{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:34:02Z","timestamp":1787337242490,"version":"build-2736575974"},"reference-count":23,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2001,1]]},"abstract":"<jats:p>\n                    Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using a Krylov subspace method, such as the conjugate gradient (CG) algorithm. Computing the estimation error variances is a more intricate task. It is difficult because the error variances are the diagonal elements of a matrix expression involving the inverse of a given matrix. This paper presents a method for using the conjugate search directions generated by the CG algorithm to obtain a convergent approximation to the estimation error variances. The algorithm for computing the error variances falls out naturally from a new estimation-theoretic interpretation of the CG algorithm. This paper discusses this interpretation and convergence issues and presents numerical examples. The examples include a 10\n                    <jats:sup>5<\/jats:sup>\n                    -dimensional estimation problem from oceanography.\n                  <\/jats:p>","DOI":"10.1137\/s1064827599357292","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1840-1864","source":"Crossref","is-referenced-by-count":17,"title":["Krylov Subspace Estimation"],"prefix":"10.1137","volume":"22","author":[{"given":"Michael K.","family":"Schneider","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alan S.","family":"Willsky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","unstructured":"A. F. Bennett,\n                      Inverse Methods and Data Assimilation\n                      , Lecture notes from the summer school at the College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR, 1999."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01029793"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01029698"},{"key":"R4","unstructured":"A. da Silva and J. Guo,\n                      Documentation of the Physical\u2010space Statistical Analysis System (PSAS) Part I: The Conjugate Gradient Solver Version PSAS\u20101.00\n                      , DAO Note 96\u201002, Data Assimilation Office, Goddard Laboratory for Atmospheres, NASA, 1996;"},{"key":"R4","unstructured":"also available online from ftp:\/\/dao.gsfc.nasa.gov\/pub\/office_notes\/on9602.ps. 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Luenberger,\n                      Optimization by Vector Space Methods\n                      , John Wiley and Sons, New York, 1969."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1030563977"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1145\/355984.355989"},{"key":"R14","unstructured":"B. N. Parlett,\n                      The Symmetric Eigenvalue Problem\n                      , Prentice\u2010Hall, Englewood Cliffs, NJ, 1980."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1979-0514820-3"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/0717059"},{"key":"R17","volume-title":"Numerical methods for large eigenvalue problems","author":"Saad Youcef","year":"1992"},{"key":"R18","unstructured":"M. K. Schneider,\n                      Krylov Subspace Estimation\n                      , Ph.D. thesis, MIT, Cambridge, MA, 2001."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1109\/78.286960"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479890183848"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1109\/78.277846"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1109\/82.318942"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827599357292","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:51:54Z","timestamp":1787334714000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827599357292"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,1]]},"references-count":23,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2001,1]]}},"alternative-id":["10.1137\/S1064827599357292"],"URL":"https:\/\/doi.org\/10.1137\/s1064827599357292","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2001,1]]}}}