{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:43:45Z","timestamp":1750308225485,"version":"3.41.0"},"reference-count":29,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2005,3,1]],"date-time":"2005-03-01T00:00:00Z","timestamp":1109635200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Math. Softw."],"published-print":{"date-parts":[[2005,3]]},"abstract":"<jats:p>BHESS uses Gaussian similarity transformations to reduce a general real square matrix to similar upper Hessenberg form. Multipliers are bounded in root mean square by a user-supplied parameter. If the input matrix is not highly nonnormal and the user-supplied tolerance on multipliers is of a size greater than ten, the returned matrix usually has small upper bandwidth. In such a case, eigenvalues of the returned matrix can be determined by the bulge-chasing BR iteration or by Rayleigh quotient iteration. BHESS followed by BR iteration determines a complete spectrum in about one-fifth the time required for orthogonal reduction to Hessenberg form followed by QR iterations. The FORTRAN 77 code provided for BHESS runs efficiently on a cache-based architecture.<\/jats:p>","DOI":"10.1145\/1055531.1055539","type":"journal-article","created":{"date-parts":[[2005,8,3]],"date-time":"2005-08-03T08:30:55Z","timestamp":1123057855000},"page":"166-185","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":5,"title":["Algorithm 841: BHESS: Gaussian reduction to a similar banded Hessenberg form"],"prefix":"10.1145","volume":"31","author":[{"given":"Gary W.","family":"Howell","sequence":"first","affiliation":[{"name":"North Carolina State University, Raleigh, NC"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nadia","family":"Diaa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2005,3]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"Bai Z. 1994. Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem. Math. Comp. 62 205 209--226. 10.2307\/2153404   Bai Z. 1994. Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem. Math. Comp. 62 205 209--226. 10.2307\/2153404","DOI":"10.1090\/S0025-5718-1994-1201066-7"},{"key":"e_1_2_2_2_1","volume-title":"ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems. Research Report 95-04, Department of Mathematics","author":"Bai Z.","year":"1995","unstructured":"Bai , Z. , Day , D. , and Ye , Q . 1995 . ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems. Research Report 95-04, Department of Mathematics , University of Kentucky . Bai, Z., Day, D., and Ye, Q. 1995. ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems. Research Report 95-04, Department of Mathematics, University of Kentucky."},{"key":"e_1_2_2_3_1","first-page":"107","article-title":"Sequential reduction to tridiagonal form","volume":"7","author":"Bauer F. L.","year":"1959","unstructured":"Bauer , F. L. 1959 . Sequential reduction to tridiagonal form . SIAM J. 7 , 107 -- 113 . Bauer, F. L. 1959. Sequential reduction to tridiagonal form. SIAM J. 7, 107--113.","journal-title":"SIAM J."},{"key":"e_1_2_2_4_1","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1090\/S0025-5718-1969-0258255-3","article-title":"Reducing a matrix to Hessenberg form","volume":"23","author":"Businger P. A.","year":"1969","unstructured":"Businger , P. A. 1969 . Reducing a matrix to Hessenberg form . Math. Comp. 23 , 819 -- 921 . Businger, P. A. 1969. Reducing a matrix to Hessenberg form. Math. Comp. 23, 819--921.","journal-title":"Math. Comp."},{"key":"e_1_2_2_5_1","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1109\/9.57015","article-title":"Efficient matrix-valued algorithms for solving stiff Riccati differential equations","volume":"35","author":"Choi C. H.","year":"1990","unstructured":"Choi , C. H. and Laub , A. J. 1990 . Efficient matrix-valued algorithms for solving stiff Riccati differential equations . IEEE Trans. Autom. Contr. 35 , 770 -- 776 . Choi, C. H. and Laub, A. J. 1990. Efficient matrix-valued algorithms for solving stiff Riccati differential equations. IEEE Trans. Autom. Contr. 35, 770--776.","journal-title":"IEEE Trans. Autom. Contr."},{"key":"e_1_2_2_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01400115"},{"key":"e_1_2_2_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/138351.138352"},{"key":"e_1_2_2_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/0728045"},{"key":"e_1_2_2_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/0914009"},{"key":"e_1_2_2_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/0612026"},{"key":"e_1_2_2_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479897317077"},{"key":"e_1_2_2_12_1","volume-title":"Tech. Rep. ORNL\/TM-11089, Mathematical Sciences Section, Oak Ridge National Laboratory.","author":"Geist G. A.","year":"1989","unstructured":"Geist , G. A. , Lu , A. , and Wachspress , E. L . 1989 . Stabilized Gaussian reduction of an arbitrary matrix to tridiagonal form. Tech. Rep. ORNL\/TM-11089, Mathematical Sciences Section, Oak Ridge National Laboratory. Geist, G. A., Lu, A., and Wachspress, E. L. 1989. Stabilized Gaussian reduction of an arbitrary matrix to tridiagonal form. Tech. Rep. ORNL\/TM-11089, Mathematical Sciences Section, Oak Ridge National Laboratory."},{"key":"e_1_2_2_13_1","article-title":"A Hessenberg-Schur method for the problem AX&plus;XB&equals;C","author":"Golub G. H.","year":"1979","unstructured":"Golub , G. H. , Nash , S. , and Van Loan , C. 1979 . A Hessenberg-Schur method for the problem AX&plus;XB&equals;C . IEEE Trans. Automa. Contr. AC-24, 909--913. Golub, G. H., Nash, S., and Van Loan, C. 1979. A Hessenberg-Schur method for the problem AX&plus;XB&equals;C. IEEE Trans. Automa. Contr. AC-24, 909--913.","journal-title":"IEEE Trans. Automa. Contr. AC-24, 909--913."},{"key":"e_1_2_2_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/114697.116805"},{"volume-title":"Accuracy and Stability of Numerical Algorithms","author":"Higham N. J.","key":"e_1_2_2_15_1","unstructured":"Higham , N. J. 1996. Accuracy and Stability of Numerical Algorithms . SIAM , Philadephia . Higham, N. J. 1996. Accuracy and Stability of Numerical Algorithms. SIAM, Philadephia."},{"key":"e_1_2_2_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(94)90155-4"},{"key":"e_1_2_2_17_1","first-page":"426","volume-title":"Proceedings of the ISCA 8th International Conference on Parallel and Distributed Computing Systems","author":"Howell G. W.","year":"1995","unstructured":"Howell , G. W. and Geist , G. A . 1995. Reduction to a similar near-tridiagonal form . In Proceedings of the ISCA 8th International Conference on Parallel and Distributed Computing Systems ( Raleigh, NC , 1995 ), pp. 426 -- 432 . ISCA. Howell, G. W. and Geist, G. A. 1995. Reduction to a similar near-tridiagonal form. In Proceedings of the ISCA 8th International Conference on Parallel and Distributed Computing Systems (Raleigh, NC, 1995), pp. 426--432. ISCA."},{"key":"e_1_2_2_18_1","volume-title":"Tech. Rep. ORNL\/TM-11097, Mathematical Sciences Section, Oak Ridge National Laboratory.","author":"Howell G. W.","year":"1997","unstructured":"Howell , G. W. , Geist , G. A. , and Rowan , T . 1997 . Stability of reduction to a similar near-tridiagonal Hessenberg for m. Tech. Rep. ORNL\/TM-11097, Mathematical Sciences Section, Oak Ridge National Laboratory. Howell, G. W., Geist, G. A., and Rowan, T. 1997. Stability of reduction to a similar near-tridiagonal Hessenberg form. Tech. Rep. ORNL\/TM-11097, Mathematical Sciences Section, Oak Ridge National Laboratory."},{"key":"e_1_2_2_19_1","first-page":"263","article-title":"Expected conditioning for eigenvalues of randomly generated matrices","volume":"3","author":"Howell G. W.","year":"1995","unstructured":"Howell , G. W. and Rekab , K. 1995 . Expected conditioning for eigenvalues of randomly generated matrices . Neural, Parallel & Scientific Computations 3 , 2, 263 -- 270 . Howell, G. W. and Rekab, K. 1995. Expected conditioning for eigenvalues of randomly generated matrices. Neural, Parallel & Scientific Computations 3, 2, 263--270.","journal-title":"Neural, Parallel & Scientific Computations"},{"key":"e_1_2_2_20_1","doi-asserted-by":"publisher","DOI":"10.1137\/0613036"},{"key":"e_1_2_2_21_1","volume-title":"Cornelius Lanczos International Centenary Conference","author":"Parlett B. N.","year":"1993","unstructured":"Parlett , B. N. December , 1993 . Do we understand the symmetric Lanczos algorithm yet? Cornelius Lanczos International Centenary Conference , North Carolina State University. Parlett, B. N. December, 1993. Do we understand the symmetric Lanczos algorithm yet? Cornelius Lanczos International Centenary Conference, North Carolina State University."},{"key":"e_1_2_2_23_1","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/BF02075304","article-title":"Beitr\u00e4ge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos","volume":"4","author":"Rutishauser H.","year":"1953","unstructured":"Rutishauser , H. 1953 . Beitr\u00e4ge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos . Zeitschrift f\u00fcr angewandte Mathematik und Physik 4 , 35 -- 56 . Rutishauser, H. 1953. Beitr\u00e4ge zur Kenntnis des Biorthogonalisierungs-Algorithmus von Lanczos. Zeitschrift f\u00fcr angewandte Mathematik und Physik 4, 35--56.","journal-title":"Zeitschrift f\u00fcr angewandte Mathematik und Physik"},{"key":"e_1_2_2_24_1","doi-asserted-by":"publisher","DOI":"10.1145\/114697.116807"},{"key":"e_1_2_2_26_1","doi-asserted-by":"publisher","DOI":"10.1137\/0611023"},{"key":"e_1_2_2_27_1","doi-asserted-by":"publisher","DOI":"10.1016\/0893-9659(88)90183-8"},{"key":"e_1_2_2_28_1","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0024-3795(91)90004-G","article-title":"Convergence of algorithms of decomposition type for the eigenvalue problem","volume":"143","author":"Watkins D. S.","year":"1991","unstructured":"Watkins , D. S. and Elsner , L. 1991 . Convergence of algorithms of decomposition type for the eigenvalue problem . Lin. Alg. and Applic. 143 , 19 -- 47 . Watkins, D. S. and Elsner, L. 1991. Convergence of algorithms of decomposition type for the eigenvalue problem. Lin. Alg. and Applic. 143, 19--47.","journal-title":"Lin. Alg. and Applic."},{"volume-title":"The Algebraic Eigenvalue Problem","author":"Wilkinson J. H.","key":"e_1_2_2_29_1","unstructured":"Wilkinson , J. H. 1965. The Algebraic Eigenvalue Problem . Clarendon Press , Oxford, England . Wilkinson, J. H. 1965. The Algebraic Eigenvalue Problem. Clarendon Press, Oxford, England."},{"key":"e_1_2_2_30_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01389751","article-title":"On neighbouring matrices with quadratic elementary divisors","volume":"44","author":"Wilkinson J. H.","year":"1984","unstructured":"Wilkinson , J. H. 1984 . On neighbouring matrices with quadratic elementary divisors . Numer. Math. 44 , 1 -- 21 . Wilkinson, J. H. 1984. On neighbouring matrices with quadratic elementary divisors. Numer. Math. 44, 1--21.","journal-title":"Numer. Math."},{"key":"e_1_2_2_31_1","doi-asserted-by":"publisher","DOI":"10.2307\/2153403"}],"container-title":["ACM Transactions on Mathematical Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1055531.1055539","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/1055531.1055539","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T16:31:27Z","timestamp":1750264287000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/1055531.1055539"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,3]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2005,3]]}},"alternative-id":["10.1145\/1055531.1055539"],"URL":"https:\/\/doi.org\/10.1145\/1055531.1055539","relation":{},"ISSN":["0098-3500","1557-7295"],"issn-type":[{"type":"print","value":"0098-3500"},{"type":"electronic","value":"1557-7295"}],"subject":[],"published":{"date-parts":[[2005,3]]},"assertion":[{"value":"2005-03-01","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}