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Optimal unit triangular factorization of symplectic matrices

LINEAR ALGEBRA AND ITS APPLICATIONS 2022年 650卷 236-247页

作者： Jin, Pengzhan Lin, Zhangli Xiao, Bo Peking Univ Sch Math Sci Beijing 100871 Peoples R China Yanqi Lake Beijing Inst Math Sci & Applicat Beijing 101408 Peoples R China Tsinghua Univ Yau Math Sci Ctr Beijing 100084 Peoples R China Chinese Acad Sci Acad Math & Syst Sci Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China

We prove that any symplectic matrix can be factored into no more than 5 unit triangular symplectic matrices, moreover, 5 is the optimal number. This result improves the existing triangular factorization of symplectic matrices which gives proof of 9 factors. We also show the corresponding improved conclusions for structured subsets of symplectic matrices. This factorization further provides an unconstrained optimization method on 2d-by-2d real symplectic group (a 2d2 + d dimensional Lie group) with 2d2 + 3d parameters. (C) 2022 Elsevier Inc. All rights reserved.

关键词： Symplectic matrix triangular factorization Unconstrained parametrization

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UNIT triangular factorization OF THE MATRIX SYMPLECTIC GROUP

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2020年第4期41卷 1630-1650页

作者： Jin, Pengzhan Tang, Yifa Zhu, Aiqing Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China

In this work, we prove that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. This structure-preserving factorization of the symplectic matrices immediately reveals two well-known features that (i) the determinant of any symplectic matrix is one and (ii) the matrix symplectic group is path connected, as well as a new feature that (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. Furthermore, this factorization yields effective methods for the unconstrained parametrization of the matrix symplectic group as well as its structured subsets. The unconstrained parametrization enables us to apply faster and more efficient unconstrained optimization algorithms to the problems with symplectic constraints under certain circumstances.

关键词： symplectic symplectic matrix triangular factorization positive definite singular symplectic matrix generators parameterization unconstrained parametrization

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Hyperspectral band selection based on triangular factorization

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JOURNAL OF APPLIED REMOTE SENSING 2017年第2期11卷

作者： Zhang, Wenqiang Li, Xiaorun Zhao, Liaoying Zhejiang Univ Dept Elect Engn Hangzhou Zhejiang Peoples R China Hangzhou Dianzi Univ Dept Comp Sci Hangzhou Zhejiang Peoples R China

We proposed an efficient unsupervised band selection method, maximum ellipsoid volume triangular factorization (MEV-TF), which is based on MEV and TF. MEV band selection regards the bands with the maximum determinant of the covariance matrix as the optimal band collection. By adopting TF, MEV-TF replaces the matrix determinant with scalar multiplication and achieves incremental calculation, which decreases the computational cost significantly. MEV-TF tries to select bands with large information and low correlation. Experimental results on different real data verify the efficiency of the proposed method. (C) 2017 Society of PhotoOptical Instrumentation Engineers (SPIE)

关键词： band selection hyperspectral remote sensing triangular factorization sequential forward search

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Parallel Elimination Based triangular factorization and its Application for Power Flow

Parallel Elimination Based Triangular Factorization and its ...

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Chinese Automation Congress (CAC)

作者： Hu, Xiaojing Guo, Jian Li, Dan Yang, Ning Tang, Weidong Li, Tianyi China Elect Power Res Inst Beijing Peoples R China Northeast China Elect Power Dispatching & Control Shenyang Liaoning Peoples R China Cent China Elect Power Dispatching & Control Ctr Wuhan Hubei Peoples R China Huaqiang Elect Power Automat Engn Co Ltd Harbin Heilongjiang Peoples R China

ISBN: (纸本)9781538635247

Parallelization is the main method to improve efficiency for solving sparse linear equations. This paper summarized the computing dependency relations of power system off/on line analysis application types. Almost all kinds of numerical methods related to power system computation can be attributed to solving Ax=b (A is sparse), and the direct method based on the Gaussian elimination is used widely, which its main process is decomposing A into upper and lower triangular matrices, getting solution through forward backward substitution. The parallel decomposition method called elimination tree based parallelization (ETP) is proposed to reduce time-consuming in triangulation process. ETP method in detail is that stores block matrices according to density of diagonal elements in symbolic factor matrix, and divides independent parallel calculation sub-tasks according to column vectors relations in elimination tree generated from symbolic factor matrix. Take the experimental analysis of power grid Jacobian matrices for example, compare the acceleration effect of above two parallel elimination methods and then apply them to the power flow. Through comparison and analysis, the method proposed in this paper can reduce the convergence time greatly and improve computational performance and practicability.

关键词： sparse linear equations solver triangular factorization elimination tree parallel elimination power flow

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triangular factorization of Strongly Row (Column) Full Rank Linear Equations 10

Triangular Factorization of Strongly Row (Column) Full Rank ...

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10th International Conference on Computational Intelligence and Security CIS 2014

作者： Liu Hongxia Feng Tianxiang Dongguan Polytech Dept Basic Courses Dongguan 523808 Peoples R China

ISBN: (纸本)9781479974344

First, the paper gives the definition of the strongly row (column) full rank matrix, the quasi-upper triangular matrix and the quasi-lower triangular matrix of the m x n matrix. Then the triangular factorization is generalized to the m x n matrix, and some factorization theorems are obtained. By using the triangular factorization, the linear equations with strongly row (column) full rank coefficient matrix are solved procedurally and the corresponding computer algorithm is obtained. Finally, a numerical example is solved by the algorithm.

关键词： triangular factorization full rank matrix algorithm linear equations

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A triangular factorization formula for the Chebyshev-Bezout Matrix

A Triangular Factorization formula for the Chebyshev-Bezout ...

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The Sixth International Conference of Matrices and Operators

作者： Bing-fei Cui Zheng-hong Yang Department of Mathematics China Agricultural University

In the present paper a triangular factorization formula is obtained for the Chebyshev-Bezout matrix,which generalizes the corresponding result for the classical Bezout matrix with respect to the standard power basis.

关键词： Chebyshev polynomials Chebyshev-Bezoutian triangular factorization

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An Improvement to Exact Reanalysis Algorithm for Local Non-topological Structural Modifications

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2020年第4期17卷

作者： Song, Qi Yang, Ren Chen, Pu Beijing Internet Based Engn Co Ltd Beijing 100083 Peoples R China Peking Univ Coll Engn Dept Mech & Engn Sci Beijing 100871 Peoples R China Peking Univ Coll Engn LTCS Beijing 100871 Peoples R China

A new direct reanalysis algorithm for local high-rank structural modifications, based on triangular factorization, has recently been developed. In this work, an improvement is proposed for further reduction of the workload so that the algorithm becomes more efficient and also suitable for low-rank modifications. Compared to the original algorithm, firstly, an alternative formula for updating the triangular factors is derived to avoid repetitive calculations for certain low-rank cases. Secondly, to maximize the efficiency, a combined algorithm is proposed that estimates the workloads of the original and alternative algorithms for each row individually before numerical calculations and selects the one with smaller workload. Numerical experiments show that compared with full factorization, the combined algorithm is significantly more efficient and expected to save up to 75% of execution time in our numerical examples. The new method can be easily implemented and applied to engineering problems, especially to local and step-by-step modification of structures.

关键词： Structural modification reanalysis sparse matrix triangular factorization graph partition

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Shifted Darboux Transformations of the Generalized Jacobi Matrices, I

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Journal of Mathematical Sciences (United States) 2019年第3期242卷 393-412页

作者： Kovalyov, Ivan M. Dragomanov National Pedagogical University Kiev Ukraine

Let ℑ be a monic generalized Jacobi matrix, i.e., a three-diagonal block matrix of a special form. We find conditions for a monic generalized Jacobi matrix ℑ to admit a factorization ℑ = 𝔏𝔘 + αI with 𝔏 and 𝔘 being lower and upper triangular two-diagonal block matrices of special forms. In this case, the shifted parameterless Darboux transformation of ℑ defined by ℑ^(p) = 𝔘𝔏 + αI is shown to be also a monic generalized Jacobi matrix. Analogs of the Christoffel formulas for polynomials of the first and second kinds corresponding to the Darboux transformation ℑ^(p) are found. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

关键词： Darboux transformation indefinite inner product m-function monic generalized Jacobi matrix orthogonal polynomials triangular factorization

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COMPACT SYMMETRIC SPACES, triangular factorization, AND CAYLEY COORDINATES

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PACIFIC JOURNAL OF MATHEMATICS 2011年第1期253卷 57-73页

作者： Habermas, Derek SUNY Coll Potsdam Dept Math Potsdam NY 13676 USA

Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi : U/K -> U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g is an element of phi(U/K) is generic, the corresponding factorization g = ld(g)u is unique, where l is an element of N-, d(g) is an element of H, and u is an element of N+. We present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spaces, and use it to identify representatives of the connected components of the generic part of phi(U/K). This formula calculates a moment map for a torus action on the highest dimensional symplectic leaves of the Evens-Lu Poisson structure on U/K.

关键词： compact symmetric space triangular factorization ldu factorization Bruhat decomposition Cayley map Cayley coordinates symplectic leaves compute computation concrete classical connected component Cartan embedding antidiagonal antitranspose

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DIFFERENTIATION OF MATRIX FUNCTIONALS USING triangular factorization

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MATHEMATICS OF COMPUTATION 2011年第275期80卷 1585-1600页

作者： de Hoog, F. R. Anderssen, R. S. Lukas, M. A. CSIRO Math Informat & Stat Canberra ACT 2601 Australia Murdoch Univ Murdoch WA 6150 Australia

In various applications, it is necessary to differentiate a matrix functional w(A(x)), where A(x) is a matrix depending on a parameter vector x. Usually, the functional itself can be readily computed from a triangular factorization of A(x). This paper develops several methods that also use the triangular factorization to efficiently evaluate the first and second derivatives of the functional. Both the full and sparse matrix situations are considered. There are similarities between these methods and algorithmic differentiation. However, the methodology developed here is explicit, leading to new algorithms. It is shown how the methods apply to several applications where the functional is a log determinant, including spline smoothing, covariance selection and restricted maximum likelihood.

关键词： Differentiation matrix functionals triangular factorization algorithmic differentiation log-cat relationships robust generalized cross validation smoothing splines REML covariance selection

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