{"title":"关于四元数矩阵的秩揭示 QR 因式分解","authors":"Qiaohua Liu, Chuge Li","doi":"10.1002/nla.2585","DOIUrl":null,"url":null,"abstract":"This work develops theories and algorithms for computing rank‐revealing QR factorizations (qRRQR) of quaternion matrices. First, by introducing a novel quaternion determinant, a quasi‐Cramer's rule is established to investigate the existence theory of the qRRQR factorization of an quaternion matrix . The proposed theory provides a systematic approach for selecting linearly independent columns of in order to ensure that the resulting diagonal blocks , of the <jats:italic>R</jats:italic>‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of in each iteration by a factor of at least (), a greedy algorithm with cyclic block pivoting strategy is derived to implement the qRRQR factorization. This algorithm costs about real floating‐point operations, which is as nearly efficient as quaternion QR with column pivoting for most problems. It also shows the effectiveness and reliability, particularly when dealing with a class of quaternion Kahan matrices. Furthermore, two improved greedy algorithms are proposed. Experiments evaluations are conducted on synthetic data as well as color image compression and quaternion signals denoising. The experimental results validate the usefulness and efficiency of our improved algorithm across various scenarios.","PeriodicalId":49731,"journal":{"name":"Numerical Linear Algebra with Applications","volume":"44 1","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On rank‐revealing QR factorizations of quaternion matrices\",\"authors\":\"Qiaohua Liu, Chuge Li\",\"doi\":\"10.1002/nla.2585\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This work develops theories and algorithms for computing rank‐revealing QR factorizations (qRRQR) of quaternion matrices. First, by introducing a novel quaternion determinant, a quasi‐Cramer's rule is established to investigate the existence theory of the qRRQR factorization of an quaternion matrix . The proposed theory provides a systematic approach for selecting linearly independent columns of in order to ensure that the resulting diagonal blocks , of the <jats:italic>R</jats:italic>‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of in each iteration by a factor of at least (), a greedy algorithm with cyclic block pivoting strategy is derived to implement the qRRQR factorization. This algorithm costs about real floating‐point operations, which is as nearly efficient as quaternion QR with column pivoting for most problems. It also shows the effectiveness and reliability, particularly when dealing with a class of quaternion Kahan matrices. Furthermore, two improved greedy algorithms are proposed. Experiments evaluations are conducted on synthetic data as well as color image compression and quaternion signals denoising. The experimental results validate the usefulness and efficiency of our improved algorithm across various scenarios.\",\"PeriodicalId\":49731,\"journal\":{\"name\":\"Numerical Linear Algebra with Applications\",\"volume\":\"44 1\",\"pages\":\"\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-08-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerical Linear Algebra with Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/nla.2585\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerical Linear Algebra with Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/nla.2585","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
这项研究开发了计算四元数矩阵的秩揭示 QR 因式分解(qRRQR)的理论和算法。首先,通过引入一个新颖的四元数行列式,建立了一个准克拉默规则来研究四元数矩阵 qRRQR 因式分解的存在性理论。所提出的理论提供了一种选择四元矩阵线性独立列的系统方法,以确保 R 因子的对角块 , 在谱规范和弗罗贝尼斯规范中都具有秩揭示特性。其次,通过在每次迭代中将四元数行列式至少增加()的因子,推导出一种具有循环块支点策略的贪婪算法,以实现 qRRQR 因式分解。该算法的浮点运算成本约为实数,对于大多数问题来说,其效率与采用列支点的四元数 QR 算法相当。它还显示了有效性和可靠性,特别是在处理一类四元数 Kahan 矩阵时。此外,还提出了两种改进的贪心算法。对合成数据以及彩色图像压缩和四元数信号去噪进行了实验评估。实验结果验证了我们的改进算法在各种场景下的实用性和效率。
On rank‐revealing QR factorizations of quaternion matrices
This work develops theories and algorithms for computing rank‐revealing QR factorizations (qRRQR) of quaternion matrices. First, by introducing a novel quaternion determinant, a quasi‐Cramer's rule is established to investigate the existence theory of the qRRQR factorization of an quaternion matrix . The proposed theory provides a systematic approach for selecting linearly independent columns of in order to ensure that the resulting diagonal blocks , of the R‐factor possess rank revealing properties in both spectral and Frobenius norms. Secondly, by increasing the quaternion determinants of in each iteration by a factor of at least (), a greedy algorithm with cyclic block pivoting strategy is derived to implement the qRRQR factorization. This algorithm costs about real floating‐point operations, which is as nearly efficient as quaternion QR with column pivoting for most problems. It also shows the effectiveness and reliability, particularly when dealing with a class of quaternion Kahan matrices. Furthermore, two improved greedy algorithms are proposed. Experiments evaluations are conducted on synthetic data as well as color image compression and quaternion signals denoising. The experimental results validate the usefulness and efficiency of our improved algorithm across various scenarios.
期刊介绍:
Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review.
Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects.
Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.