplex orders, the library uses by default a multiplication method based on binary bitwise operations. The computation of algebraic and transcendental functions is achieved using a Taylor series approximation. Fortran and Python versions were developed, and extensions to other languages are self-evident.<\/jats:p>","DOI":"10.1145\/3378538","type":"journal-article","created":{"date-parts":[[2020,7,7]],"date-time":"2020-07-07T12:37:18Z","timestamp":1594125438000},"page":"1-30","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":21,"title":["MultiZ"],"prefix":"10.1145","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0729-1093","authenticated-orcid":false,"given":"Andres M.","family":"Aguirre-Mesa","sequence":"first","affiliation":[{"name":"Universidad EAFIT, Colombia., Medellin, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manuel J.","family":"Garcia","sequence":"additional","affiliation":[{"name":"Angelo State University, USA."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7097-9283","authenticated-orcid":false,"given":"Harry","family":"Millwater","sequence":"additional","affiliation":[{"name":"University of Texas at San Antonio, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2020,7,13]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/09074721X"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/110852553"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.2514\/2.1270"},{"key":"e_1_2_1_4_1","article-title":"Application of hyper-dual numbers to multibody kinematics","volume":"8","author":"Cohen A.","year":"2015","unstructured":"A. Cohen and M. Shoham . 2015 . Application of hyper-dual numbers to multibody kinematics . J. Mechan. Robot. 8 , 1 (08 2015), 011015\u2013011015\u20134. DOI:https:\/\/doi.org\/10.1115\/1.4030588 10.1115\/1.4030588 A. Cohen and M. Shoham. 2015. Application of hyper-dual numbers to multibody kinematics. J. Mechan. Robot. 8, 1 (08 2015), 011015\u2013011015\u20134. DOI:https:\/\/doi.org\/10.1115\/1.4030588","journal-title":"J. Mechan. Robot."},{"key":"#cr-split#-e_1_2_1_5_1.1","doi-asserted-by":"crossref","unstructured":"E. Deadman N. J. Higham and R. Ralha. 2013. Blocked Schur algorithms for computing the matrix square root. In Applied Parallel and Scientific Computing Pekka Manninen and Per \u00d6ster (Eds.). Springer Berlin 171--182. DOI:https:\/\/doi.org\/10.1007\/978-3-642-36803-5_12 10.1007\/978-3-642-36803-5_12","DOI":"10.1007\/978-3-642-36803-5_12"},{"key":"#cr-split#-e_1_2_1_5_1.2","doi-asserted-by":"crossref","unstructured":"E. Deadman N. J. Higham and R. Ralha. 2013. Blocked Schur algorithms for computing the matrix square root. In Applied Parallel and Scientific Computing Pekka Manninen and Per \u00d6ster (Eds.). Springer Berlin 171--182. DOI:https:\/\/doi.org\/10.1007\/978-3-642-36803-5_12","DOI":"10.1007\/978-3-642-36803-5_12"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(76)90020-5"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.2514\/6.2011-886"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.2514\/1.J053282"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.probengmech.2015.12.001"},{"key":"e_1_2_1_11_1","volume-title":"Functions of Matrices: Theory and Computation","author":"Higham N. J.","unstructured":"N. J. Higham . 2008. Functions of Matrices: Theory and Computation . Society for Industrial and Applied Mathematics . N. J. Higham. 2008. Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics."},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1038\/s41592-019-0686-2"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3650-4"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/2168773.2168774"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/0704019"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/838250.838251"},{"key":"e_1_2_1_17_1","unstructured":"M. Metcalf J. Reid and M. Cohen. 2011. Modern Fortran Explained. OUP Oxford.  M. Metcalf J. Reid and M. Cohen. 2011. Modern Fortran Explained. OUP Oxford."},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.engfracmech.2016.04.002"},{"key":"e_1_2_1_19_1","unstructured":"G. B. Price. 1990. An Introduction to Multicomplex Spaces and Functions. Taylor 8 Francis.  G. B. Price. 1990. An Introduction to Multicomplex Spaces and Functions. Taylor 8 Francis."},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1137\/S003614459631241X"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.12.010"},{"key":"e_1_2_1_22_1","volume-title":"MATLAB version 9.3.0 (R2017b)","author":"The MathWorks Inc. 2017.","unstructured":"The MathWorks Inc. 2017. MATLAB version 9.3.0 (R2017b) . Natick, Massachusetts . The MathWorks Inc. 2017. MATLAB version 9.3.0 (R2017b). Natick, Massachusetts."},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1080\/174159794088027573"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2011.05.003"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.2514\/1.19225"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3378538","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3378538","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3378538","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:45:04Z","timestamp":1750203904000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3378538"}},"subtitle":["A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers"],"short-title":[],"issued":{"date-parts":[[2020,7,13]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9,30]]}},"alternative-id":["10.1145\/3378538"],"URL":"https:\/\/doi.org\/10.1145\/3378538","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"value":"0098-3500","type":"print"},{"value":"1557-7295","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,13]]},"assertion":[{"value":"2018-10-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-01-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2020-07-13","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}