{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T11:05:45Z","timestamp":1760267145105},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2011,12,21]],"date-time":"2011-12-21T00:00:00Z","timestamp":1324425600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2012,9]]},"DOI":"10.1007\/s11075-011-9524-z","type":"journal-article","created":{"date-parts":[[2011,12,20]],"date-time":"2011-12-20T11:17:45Z","timestamp":1324379865000},"page":"1-34","source":"Crossref","is-referenced-by-count":14,"title":["Fast interval matrix multiplication"],"prefix":"10.1007","volume":"61","author":[{"given":"Siegfried M.","family":"Rump","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2011,12,21]]},"reference":[{"key":"9524_CR1","unstructured":"ACRITH: IBM High-Accuracy Arithmetic Subroutine Library, 3rd edn. IBM Deutschland GmbH, Sch\u00f6naicher Strasse 220, 71032 B\u00f6blingen (1986)"},{"key":"9524_CR2","unstructured":"ARITHMOS, Benutzerhandbuch, Siemens AG, Bibl.-Nr. U 2900-I-Z87-1 edition (1986)"},{"key":"9524_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/77626.79170","volume":"16","author":"JJ Dongarra","year":"1990","unstructured":"Dongarra, J.J., Du\u00a0Croz, J.J., Duff, I.S., Hammarling, S.J.: A set of level 3 basic linear algebra subprograms. ACM Trans. Math. Softw. 16, 1\u201317 (1990)","journal-title":"ACM Trans. Math. Softw."},{"key":"9524_CR4","doi-asserted-by":"crossref","first-page":"931","DOI":"10.2307\/2318109","volume":"77","author":"GE Forsythe","year":"1970","unstructured":"Forsythe, G.E.: Pitfalls in computation, or why a math book isn\u2019t enough. Am. Math. Mon. 77, 931\u2013956 (1970)","journal-title":"Am. Math. Mon."},{"key":"9524_CR5","doi-asserted-by":"crossref","unstructured":"Frommer, A.: Proving conjectures by use of interval arithmetic. In: Kulisch, U., et al. (ed.) Perspectives on Enclosure Methods. SCAN 2000, GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Univ. Karlsruhe, Germany, 19\u201322 September , 2000, Wien. Springer (2001)","DOI":"10.1007\/978-3-7091-6282-8_1"},{"key":"9524_CR6","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/BF01404681","volume":"18","author":"I Gargantini","year":"1972","unstructured":"Gargantini, I., Henrici, P.: Circular arithmetic and the determination of polynomial zeros. Numer. Math. 18, 305\u2013320 (1972)","journal-title":"Numer. Math."},{"issue":"1","key":"9524_CR7","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1145\/103162.103163","volume":"23","author":"D Goldberg","year":"1991","unstructured":"Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv 23(1), 5\u201347 (1991)","journal-title":"ACM Comput. Surv"},{"key":"9524_CR8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1377603.1377607","volume":"35","author":"K Goto","year":"2008","unstructured":"Goto, K., Van De\u00a0Geijnm, R. High-performance implementation of the level-3 blas. ACM Trans. Math. Softw. 35 1\u201314 (2008)","journal-title":"ACM Trans. Math. Softw."},{"key":"9524_CR9","unstructured":"Goualard, F.: How do you compute the midpoint of an interval? Technical report, CNRS, LINA, UMR 6241, Universit\u00e9 de Nantes (2011)"},{"issue":"2","key":"9524_CR10","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1145\/227699.227701","volume":"18","author":"JR Hauser","year":"1996","unstructured":"Hauser, J.R.: Handling floating-point exceptions in numeric programs. ACM Trans. Program. Lang. Syst. 18(2), 139\u2013174 (1996)","journal-title":"ACM Trans. Program. Lang. Syst."},{"key":"9524_CR11","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718027","volume-title":"Accuracy and stability of numerical algorithms","author":"NJ Higham","year":"2002","unstructured":"Higham, N.J.: Accuracy and stability of numerical algorithms, 2nd edn. SIAM Publications, Philadelphia (2002)","edition":"2"},{"key":"9524_CR12","unstructured":"ANSI\/IEEE 754-1985: IEEE Standard for Binary Floating-Point Arithmetic. New York (1985)"},{"key":"9524_CR13","unstructured":"ANSI\/IEEE 754-2008: IEEE Standard for Floating-Point Arithmetic. New York (2008)"},{"key":"9524_CR14","unstructured":"Jacobi, C., Oh, H.J., Tran, K.D., Cottier, S.R., Michael, B.W., Nishikawa, H., Totsuka, Y., Namatame, T., Yano, N.: The vector floating-point unit in a synergistic processor element of a cell processor. In: ARITH 2005: Proceedings of the 17th IEEE Symposium on Computer Arithmetic, pp. 59\u201367, Washington (2005)"},{"key":"9524_CR15","unstructured":"Jacobi, C., Oh, H.-J., Tran, K.D., Cottier, S.R., Michael, B.W., Nishikawa, H., Totsuka, Y., Namatame, T., Yano, N.: The vector floating-point unit in a synergistic processor element of a cell processor. In: ARITH \u201905: Proceedings of the 17th IEEE Symposium on Computer Arithmetic, pp. 59\u201367, Washington, DC, USA. IEEE Computer Society (2005)"},{"issue":"5","key":"9524_CR16","first-page":"96","volume":"3","author":"RB Kearfott","year":"1992","unstructured":"Kearfott, R.B., Dawande, M., Du, K., Hu, C.: Intlib: a portable Fortran-77 elementary function library. Interval Comput. 3(5), 96\u2013105 (1992)","journal-title":"Interval Comput."},{"key":"9524_CR17","volume-title":"C-XSC: A C++ Class Library for Extended Scientific Computing","author":"R Klatte","year":"1993","unstructured":"Klatte, R., Kulisch, U., Wiethoff, A., Lawo, C., Rauch, M.: C-XSC: A C++ Class Library for Extended Scientific Computing. Springer, Berlin (1993)"},{"key":"9524_CR18","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1007\/BF02307379","volume":"53","author":"O Kn\u00fcppel","year":"1994","unstructured":"Kn\u00fcppel, O.: PROFIL\/BIAS\u2014a fast interval library. Computing 53, 277\u2013287 (1994)","journal-title":"Computing"},{"key":"9524_CR19","unstructured":"Kn\u00fcppel, O.: PROFIL\/BIAS and extensions, version 2.0. Technical report, Inst. f. Informatik III, Technische Universit\u00e4t Hamburg-Harburg (1998)"},{"issue":"11","key":"9524_CR20","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1145\/362854.362889","volume":"14","author":"M Malcolm","year":"1971","unstructured":"Malcolm, M.: On accurate floating-point summation. Commun. ACM 14(11), 731\u2013736 (1971)","journal-title":"Commun. ACM"},{"key":"9524_CR21","unstructured":"Markoff, J.: Writing the Fastest Code, by Hand, for Fun: A Human Computer Keeps Speeding Up Chips. New York Times (2005)"},{"key":"9524_CR22","unstructured":"Moore, R.E.: Interval Arithmetic and Automatic Error Analysis in Digital Computing. Dissertation, Stanford University (1963)"},{"key":"9524_CR23","doi-asserted-by":"crossref","unstructured":"Muller, J.M., Brisebarre, N., de\u00a0Dinechin, F., Jeannerod, C.P., Lef\u00e8vre, V., Melquiond, G., Revol, N., Stehl\u00e9, D., Torres, S.: Handbook of Floating-Point Arithmetic. Birkh\u00e4user Boston (2010)","DOI":"10.1007\/978-0-8176-4705-6"},{"key":"9524_CR24","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/978-3-7091-6918-6_13","volume":"9","author":"MT Nakao","year":"1993","unstructured":"Nakao, M.T.: Solving nonlinear elliptic problems with result verification using an H \u2009\u2212\u20091type residual iteration. Computing Supplementum 9 161\u2013173 (1993)","journal-title":"Computing Supplementum"},{"issue":"2","key":"9524_CR25","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF03167877","volume":"5","author":"MT Nakao","year":"1988","unstructured":"Nakao, M.T.: A numerical approach to the proof of existence of solutions for elliptic problems. Jpn. J. Appl. Math. 5(2), 313\u2013332 (1988)","journal-title":"Jpn. J. Appl. Math."},{"key":"9524_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00607-004-0111-1","volume":"75","author":"MT Nakao","year":"2005","unstructured":"Nakao, M.T., Hashimoto, K., Watanabe, Y.: A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems. Computing 75, 1\u201314 (2005)","journal-title":"Computing"},{"key":"9524_CR27","unstructured":"Neumaier, A.: Interval Methods for Systems of Equations. Encyclopedia of Mathematics and its Applications. Cambridge University Press (1990)"},{"key":"9524_CR28","unstructured":"Neumaier, A.: Vienna proposal for interval standardization (2008). http:\/\/www.mat.univie.ac.at\/~neum\/papers.html . Accessed 6 Dec 2011"},{"key":"9524_CR29","doi-asserted-by":"crossref","unstructured":"Nguyen, H.D., Revol, N.: Accuracy issues in linear algebra using interval arithmetic. In: SCAN Conference Lyon (2010)","DOI":"10.1145\/1837210.1837236"},{"issue":"4","key":"9524_CR30","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s002110100310","volume":"90","author":"S Oishi","year":"2002","unstructured":"Oishi, S., Rump, S.M.: Fast verification of solutions of matrix equations. Numer. Math. 90(4), 755\u2013773 (2002)","journal-title":"Numer. Math."},{"key":"9524_CR31","unstructured":"Ozaki, K., Ogita, T., Rump, S.M., Oishi, S.: Accurate matrix multiplication by using level 3 BLAS operation. In: Proceedings of the 2008 International Symposium on Nonlinear Theory and its Applications, pp. 508\u2013511, NOLTA\u201908, Budapest, Hungary. IEICE (2008)"},{"issue":"1","key":"9524_CR32","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/BF02238648","volume":"49","author":"M Plum","year":"1992","unstructured":"Plum, M.: Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing 49(1), 25\u201344 (1992)","journal-title":"Computing"},{"issue":"1","key":"9524_CR33","first-page":"19","volume":"110","author":"M Plum","year":"2008","unstructured":"Plum, M.: Existence and multiplicity proofs for semilinear elliptic boundary value problems by computer assistance. DMV Jahresbericht 110(1), 19\u201354 (2008)","journal-title":"DMV Jahresbericht"},{"key":"9524_CR34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01213466","volume":"6","author":"S Poljak","year":"1993","unstructured":"Poljak, S., Rohn, J.: Checking Robust Nonsingularity Is NP-Hard. Math. Control Signals Syst. 6, 1\u20139 (1993)","journal-title":"Math. Control Signals Syst."},{"key":"9524_CR35","unstructured":"Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen. PhD thesis, Universit\u00e4t Karlsruhe (1980)"},{"issue":"3","key":"9524_CR36","first-page":"539","volume":"39","author":"SM Rump","year":"1999","unstructured":"Rump, S.M.: Fast and parallel interval arithmetic. BIT Numer. Math. 39(3), 539\u2013560 (1999)","journal-title":"BIT Numer. Math."},{"key":"9524_CR37","first-page":"77","volume-title":"Developments\u00a0in\u00a0Reliable Computing","author":"SM Rump","year":"1999","unstructured":"Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments\u00a0in\u00a0Reliable Computing, pp. 77\u2013104. Kluwer Academic Publishers, Dordrecht (1999)"},{"key":"9524_CR38","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF03167543","volume":"26","author":"SM Rump","year":"2009","unstructured":"Rump, S.M.: Inversion of extremely ill-conditioned matrices in floating-point. Jpn. J. Ind. Appl. Math. 26, 1\u201329 (2009)","journal-title":"Jpn. J. Ind. Appl. Math."},{"key":"9524_CR39","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1017\/S096249291000005X","volume":"19","author":"SM Rump","year":"2010","unstructured":"Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287\u2013449 (2010)","journal-title":"Acta Numer."},{"key":"9524_CR40","doi-asserted-by":"crossref","unstructured":"Rump, S.M.: Error estimation of floating-point summation and dot product. BIT (2011, in press)","DOI":"10.1007\/s10543-011-0342-4"},{"issue":"1","key":"9524_CR41","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1137\/050645671","volume":"31","author":"SM Rump","year":"2008","unstructured":"Rump, S.M., Ogita, T., Oishi, S.: Accurate floating-point summation part I: faithful rounding. SIAM J. Sci. Comput. 31(1), 189\u2013224 (2008)","journal-title":"SIAM J. Sci. Comput."},{"key":"9524_CR42","volume-title":"Floating-point Computation","author":"PH Sterbenz","year":"1974","unstructured":"Sterbenz, P.H.: Floating-point Computation. Prentice Hall, Englewood Cliffs, NJ (1974)"},{"key":"9524_CR43","unstructured":"Sunaga, T.: Geometry of Numerals. Master\u2019s thesis, University of Tokyo (1956)"},{"key":"9524_CR44","unstructured":"Takayasu, A., Oishi, S., Kubo, T.: Guaranteed error estimate for solutions to two-point boundary value problem. In: Proceedings of the International Symposium on Nonlinear Theory and its Applications (NOLTA2009), pp. 214\u2013217. Sapporo, Japan (2009)"},{"key":"9524_CR45","doi-asserted-by":"crossref","first-page":"953","DOI":"10.1016\/j.cam.2008.03.034","volume":"223","author":"Y Watanabe","year":"2009","unstructured":"Watanabe, Y.: A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math. 223:953\u2013966 (2009)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"9524_CR46","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1002\/zamm.200700158","volume":"89","author":"Y Watanabe","year":"2009","unstructured":"Watanabe, Y., Plum, M., Nakao, M.T.: A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow. Z. angew. Math. Mech. 89(1), 5\u201318 (2009)","journal-title":"Z. angew. Math. Mech."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-011-9524-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-011-9524-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-011-9524-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,21]],"date-time":"2019-06-21T03:23:34Z","timestamp":1561087414000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-011-9524-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,12,21]]},"references-count":46,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["9524"],"URL":"https:\/\/doi.org\/10.1007\/s11075-011-9524-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,12,21]]}}}