{"title":"A simple division-free algorithm for computing Pfaffians","authors":"Adam J. Przeździecki","doi":"10.1016/j.ipl.2024.106550","DOIUrl":null,"url":null,"abstract":"<div><div>We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.</div><div>Its complexity, for a <span><math><mn>2</mn><mi>n</mi><mo>×</mo><mn>2</mn><mi>n</mi></math></span> matrix, is <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of matrix multiplication. In case of a sparse matrix, <span><math><mi>M</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the cost of the dense-sparse matrix multiplication.</div><div>The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.</div></div>","PeriodicalId":56290,"journal":{"name":"Information Processing Letters","volume":"189 ","pages":"Article 106550"},"PeriodicalIF":0.7000,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Information Processing Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0020019024000802","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
We present a very simple algorithm for computing Pfaffians which uses no division operations. Essentially, it amounts to iterating matrix multiplication and truncation.
Its complexity, for a matrix, is , where is the cost of matrix multiplication. In case of a sparse matrix, is the cost of the dense-sparse matrix multiplication.
The algorithm is an adaptation of the Bird algorithm for determinants. We show how to extract, with practically no additional work, the characteristic polynomial and the Pfaffian characteristic polynomial from these algorithms.
期刊介绍：
Information Processing Letters invites submission of original research articles that focus on fundamental aspects of information processing and computing. This naturally includes work in the broadly understood field of theoretical computer science; although papers in all areas of scientific inquiry will be given consideration, provided that they describe research contributions credibly motivated by applications to computing and involve rigorous methodology. High quality experimental papers that address topics of sufficiently broad interest may also be considered.
Since its inception in 1971, Information Processing Letters has served as a forum for timely dissemination of short, concise and focused research contributions. Continuing with this tradition, and to expedite the reviewing process, manuscripts are generally limited in length to nine pages when they appear in print.
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