{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:26:51Z","timestamp":1786580811630,"version":"build-2736575974"},"reference-count":43,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/501100003392","name":"Fujian Provincial Natural Science Foundation","doi-asserted-by":"publisher","award":["2026J001412"],"award-info":[{"award-number":["2026J001412"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171168"],"award-info":[{"award-number":["12171168"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110551","type":"journal-article","created":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T06:08:39Z","timestamp":1783922919000},"page":"110551","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P2","title":["Structure-preserving randomized quaternion full orthogonalization method for non-Hermitian quaternion linear systems"],"prefix":"10.1016","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5011-0801","authenticated-orcid":false,"given":"Baohua","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaojian","family":"Ding","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changfeng","family":"Ma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.110551_bib0001","series-title":"Elements of Quaternions","author":"Hamilton","year":"1866"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110551_bib0002","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1137\/17M1136699","article-title":"SVD-based algorithms for the best rank-1 approximation of a symmetric tensor","volume":"39","author":"Guan","year":"2018","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0003","doi-asserted-by":"crossref","first-page":"2971","DOI":"10.1088\/0305-4470\/33\/15\/306","article-title":"Right eigenvalue equation in quaternionic quantum mechanics","volume":"33","author":"Leo","year":"2000","journal-title":"J Phys A: Math Gen"},{"key":"10.1016\/j.cnsns.2026.110551_bib0004","doi-asserted-by":"crossref","first-page":"1967","DOI":"10.1109\/TNN.2011.2171362","article-title":"Fast independent component analysis algorithm for quaternion valued signals","volume":"22","author":"Javidi","year":"2011","journal-title":"IEEE T Neural Netw"},{"key":"10.1016\/j.cnsns.2026.110551_bib0005","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.neucom.2016.09.060","article-title":"Cognitive quaternion valued neural network and some applications","volume":"221","author":"Saoud","year":"2017","journal-title":"Neurocomputing"},{"key":"10.1016\/j.cnsns.2026.110551_bib0006","doi-asserted-by":"crossref","DOI":"10.1002\/nla.2245","article-title":"Robust quaternion matrix completion with applications to image inpainting","volume":"26","author":"Jia","year":"2019","journal-title":"Numer Linear Algebr"},{"key":"10.1016\/j.cnsns.2026.110551_bib0007","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1137\/18M1230451","article-title":"Color image restoration by saturation-value total variation","volume":"12","author":"Jia","year":"2019","journal-title":"SIAM J Imaging Sci"},{"key":"10.1016\/j.cnsns.2026.110551_bib0008","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.neucom.2016.08.006","article-title":"Color image classification via quaternion principal component analysis network","volume":"216","author":"Zeng","year":"2016","journal-title":"Neurocomputing"},{"key":"10.1016\/j.cnsns.2026.110551_bib0009","doi-asserted-by":"crossref","first-page":"23483","DOI":"10.1007\/s11042-018-5652-y","article-title":"Optimal blind watermarking for color images based on the U matrix of quaternion singular value decomposition","volume":"77","author":"Liu","year":"2018","journal-title":"Multimed Tools Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0010","first-page":"709","article-title":"Total variation based pure quaternion dictionary learning method for color image denoising","volume":"19","author":"Wu","year":"2022","journal-title":"Int J Numer Anal Mod"},{"key":"10.1016\/j.cnsns.2026.110551_bib0011","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1137\/24M1715532","article-title":"Multilinear analysis of quaternion arrays: theory and computation","volume":"47","author":"Flamant","year":"2026","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0012","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.1109\/TSP.2008.2010600","article-title":"The quaternion lms algorithm for adaptive filtering of hypercomplex processes","volume":"57","author":"Took","year":"2009","journal-title":"IEEE T Signal Proces"},{"key":"10.1016\/j.cnsns.2026.110551_bib0013","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/MSP.2023.3278071","article-title":"Quaternions in signal and image processing: acomprehensive and objective overview","volume":"40","author":"Miron","year":"2023","journal-title":"IEEE Signal Proc Mag"},{"key":"10.1016\/j.cnsns.2026.110551_bib0014","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1137\/20M133751X","article-title":"Structure preserving quaternion generalized minimal residual method","volume":"42","author":"Jia","year":"2021","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0015","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1137\/23M1552115","article-title":"Block diagonalization of quaternion circulant matrices with applications to quaternion tensor singular value decomposition","volume":"45","author":"Pan","year":"2024","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0016","doi-asserted-by":"crossref","DOI":"10.1002\/nla.2495","article-title":"Structure preserving quaternion full orthogonalization method with applications","volume":"30","author":"Li","year":"2023","journal-title":"Numer Linear Algebr"},{"key":"10.1016\/j.cnsns.2026.110551_bib0017","doi-asserted-by":"crossref","DOI":"10.1002\/nla.70008","article-title":"Gl-qfom and gl-qgmres: two efficient algorithms for quaternion linear systems with multiple right-hand sides","volume":"32","author":"Li","year":"2025","journal-title":"Numer Linear Algebr"},{"key":"10.1016\/j.cnsns.2026.110551_bib0018","doi-asserted-by":"crossref","DOI":"10.1007\/s11075-026-02361-4","article-title":"Preconditioned generalized minimal residual method for quaternion linear systems with structured coefficient matrices and its applications","author":"Hu","year":"2026","journal-title":"Numer Algorithms"},{"key":"10.1016\/j.cnsns.2026.110551_bib0019","doi-asserted-by":"crossref","first-page":"2957","DOI":"10.1007\/s10462-019-09752-1","article-title":"A survey of quaternion neural networks","volume":"53","author":"Parcollet","year":"2020","journal-title":"Artif Intell Rev"},{"key":"10.1016\/j.cnsns.2026.110551_bib0020","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1016\/j.cam.2018.04.019","article-title":"A new real structure-preserving quaternion qr algorithm","volume":"343","author":"Jia","year":"2018","journal-title":"J Comput Appl Math"},{"key":"10.1016\/j.cnsns.2026.110551_bib0021","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.cam.2012.09.018","article-title":"A new structure-preserving method for quaternion hermitian eigenvalue problems","volume":"239","author":"Jia","year":"2013","journal-title":"J Comput Appl Math"},{"key":"10.1016\/j.cnsns.2026.110551_bib0022","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1137\/23M1547299","article-title":"Structure preserving quaternion biconjugate gradient method","volume":"45","author":"Li","year":"2024","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0023","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1090\/mcom\/4074","article-title":"QQMR: a structure preserving quaternion quasi-minimal residual method for non-Hermitian quaternion linear systems","volume":"95","author":"Li","year":"2026","journal-title":"Math Comput"},{"key":"10.1016\/j.cnsns.2026.110551_bib0024","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1137\/24M1658723","article-title":"Reorthogonalized pythagorean variants of block classical gram-schmidt","volume":"46","author":"Carson","year":"2026","journal-title":"SIAM J Matrix Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0025","doi-asserted-by":"crossref","first-page":"A1450","DOI":"10.1137\/20M138870X","article-title":"Randomized gram-schmidt process with application to gmres","volume":"44","author":"Balabanov","year":"2022","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110551_bib0026","doi-asserted-by":"crossref","first-page":"A553","DOI":"10.1137\/22M1518396","article-title":"Randomized block gram-schmidt process for solution of linear systems and eigenvalue problems","volume":"47","author":"Balabanov","year":"2025","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110551_bib0027","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s11075-024-01801-3","article-title":"Randomized flexible gmres with deflated restarting","volume":"98","author":"Jang","year":"2025","journal-title":"Numer Algorithms"},{"key":"10.1016\/j.cnsns.2026.110551_bib0028","unstructured":"Rimsit E., Grigori L., Balabanov O.. Randomized orthogonal projection methods for krylov subspace solvers.2023; arXiv preprint arXiv: 2302.07466v3."},{"key":"10.1016\/j.cnsns.2026.110551_bib0029","doi-asserted-by":"crossref","DOI":"10.1142\/S0129055X13500062","article-title":"Continuous slice functional calculus in quaternionic hilbert spaces","volume":"25","author":"Ghiloni","year":"2013","journal-title":"Rev Math Phys"},{"key":"10.1016\/j.cnsns.2026.110551_bib0030","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1023\/B:BITN.0000014561.58141.2c","article-title":"Givens\u2019 transformation applied to quaternion valued matrices","volume":"43","author":"Janovsk\u00e1","year":"2003","journal-title":"BIT Numer Math"},{"key":"10.1016\/j.cnsns.2026.110551_bib0031","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1016\/S0022-0000(03)00025-4","article-title":"Database-friendly random projections: Johnson-Lindenstrauss with binary coins","volume":"66","author":"Achlioptas","year":"2003","journal-title":"J Comput Syst Sci"},{"key":"10.1016\/j.cnsns.2026.110551_bib0032","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1137\/090771806","article-title":"Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions","volume":"53","author":"Halko","year":"2010","journal-title":"SIAM Rev"},{"key":"10.1016\/j.cnsns.2026.110551_bib0033","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/0400000060","article-title":"Sketching as a tool for numerical linear algebra","volume":"10","author":"Woodruff","year":"2014","journal-title":"Found Trends Theor Comput Sci"},{"key":"10.1016\/j.cnsns.2026.110551_bib0034","doi-asserted-by":"crossref","first-page":"2969","DOI":"10.1007\/s10444-019-09725-6","article-title":"Randomized linear algebra for model reduction. part i: galerkin methods and error estimation","volume":"45","author":"Balabanov","year":"2019","journal-title":"Adv Comput Math"},{"key":"10.1016\/j.cnsns.2026.110551_bib0035","doi-asserted-by":"crossref","first-page":"3895","DOI":"10.1016\/j.optcom.2011.04.001","article-title":"Color image encryption using spatial bit-level permutation and high-dimension chaotic system","volume":"284","author":"Liu","year":"2011","journal-title":"Opt Commun"},{"key":"10.1016\/j.cnsns.2026.110551_bib0036","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.sigpro.2017.03.011","article-title":"A new color image encryption using combination of the 1d chaotic map","volume":"138","author":"Pak","year":"2017","journal-title":"Signal Process"},{"key":"10.1016\/j.cnsns.2026.110551_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.dsp.2025.105791","article-title":"Novel 4d chaotic system based on memristor and its application in secure color image encryption","volume":"170","author":"Zhang","year":"2026","journal-title":"Digit Signal Process"},{"key":"10.1016\/j.cnsns.2026.110551_bib0038","doi-asserted-by":"crossref","DOI":"10.1088\/1674-1056\/ae0bff","article-title":"Color image encryption algorithm using a new four-dimensional hyperchaotic system","volume":"35","author":"Lin","year":"2026","journal-title":"Chinese Phys B"},{"key":"10.1016\/j.cnsns.2026.110551_bib0039","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s11071-025-11972-w","article-title":"Visually secure medical image encryption algorithm based on improved yolov11 and replacement strategy for region of interest","volume":"114","author":"Zhu","year":"2026","journal-title":"Nonlinear Dyn"},{"key":"10.1016\/j.cnsns.2026.110551_bib0040","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1080\/09500340.2026.2623157","article-title":"Image encryption-authentication scheme based on pixel adaptive diffusion","volume":"73","author":"Gong","year":"2026","journal-title":"J Mod Opt"},{"key":"10.1016\/j.cnsns.2026.110551_bib0041","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/s10773-024-05630-x","article-title":"A quantum image encryption and watermarking algorithm based on QDCT and Baker map","volume":"63","author":"Zhou","year":"2024","journal-title":"Int J Theor Phys"},{"key":"10.1016\/j.cnsns.2026.110551_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2025.130442","article-title":"Color image encryption and authentication algorithm integrating computational ghost imaging with quaternion multi-parameter discrete fractional angular transform","volume":"301","author":"Hu","year":"2026","journal-title":"Expert Syst Appl"},{"key":"10.1016\/j.cnsns.2026.110551_bib0043","doi-asserted-by":"crossref","first-page":"1429","DOI":"10.1007\/s11075-024-01920-x","article-title":"QMGI algorithm for solving quaternion equation and its application in color image encryption","volume":"99","author":"Li","year":"2025","journal-title":"Numer Algorithms"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009056?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009056?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T23:26:56Z","timestamp":1786577216000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426009056"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":43,"alternative-id":["S1007570426009056"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110551","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Structure-preserving randomized quaternion full orthogonalization method for non-Hermitian quaternion linear systems","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110551","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110551"}}