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Math. Softw."],"published-print":{"date-parts":[[2016,3]]},"abstract":"<jats:p>\n            Algorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit roundoff. The forward error is computed with the aid of a reference solution, typically computed at high precision. An alternative approach is to use functional identities such as the \u201cround-trip tests\u201d\n            <jats:italic>e<\/jats:italic>\n            <jats:sup>\n              log\n              <jats:italic>A<\/jats:italic>\n            <\/jats:sup>\n            =\n            <jats:italic>A<\/jats:italic>\n            and (\n            <jats:italic>A<\/jats:italic>\n            <jats:sup>\n              1\/\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sup>\n            )\n            <jats:sup>\n              <jats:italic>p<\/jats:italic>\n            <\/jats:sup>\n            =\n            <jats:italic>A<\/jats:italic>\n            , as are currently employed in a SciPy test module. We show how a linearized perturbation analysis for a functional identity allows the determination of a maximum residual consistent with backward stability of the constituent matrix function evaluations. Comparison of this maximum residual with a computed residual provides a necessary test for backward stability. We also show how the actual linearized backward error for these relations can be computed. Our approach makes use of Fr\u00e9chet derivatives and estimates of their norms. Numerical experiments show that the proposed approaches are able both to detect instability and to confirm stability.\n          <\/jats:p>","DOI":"10.1145\/2723157","type":"journal-article","created":{"date-parts":[[2016,2,1]],"date-time":"2016-02-01T20:37:54Z","timestamp":1454359074000},"page":"1-15","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":4,"title":["Testing Matrix Function Algorithms Using Identities"],"prefix":"10.1145","volume":"42","author":[{"given":"Edvin","family":"Deadman","sequence":"first","affiliation":[{"name":"The University of Manchester, Manchester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas J.","family":"Higham","sequence":"additional","affiliation":[{"name":"The University of Manchester, Manchester, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2016,1,29]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/080716426"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/09074721X"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-009-9323-y"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1137\/110852553"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/120885991"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/130920137"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-012-9665-8"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/151271.151272"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479802410815"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/0096-3003(76)90020-5"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894273614"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1137\/10079687X"},{"key":"e_1_2_1_13_1","volume-title":"Van Loan","author":"Golub Gene H.","year":"2013","unstructured":"Gene H. 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Version 1.0. MIMS EPrint 2014.8. Manchester Institute for Mathematical Sciences, The University of Manchester, UK. 19 pages."},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1137\/10081232X"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1137\/130906118"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1137\/130950082"},{"key":"e_1_2_1_24_1","doi-asserted-by":"publisher","DOI":"10.1137\/130945259"},{"key":"e_1_2_1_25_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899356080"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.5555\/19572"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.1137\/120877398"},{"key":"e_1_2_1_28_1","unstructured":"Eric Jones Travis Oliphant Pearu Peterson and others. 2001. SciPy: Open Source Scientific Tools for Python. (2001). http:\/\/www.scipy.org\/."},{"key":"e_1_2_1_29_1","unstructured":"Cleve B. Moler. 2012. 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