{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,5]],"date-time":"2024-01-05T00:02:59Z","timestamp":1704412979937},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2001,6,27]],"date-time":"2001-06-27T00:00:00Z","timestamp":993600000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S upper R\"><mml:semantics><mml:mrow><mml:mi>S<\/mml:mi><mml:mi>R<\/mml:mi><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">SR<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>algorithm is a structure-preserving algorithm for computing the spectrum of symplectic matrices. Any symplectic matrix can be reduced to symplectic butterfly form. A symplectic matrix<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\"><mml:semantics><mml:mi>B<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>in butterfly form is uniquely determined by<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 n minus 1\"><mml:semantics><mml:mrow><mml:mn>4<\/mml:mn><mml:mi>n<\/mml:mi><mml:mo>\u2212<!-- \u2212 --><\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">4n-1<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>parameters. Using these<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"4 n minus 1\"><mml:semantics><mml:mrow><mml:mn>4<\/mml:mn><mml:mi>n<\/mml:mi><mml:mo>\u2212<!-- \u2212 --><\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">4n-1<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>parameters, we show how one step of the symplectic<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper S upper R\"><mml:semantics><mml:mrow><mml:mi>S<\/mml:mi><mml:mi>R<\/mml:mi><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">SR<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>algorithm for<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper B\"><mml:semantics><mml:mi>B<\/mml:mi><mml:annotation encoding=\"application\/x-tex\">B<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>can be carried out in<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis n right-parenthesis\"><mml:semantics><mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">\\mathcal {O}(n)<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>arithmetic operations compared to<inline-formula content-type=\"math\/mathml\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper O left-parenthesis n cubed right-parenthesis\"><mml:semantics><mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>n<\/mml:mi><mml:mn>3<\/mml:mn><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:mrow><mml:annotation encoding=\"application\/x-tex\">\\mathcal {O}(n^3)<\/mml:annotation><\/mml:semantics><\/mml:math><\/inline-formula>arithmetic operations when working on the actual symplectic matrix. Moreover, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors when working with a symplectic (butterfly) matrix, will be forced by working just with the parameters.<\/p>","DOI":"10.1090\/s0025-5718-00-01265-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:14:28Z","timestamp":1027721668000},"page":"1515-1541","source":"Crossref","is-referenced-by-count":7,"title":["The parameterized \ud835\udc46\ud835\udc45 algorithm for symplectic (butterfly) matrices"],"prefix":"10.1090","volume":"70","author":[{"given":"H.","family":"Fa\u00dfbender","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,6,27]]},"reference":[{"key":"1","unstructured":"G. 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