{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,6]],"date-time":"2025-10-06T18:03:16Z","timestamp":1759773796568},"reference-count":20,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2014,10,22]],"date-time":"2014-10-22T00:00:00Z","timestamp":1413936000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2015,8]]},"DOI":"10.1007\/s11075-014-9931-z","type":"journal-article","created":{"date-parts":[[2014,10,22]],"date-time":"2014-10-22T03:00:45Z","timestamp":1413946845000},"page":"893-912","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A new subtraction-free formula for lower bounds of the minimal singular value of an upper bidiagonal matrix"],"prefix":"10.1007","volume":"69","author":[{"given":"Takumi","family":"Yamashita","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kinji","family":"Kimura","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yusaku","family":"Yamamoto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,10,22]]},"reference":[{"key":"9931_CR1","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0377-0427(96)00018-0","volume":"74","author":"Z Bai","year":"1996","unstructured":"Bai, Z., Fahey, M., Golub, G.H.: Some large-scale matrix computation problems. J. Comput. Appl. Math. 74, 71\u201389 (1996)","journal-title":"J. Comput. Appl. Math."},{"issue":"3","key":"9931_CR2","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1109\/MCSE.2005.52","volume":"7","author":"DH Bailey","year":"2005","unstructured":"Bailey, D.H.: High-precision floating-point arithmetic in scientific computation. Comput. Sci. Eng. 7(3), 54\u201361 (2005)","journal-title":"Comput. Sci. Eng."},{"key":"9931_CR3","first-page":"144","volume":"39","author":"C Brezinski","year":"2012","unstructured":"Brezinski, C., Fika, P., Mitrouli, M.: Estimations of the trace of powers of positive self-adjoint operators by extrapolation of the moments. Electron. Trans. Numer. Anal. 39, 144\u2013155 (2012)","journal-title":"Electron. Trans. Numer. Anal."},{"key":"9931_CR4","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1007\/BF01397083","volume":"18","author":"TJ Dekker","year":"1971","unstructured":"Dekker, T.J.: A floating-point technique for extending the available precision. Numer. Math. 18, 224\u2013242 (1971)","journal-title":"Numer. Math."},{"key":"9931_CR5","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/s002110050024","volume":"67","author":"KV Fernando","year":"1994","unstructured":"Fernando, K.V., Parlett, B.N.: Accurate singular values and differential qd algorithms. Numer. Math. 67, 191\u2013229 (1994)","journal-title":"Numer. Math."},{"key":"9931_CR6","doi-asserted-by":"crossref","DOI":"10.1515\/9781400833887","volume-title":"Matrices, Moments and Quadrature with Applications","author":"GH Golub","year":"2010","unstructured":"Golub, G.H., Meurant, M.: Matrices, Moments and Quadrature with Applications. Princeton University Press, Princeton and Oxford (2010)"},{"key":"9931_CR7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/0024-3795(92)90016-4","volume":"172","author":"YP Hong","year":"1992","unstructured":"Hong, Y.P., Pan, C.-T.: A lower bound for the smallest singular value. Linear Algebra Appl. 172, 27\u201332 (1992)","journal-title":"Linear Algebra Appl."},{"key":"9931_CR8","doi-asserted-by":"crossref","unstructured":"Iwasaki, M., Nakamura, Y.: Accurate computation of singular values in terms of shifted integrable schemes. Japan J. Indust. Appl. Math. 23, 239\u2013259 (2006)","DOI":"10.1007\/BF03167593"},{"key":"9931_CR9","doi-asserted-by":"crossref","unstructured":"Johnson, C.R.: A Gersgorin-type lower bound for the smallest singular value. Linear Algebra Appl. 112, 1\u20137 (1989)","DOI":"10.1016\/0024-3795(89)90583-1"},{"key":"9931_CR10","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/S0024-3795(97)00330-3","volume":"272","author":"CR Johnson","year":"1998","unstructured":"Johnson, C.R., Szulc, T.: Further lower bounds for the smallest singular value. Linear Algebra Appl. 272, 169\u2013179 (1998)","journal-title":"Linear Algebra Appl."},{"key":"9931_CR11","doi-asserted-by":"crossref","unstructured":"Kimura, K., Yamashita, T., Nakamura, Y.: Conserved quantities of the discrete finite Toda equation and lower bounds of the minimal singular value of upper bidiagonal matrices. J. Phys. A: Math. Theor. 44(285207), 12 (2011)","DOI":"10.1088\/1751-8113\/44\/28\/285207"},{"key":"9931_CR12","unstructured":"Knuth, D.E.: The Art of Computer Programming: Seminumerical Algorithms, volume 2, Addison-Wesley, Reading, MA (1969)"},{"key":"9931_CR13","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/S0898-1221(00)00289-3","volume":"41","author":"L Li","year":"2001","unstructured":"Li, L.: Lower bounds for the smallest singular value. Comput. Math. Appl. 41, 483\u2013487 (2001)","journal-title":"Comput. Math. Appl."},{"key":"9931_CR14","doi-asserted-by":"crossref","first-page":"1163","DOI":"10.1137\/S1064827594274887","volume":"18","author":"U Von Matt","year":"1997","unstructured":"Von Matt, U.: The orthogonal qd-algorithm. SIAM J. Sci. Comput. 18, 1163\u20131186 (1997)","journal-title":"SIAM J. Sci. Comput."},{"key":"9931_CR15","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s11075-012-9607-5","volume":"61","author":"M Nagata","year":"2012","unstructured":"Nagata, M., Iwasaki, M., Nakamura, Y.: Error analysis of the mdLVs algorithm for computing bidiagonal singular values. Numer. Algor. 61, 261\u2013274 (2012)","journal-title":"Numer. Algor."},{"key":"9931_CR16","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0898-1221(99)00252-7","volume":"38","author":"O Rojo","year":"1999","unstructured":"Rojo, O.: Further bounds for the smallest singular value and the spectral condition number. Comput. Math. Appl. 38, 215\u2013228 (1999)","journal-title":"Comput. Math. Appl."},{"key":"9931_CR17","volume-title":"The Algebraic Eigenvalue Problem","author":"JH Wilkinson","year":"1965","unstructured":"Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1965)"},{"key":"9931_CR18","first-page":"55","volume":"4","author":"T Yamashita","year":"2012","unstructured":"Yamashita, T., Kimura, K., Nakamura, Y.: Subtraction-free recurrence relations for lower bounds of the minimal singular value of an upper bidiagona matrix. J. Math-for-Ind. 4, 55\u201371 (2012)","journal-title":"J. Math-for-Ind."},{"key":"9931_CR19","doi-asserted-by":"crossref","unstructured":"Yamashita, T., Kimura, K., Takata, M., Nakamura, Y.: An application of the Kato-Temple inequality on matrix eigenvalues to the dqds algorithm for singular values. JSIAM Letters 5, 21\u201324 (2013)","DOI":"10.14495\/jsiaml.5.21"},{"key":"9931_CR20","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/0024-3795(95)00784-9","volume":"253","author":"Y-S Yu","year":"1997","unstructured":"Yu, Y.-S., Gu, D.-H.: A note on a lower bound for the smallest singular value. Linear Algebra Appl. 253, 25\u201338 (1997)","journal-title":"Linear Algebra Appl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9931-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-014-9931-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-014-9931-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,26]],"date-time":"2020-08-26T01:29:30Z","timestamp":1598405370000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-014-9931-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,10,22]]},"references-count":20,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,8]]}},"alternative-id":["9931"],"URL":"https:\/\/doi.org\/10.1007\/s11075-014-9931-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,10,22]]}}}