{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:24:14Z","timestamp":1787383454256,"version":"3.56.0"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[1996,7]]},"abstract":"<jats:p>This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with $m \\geq n$ by $QR$ factorization of A and transformation of the right-hand side vector b to $Q^T b$. A multifrontal-based method for computing $Q^T b$ using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid model problem and problems defined on graphs with $\\sqrt n $-separators shows that the proposed method requires $O( N_R )$ storage and multiplications to compute $Q^T b$, where $N_R = O( n \\log n )$ is the number of nonzeros of the upper triangular factor R of A. In order to introduce BLAS-2 operations, Schreiber and Van Loan\u2019s storage-efficient WY representation [SIAM J. Sci. Stat. Comput., 10 (1989), pp. 53\u201357] is applied for the orthogonal factor $Q_i $ of each frontal matrix $F_i $. If this technique is used, the bound on storage increases to $O( ( n \\log n )^2 )$. Some numerical results for the grid model problems as well as Harwell\u2013Boeing problems are provided.<\/jats:p>","DOI":"10.1137\/s0895479893259509","type":"journal-article","created":{"date-parts":[[2005,2,27]],"date-time":"2005-02-27T07:15:07Z","timestamp":1109488507000},"page":"658-679","source":"Crossref","is-referenced-by-count":13,"title":["Multifrontal Computation with the Orthogonal Factors of Sparse Matrices"],"prefix":"10.1137","volume":"17","author":[{"given":"Szu-Min","family":"Lu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jesse L.","family":"Barlow","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,2,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0909046"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0725076"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/0729016"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0908009"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90101-7"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971811"},{"key":"R7","unstructured":"E. Chu,  Orthogonal Decomposition of Dense and Sparse Matrices on Multiprocessors, Tech. report, CS-88-08, University of Waterloo, Waterloo, Ontario, Canada,  1988"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1145\/356044.356047"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1137\/0710032"},{"key":"R10","volume-title":"Computer solution of large sparse positive definite systems","author":"George A.","year":"1981"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0909008"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396660"},{"key":"R13","unstructured":"J. R. Gilbert, E. G. Ng, B. W. Peyton,  Separators and Structure Prediction in Sparse Orthogonal Factorization, Tech. report, Xerox Palo Alto Research Center, Palo Alto, CA,  1993"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF01436075"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/0907053"},{"key":"R16","unstructured":"J. G. Lewis, D. J. Pierce, D. K. Wah,  Multifrontal Householder  QR Factorization, Tech. report, ECA-TR-127, Boeing Computer Services, Seattle, WA,  1989"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0716027"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0907081"},{"key":"R19","first-page":"486","volume-title":"Parallel processing for scientific computing, Vol. I, II (Norfolk, VA, 1993)","author":"Lu S. M.","year":"1993"},{"key":"R20","unstructured":"P. Matstoms, Masters Thesis,  The Multifrontal Solution of Sparse Linear Least Squares Problems, Thesis No. 293, LIU-TEK-LIC-1991:33, Department of Mathematics, Link\u00f6ping University, Sweden,  1991"},{"key":"R21","unstructured":"P. Matstoms,  Parallel Sparse  QR Factorization on Shared Memory Architectures, LiTH-MAT-R-1993-18, Department of Mathematics, Link\u00f6ping University, Sweden,  1993"},{"key":"R22","unstructured":"E. G. Ng, B. W. Peyton,  A Tight and Explicit Representation of  Q in Sparse  QR Factorization, Tech. report, ORNL\/TM-12059, Oak Ridge National Laboratory, Argonne, IL,  1992"},{"key":"R23","unstructured":"C. Puglisi, Ph.D. Thesis,  QR Factorization of Large Sparse Overdetermined and Square Matrices Using the Multifrontal Method in a Multiprocessor Environment, De L'institut National Polytechnique de Toulouse, Toulouse, France,  1993"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/0910005"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479892230948"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0895479893259509","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:48:21Z","timestamp":1787320101000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0895479893259509"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,7]]},"references-count":25,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["10.1137\/S0895479893259509"],"URL":"https:\/\/doi.org\/10.1137\/s0895479893259509","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,7]]}}}