{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T14:01:11Z","timestamp":1754488871461},"reference-count":43,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1066471"],"award-info":[{"award-number":["1066471"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,12,1]]},"abstract":"Abstract<\/jats:title>\n In this article, we consider the general problem of checking the correctness of matrix multiplication.\nGiven three \n \n \n \n n<\/m:mi>\n \u00d7<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n n\\times n<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> matrices \ud835\udc34, \ud835\udc35 and \ud835\udc36, the goal is to verify that \n \n \n \n \n A<\/m:mi>\n \u00d7<\/m:mo>\n B<\/m:mi>\n <\/m:mrow>\n =<\/m:mo>\n C<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n A\\times B=C<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> without carrying out the computationally costly operations of matrix multiplication and comparing the product \n \n \n \n A<\/m:mi>\n \u00d7<\/m:mo>\n B<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n A\\times B<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \ud835\udc36, term by term.\nThis is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors.\nHere we extend Freivalds\u2019 algorithm to a Gaussian Variant of Freivalds\u2019 Algorithm (GVFA) by projecting the product \n \n \n \n A<\/m:mi>\n \u00d7<\/m:mo>\n B<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n A\\times B<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> as well as \ud835\udc36 onto a Gaussian random vector and then comparing the resulting vectors.\nThe computational complexity of GVFA is consistent with that of Freivalds\u2019 algorithm, which is \n \n \n \n O<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n n<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n O(n^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nHowever, unlike Freivalds\u2019 algorithm, whose probability of a false positive is \n \n \n \n 2<\/m:mn>\n \n -<\/m:mo>\n k<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n <\/m:math>\n \n 2^{-k}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \ud835\udc58 is the number of iterations, our theoretical analysis shows that, when \n \n \n \n \n A<\/m:mi>\n \u00d7<\/m:mo>\n B<\/m:mi>\n <\/m:mrow>\n \u2260<\/m:mo>\n C<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n \n A\\times B\\neq C<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic.\nWhen we introduce round-off error and floating-point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives.\nMoreover, by iterating GVFA \ud835\udc58 times, the probability of a false positive decreases as \n \n \n \n p<\/m:mi>\n k<\/m:mi>\n <\/m:msup>\n <\/m:math>\n \n p^{k}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \ud835\udc5d is a very small value depending on the nature of the fault on the result matrix and the arithmetic system\u2019s floating-point precision.\nUnlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA.\nThus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures.\nIn particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations.<\/jats:p>","DOI":"10.1515\/mcma-2020-2076","type":"journal-article","created":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T21:45:29Z","timestamp":1602711929000},"page":"273-284","source":"Crossref","is-referenced-by-count":3,"title":["Gaussian variant of Freivalds\u2019 algorithm for efficient and reliable matrix product verification"],"prefix":"10.1515","volume":"26","author":[{"given":"Hao","family":"Ji","sequence":"first","affiliation":[{"name":"Department of Computer Science , California State Polytechnic University Pomona , Pomona , CA 91768 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Mascagni","sequence":"additional","affiliation":[{"name":"Department of Computer Science , Florida State University , Tallahassee , FL 32306-4530; and Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards & Technology, ITL, Gaithersburg, MD 20899-8910 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yaohang","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Computer Science , Old Dominion University , Norfolk , VA 23529 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2020,10,8]]},"reference":[{"key":"2023040102281283181_j_mcma-2020-2076_ref_001_w2aab3b7e1030b1b6b1ab2b1b1Aa","unstructured":"N. Alon, O. Goldreich, J. H\u00e5 stad and R. Peralta,\nSimple constructions of almost \ud835\udc58-wise independent random variables,\n31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis 1990),\nIEEE Press, Los Alamitos (1990), 544\u2013553."},{"key":"2023040102281283181_j_mcma-2020-2076_ref_002_w2aab3b7e1030b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"P. Banerjee and J. A. Abraham,\nBounds on algorithm-based fault tolerance in multiple processor systems,\nIEEE Trans. Comput. 100 (1986), no. 4, 296\u2013306.","DOI":"10.1109\/TC.1986.1676762"},{"key":"2023040102281283181_j_mcma-2020-2076_ref_003_w2aab3b7e1030b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"P. Banerjee, J. T. Rahmeh, C. Stunkel, V. S. Nair, K. Roy, V. Balasubramanian and J. A. Abraham,\nAlgorithm-based fault tolerance on a hypercube multiprocessor,\nIEEE Trans. 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