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Revisiting the Adjoint matrix for FPGA Calculating the triangular matrix Inversion

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2021年第6期68卷 2127-2131页

作者： Yan, Di Wang, Wei-Xing Zuo, Lei Zhang, Xiao-Wei Changan Univ Sch Informat & Commun Xian 710064 Peoples R China Xidian Univ Natl Lab Radar Signal Proc Xian 710071 Peoples R China Xian Univ Posts & Telecommun Sch Commun & Informat Engn Xian 710121 Peoples R China

Due to the robustness, the matrix inversion methods based on matrix factorization are often adopted in engineering while the triangular matrix inversion is one key part of those methods. This brief reviews the adjoint matrix and presents a novel scheme for field-programmable gate arrays (FPGAs) calculating the triangular matrix inversion. Employing the more characteristics of both the triangular matrix and its inversion, the proposed diagonal-wise algorithm for the triangular matrix inversion has the high parallelism and extensibility in the hardware implementation and is suitable for the different matrix triangular factorization (QR, LDL, Cholesky and LU). Meanwhile, the recursive diagonal-wise algorithm is designed for the large scale triangular matrices. Compared with the traditional row-/column-wise methods, our algorithm has a good performance at the low computation load. Finally, the experiments are conducted on one Xilinx Virtex-7 FPGA to illustrate the performances of the four different methods.

关键词： matrix decomposition Field programmable gate arrays Mathematical model Signal processing algorithms Hardware Adjoint matrix field-programmable gate array (FPGA) matrix inversion triangular matrix

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triangular matrix Formulation for Power Flow Analysis in Radial DC Resistive Grids With CPLs

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2020年第6期67卷 1094-1098页

作者： Montoya, Oscar Danilo Grisales-Norena, L. F. Gil-Gonzalez, Walter Univ Tecnol Bolivar Program Elect & Elect Engn Cartagena 131001 Colombia Inst Tecnol Metropolitano Dept Mechatron & Electromech Engn Medellin 050012 Colombia Univ Tecnol Pereira Dept Elect Power Engn Pereira 660003 Colombia

This brief briefly addresses the problem of power flow solution for direct-current (dc) networks with radial configuration and constant power loads (CPLs). It proposes a novel iterative method based on the upper triangular relationship between nodal and branch currents, it also uses a primitive impedance matrix. The main advantage of this method lies in the possibility of avoiding inversions of non-diagonal matrices, which allows its convergence to be improved in terms of the number of iterations and processing times required in comparison to classical admittance-based methods. Three different radial dc resistive networks composed by 21, 33, and 69 nodes are employed to validate the effectiveness of the proposed power flow solution method. For comparison purposes, the Newton-Raphson method, and also successive approximations and Taylor-based approaches are implemented. All simulations have performed in MATLAB software.

关键词： Voltage control Impedance Convergence Load modeling Iterative methods Power demand Load flow analysis Direct-current radial distribution networks numerical methods power flow method primitive impedance matrix triangular matrix

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A triangular-matrix-Based Spectral Encoding Method for Broadband Filtering and Reconstruction-Based Spectral Measurement

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SENSORS 2024年第4期24卷 1215页

作者： Yue, Pinliang Wang, Xiaoxu Chinese Acad Sci Changchun Inst Opt Fine Mech & Phys Changchun 130033 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China

Broadband filtering and reconstruction-based spectral measurement represent a hot technical route for miniaturized spectral measurement;the measurement encoding scheme has a great effect on the spectral reconstruction fidelity. The existing spectral encoding schemes are usually complex and hard to implement;thus, the applications are severely limited. Considering this, here, a simple spectral encoding method based on a triangular matrix is designed. The condition number of the proposed spectral encoding system is estimated and demonstrated to be relatively low theoretically;then, verification experiments are carried out, and the results show that the proposed encoding can work well under precise or unprecise encoding and measurement conditions;therefore, the proposed scheme is demonstrated to be an effective trade-off of the spectral encoding efficiency and implementation cost.

关键词： spectral encoding spectral reconstruction triangular matrix spectral imaging miniaturized spectrometer

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CONDITIONING OF matrix FUNCTIONS AT QUASI-triangular MATRICES

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2024年第2期45卷 954-966页

作者： Al-Mohy, Awad H. King Khalid Univ Dept Math Abha Saudi Arabia

The area of matrix functions has received growing interest for a long period of time due to their growing applications. Having a numerical algorithm for a matrix function, the ideal situation is to have an estimate or bound for the error returned alongside the solution. Condition numbers serve this purpose;they measure the first -order sensitivity of matrix functions to perturbations in the input data. We have observed that the existing unstructured condition number leads most of the time to inferior bounds of relative forward errors for some matrix functions at triangular and quasi -triangular matrices. We propose a condition number of matrix functions exploiting the structure of triangular and quasi -triangular matrices. We then adapt an existing algorithm for exact computation of the unstructured condition number to an algorithm for exact evaluation of the structured condition number. Although these algorithms are direct rather than iterative and useful for testing the numerical stability of numerical algorithms, they are less practical for relatively large problems. Therefore, we use an implicit power method approach to estimate the structured condition number. Our numerical experiments show that the structured condition number captures the behavior of the numerical algorithms and provides sharp bounds for the relative forward errors. In addition, the experiment indicates that the power method algorithm is reliable to estimate the structured condition number.

关键词： structured condition number Frechet derivative matrix function triangular matrix quasi-triangular matrix power method Kronecker form

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How to construct a triangular matrix of a given order

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LINEAR & MULTILINEAR ALGEBRA 2014年第1期62卷 28-38页

作者： Slowik, Roksana Silesian Tech Univ Inst Math Gliwice Poland

In this article we consider the elements in upper triangular finite and infinite dimensional matrix groups over fields, whose order is equal to k (k is an element of N). For the case when the characteristic of K does not divide k, we give a description of such elements. Next using this criterion, we show how to find the number of these elements in finite dimensional groups when K is a finite field.

关键词： order of a group element triangular matrix infinite triangular matrix 20H20 15A99

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n-dimensional hyperchaotic discrete map with desired positive Lyapunov exponents and application to UART secure communication

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NONLINEAR DYNAMICS 2025年第8期113卷 9061-9079页

作者： Xu, Bo Tang, Zhongmin Ye, Xiaoxuan Chen, Kai Gou, Xuan Zhao, Jia Univ Elect Sci & Technol China Sch Automat Engn Chengdu 611731 Peoples R China Chengdu Med Coll Sch Big Hlth & Intelligent Engn Chengdu 610500 Peoples R China

When applying chaotic sequences in engineering applications, it is commonly imperative for chaotic systems to possess continuous hyperchaotic intervals and adjustable dimensions. This study proposes a n-dimensional hyperchaotic discrete map (nD-HDM) utilizing triangular matrices and mode operations. The diagonal elements and mode values can effectively adjust Lyapunov exponents(LEs) and amplitude intervals of the nD-HDM. To illustrate the effectiveness of nD-HDM, two typical 4D-HDMs are examined by numerical computations. The findings indicate that both 4D-HDMs possess four positive LEs, continuous hyperchaotic intervals, controlled amplitude ranges, and controllable distribution characteristics. The generated state variables exhibit exceptional stochastic characteristics and successfully pass the TestU01 test. To further improve the attractor fractal structure of nD-HDM, strategies for controlling state matrix-based heterogeneous attractors and graph transformation-based homogeneous attractors are given. The offset boosting, amplitude, and rotational control of the attractor can also be achieved. Finally, we validate 4D-HDMs on an ARM-based hardware platform and design a pseudorandom number-based universal asynchronous receiver/transmitter(UART) physical layer secure communication.

关键词： n-dimensional hyperchaotic discrete map Modal operation triangular matrix Attractor hybrid control Probability mass function Hardware implementation Secure communication

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Groups generated by matrices of finite order

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SEMIGROUP FORUM 2025年 1-15页

作者： Bien, Mai Hoang Huy, Trinh Quoc Sy, Nguyen Anh Thinh, Nguyen Phu Truong, Le Quang Univ Sci Fac Math & Comp Sci Ho Chi Minh City Vietnam Vietnam Natl Univ Ho Chi Minh City Vietnam Ho Chi Minh City Univ Educ Dept Math Informat Ho Chi Minh City Vietnam

Let R be an associative ring with identity, 1 <= kappa <= N-0, and n, k be two positive integers. Denote by M-kappa(R) the ring of all kappa x kappa square matrices over R if kappa < N-0, and the ring of all column-finite infinite matrices over R if kappa = N-0. The ring of all upper triangular matrices in M-kappa(R) is denoted by T-kappa(R). In this paper, we first establish the necessary and sufficient conditions for a matrix to be expressible as a product of matrices whose orders are divisors of k in T-kappa(R). As an application, we then describe the subgroup of the general linear group of degree n over a field generated by matrices whose orders are divisors of k. Naturally, we also extend this study to the subgroup of the Vershik-Kerov group, that is, the group of infinite matrices with only finitely many nonzero entries below the main diagonal.

关键词： matrix of finite order Generators of matrix groups Product of matrices triangular matrix Infinite matrix

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A certain decomposition of infinite invertible matrices over division algebras

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LINEAR & MULTILINEAR ALGEBRA 2023年第12期71卷 1948-1956页

作者： Mai Hoang Bien Truong Huu Dung Nguyen Thi Thai Ha Univ Sci Fac Math & Comp Sci Ho Chi Minh City Vietnam Vietnam Natl Univ Ho Chi Minh City Vietnam Dong Nai Univ Dong Nai Vietnam Univ Transport & Commun Campus Ho Chi Minh City Ho Chi Minh City Vietnam

Let D be an infinite division ring with centre F and T(infinity, D) the group of infinite upper triangular invertible matrices indexed by N x N over D. In this paper, we first show that if D is finite dimensional over F, then every element in T(infinity, D) whose diagonal entries are commutators of D \ triangular matrix is a commutator of an infinite diagonal matrix and another infinite upper triangular matrix. Some applications are also shown. For example, if dim(F) D <= 4, then every element in the commutator subgroup [GL(VK)(infinity, D), GL(VK)(infinity, D)] of the Vershik-Kerov group GL(VK)(infinity, D) is a commutator.

关键词： Infinite matrix matrix decomposition commutator triangular matrix

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Tails of bivariate stochastic recurrence equation with triangular matrices

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2022年 150卷 147-191页

作者： Damek, Ewa Matsui, Muneya Univ Wroclaw Inst Math Pl Grunwaldzki 2 PL-50384 Wroclaw Poland Nanzan Univ Dept Business Adm 18 Yamazato Cho Showa Ku Nagoya 4668673 Japan

We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions W = (W-1, W-2). Recently it has been observed that W-1, W-2 may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical "Kesten-Goldie" and "Grey " conditions, we completely characterize tail behavior of W-1, W-2. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equations. (C) 2022 Elsevier B.V. All rights reserved.

关键词： Stochastic recurrence equation Regular variation Kesten's theorem Autoregressive models triangular matrix

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matrix computations with the Omega calculus

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LINEAR & MULTILINEAR ALGEBRA 2022年第20期70卷 5075-5106页

作者： Neto, Antonio Francisco Univ Fed Ouro Preto Depro Escola Minas Ouro Preto Brazil

In this work, we explore an extension of the Omega calculus in the context of matrix analysis introduced recently by Neto [matrix analysis and Omega calculus. SIAM Rev. 2020;62(1):264- 280]. We obtain Omega representations of analytic functions of three important classes of matrices: companion, tridiagonal, and triangular. Our representation recovers the main results of Chen and Louck [The combinatorial power of the companion matrix. Linear Algebra Appl. 1996;232:261-278] on the powers of the companion matrix. Furthermore, we generalize previous work on the powers of tridiagonal matrices due to Gutierrez- Gutierrez in [Powers of tridiagonal matrices with constant diagonals. Appl Math Comput. 2008;206(2):885-891], Oteles and Akbulak [Positive integer powers of certain complex tridiagonal matrices. Appl Math Comput. 2013;219(21):10448-10455], and triangular matrices following Shur [A simple closed form for triangular matrix powers. Electron J Linear Algebra. 2011;22:1000- 1003].

关键词： Functions of matrices companion matrix tridiagonal matrix triangular matrix omega calculus

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