{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T20:28:23Z","timestamp":1765484903646},"reference-count":28,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2005,7,8]],"date-time":"2005-07-08T00:00:00Z","timestamp":1120780800000},"content-version":"vor","delay-in-days":3721,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[1995,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Many researchers are now working on computing the product of a matrix function and a vector, using approximations in a Krylov subspace. We review our results on the analysis of one implementation of that approach for symmetric matrices, which we call the spectral lanczos decomposition method (SLDM).<\/jats:p><jats:p>We have proved a general convergence estimate, relating SLDM error bounds to those obtained through approximation of the matrix function by a part of its Chebyshev series. Thus, we arrived at effective estimates for matrix functions arising when solving parabolic, hyperbolic and elliptic partial differential equations. We concentrate on the parabolic case, where we obtain estimates that indicate superconvergence of SLDM. For this case a combination of SLDM and splitting methods is also considered and some numerical results are presented.<\/jats:p><jats:p>We implement our general estimates to obtain convergence bounds of Lanczos approximations to eigenvalues in the internal part of the spectrum. Unlike Kaniel\u2010Saad estimates, our estimates are independent of the set of eigenvalues between the required one and the nearest spectrum bound.<\/jats:p><jats:p>We consider an extension of our general estimate to the case of the simple Lanczos method (without reorthogonalization) in finite computer arithmetic which shows that for a moderate dimension of the Krylov subspace the results, proved for the exact arithmetic, are stable up to roundoff.<\/jats:p>","DOI":"10.1002\/nla.1680020303","type":"journal-article","created":{"date-parts":[[2005,11,1]],"date-time":"2005-11-01T19:21:24Z","timestamp":1130872884000},"page":"205-217","source":"Crossref","is-referenced-by-count":85,"title":["Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic"],"prefix":"10.1002","volume":"2","author":[{"given":"Vladimir","family":"Druskin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leonid","family":"Knizhnerman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2005,7,8]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90231-1"},{"key":"e_1_2_1_3_2","unstructured":"V. L.DruskinandL. A.Knizhnerman.Using operational series in orthogonal polynomials for computation of functions of self\u2010adjoint operators and background of Lanczos phenomenon. Dep. at VINITI 02.03. 1987 No. 1535 \u2013 B87. (Russian.)"},{"key":"e_1_2_1_4_2","first-page":"63+","article-title":"A spectral semi\u2010discrete method for the numerical solution of 3D nonstationary problems in electrical prospecting","volume":"8","author":"Druskin V. L.","year":"1988","journal-title":"Izv. Ac. Sci. U.S.S.R., Physics of Solid Earth"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0041-5553(89)80020-5"},{"issue":"7","key":"e_1_2_1_6_2","first-page":"20+","article-title":"Error bounds for the simple Lanczos procedure for computing functions of symmetric matrices and eigenvalues","volume":"31","author":"Druskin V.","year":"1991","journal-title":"U.S.S.R. Comput. Maths. Math. Phys."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(92)90076-A"},{"key":"e_1_2_1_8_2","unstructured":"V.DruskinandL.Knizhnerman.Evaluation for Krylov subspace approximation to internal eigenvalues of large symmetric matrices and bounded self\u2010adjoint operators with continous spectrum. Schlumberger\u2010Doll Research research note 2 September1992."},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01060992"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1137\/0913071"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90285-1"},{"key":"e_1_2_1_12_2","series-title":"Applied Mathematics series, No. 55","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","year":"1964"},{"key":"e_1_2_1_13_2","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443327"},{"key":"e_1_2_1_14_2","unstructured":"V. O.LokutsievskiyandO. V.Lokutsievskiy. Using the Chebyshev parameters for the numerical solution of some evolutionary problems.Inst. Appl. Math. Acad. Sci. USSR Moscow preprint No. 99 1984. (Russian.)"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.51.2238"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1002\/eqe.4290120410"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620240117"},{"key":"e_1_2_1_18_2","unstructured":"C. C.Paige.The computation of eigenvalues and eigenvectors of very large sparse matrices. Ph.D. thesis University of London 1971."},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/18.3.341"},{"key":"e_1_2_1_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90167-6"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1137\/0712047"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.451548"},{"key":"e_1_2_1_23_2","doi-asserted-by":"publisher","DOI":"10.2307\/2006037"},{"key":"e_1_2_1_24_2","volume-title":"The Symmetric Eigenvalue Problem","author":"Parlett B. N.","year":"1980"},{"key":"e_1_2_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90248-7"},{"key":"e_1_2_1_26_2","unstructured":"H. D.Simon.The Lanczos algorithm for solving linear systems. Ph.D. thesis University of California 1982."},{"key":"e_1_2_1_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(84)90025-9"},{"key":"e_1_2_1_28_2","unstructured":"A. V.Sobyanin. The roundoff error analysis and stability in Lanczos\u2010like methods.Dep. Comput. Math. Acad. Sci. USSR Moscow preprint No. 87 1985. (Russian.)"},{"key":"e_1_2_1_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(87)90020-3"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.1680020303","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.1680020303","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T02:39:47Z","timestamp":1694399987000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.1680020303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,5]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,5]]}},"alternative-id":["10.1002\/nla.1680020303"],"URL":"https:\/\/doi.org\/10.1002\/nla.1680020303","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,5]]}}}