{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:28:47Z","timestamp":1787336927649,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>An important step in the solution of a matrix nearness problem that arises in certain machine learning applications is finding the zero of $f(\\alpha) = \\bm{z}^T \\exp(\\log X + \\alpha\\bm{z}\\bm{z}^T)\\bm{z} - b$. The matrix valued exponential and logarithm in $f(\\alpha)$ arises from the use of the von Neumann matrix divergence $\\operatorname{tr}(X \\log X - X \\log Y - X + Y)$ to measure the nearness between the positive definite matrices $X$ and $Y$. A key step of an iterative algorithm used to solve the underlying matrix nearness problem requires the zero of $f(\\alpha)$ to be repeatedly computed. In this paper we propose zero-finding algorithms that gain their advantage by exploiting the special structure of the objective function. We show how to efficiently compute the derivative of $f$, thereby allowing the use of Newton-type methods. In numerical experiments we establish the advantage of our algorithms.<\/jats:p>","DOI":"10.1137\/11082498x","type":"journal-article","created":{"date-parts":[[2012,11,20]],"date-time":"2012-11-20T14:45:57Z","timestamp":1353422757000},"page":"1237-1249","source":"Crossref","is-referenced-by-count":0,"title":["On a Zero-Finding Problem Involving the Matrix Exponential"],"prefix":"10.1137","volume":"33","author":[{"given":"M\u00e1ty\u00e1s A.","family":"Sustik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Inderjit S.","family":"Dhillon","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,11,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2007.06.002"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen,\n                      LAPACK Users' Guide\n                      , 3rd ed., SIAM, Philadelphia, 1999.","DOI":"10.1137\/1.9780898719604"},{"key":"atypb3","unstructured":"R. Bhatia,\n                      Positive Definite Matrices\n                      , Princeton University Press, Princeton, NJ, 2006."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1145\/567806.567807"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(67)90040-7"},{"key":"atypb6","unstructured":"R. P. Brent,\n                      Algorithms for Minimization without Derivatives\n                      , Prentice-Hall, Englewood Cliffs, NJ, 1973."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396012"},{"key":"atypb8","unstructured":"Y. Censor and S. 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