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A new method for consistency test in fuzzy AHP

JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2013年第2期25卷 457-461页

作者： Mahmoudzadeh, Morteza Bafandeh, Ali R. Islamic Azad Univ East Azarbaijan Sci & Res Branch Dept Management Tabriz Iran Islamic Azad Univ Tabriz Branch Dept Management Tabriz Iran

This paper presents that the consistency test with consideration of a tolerance deviation in fuzzy AHP proposed by L. C. Leung and D. Cao (2000) is not efficient and has some errors, hence a new method of fuzzy consistency test by direct fuzzification of (Quick Response) qr algorithm - which is one of numerical methods for calculating eigenvalues of an arbitrary matrix - has been proposed.

关键词： Fuzzy AHP consistency test qr algorithm genetic algorithm

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FAST COMPUTATION OF THE ZEROS OF A POLYNOMIAL VIA FACTORIZATION OF THE COMPANION MATRIX

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第1期35卷 A255-A269页

作者： Aurentz, Jared L. Vandebril, Raf Watkins, David S. Washington State Univ Dept Math Pullman WA 99164 USA Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium

A new fast algorithm for computing the zeros of a polynomial in O(n(2)) time using O(n) memory is developed. The eigenvalues of the Frobenius companion matrix are computed by applying a nonunitary analogue of Francis's implicitly shifted qr algorithm to a factored form of the matrix. The algorithm achieves high speed and low memory use by preserving the factored form. It also provides a residual and an error estimate for each root. Numerical tests confirm the high speed of the algorithm.

关键词： polynomial root companion matrix qr algorithm

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Zero-preserving iso-spectral flows based on parallel sums

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LINEAR ALGEBRA AND ITS APPLICATIONS 2007年第1期421卷 69-84页

作者： Driessel, Kenneth R. Gerisch, Alf Colorado State Univ Dept Math Ft Collins CO 80523 USA Univ Halle Wittenberg Fachbereich Math & Informat D-06099 Halle Saale Germany

Driessel [K.R. Driessel, Computing canonical forms using flows, Linear Algebra Appl. 379 (2004) 353379] introduced the notion of quasi-projection onto the range of a linear transformation from one inner product space into another inner product space. Here we introduce the notion of quasi-projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space. As an application, we design a new family of iso-spectral flows on the space of symmetric matrices that preserves zero patterns. We discuss the equilibrium points of these flows. We conjecture that these flows generically converge to diagonal matrices. We perform some numerical experiments with these flows which support this conjecture. We also compare our zero-preserving flows with the Toda flow. (c) 2006 Elsevier Inc. All rights reserved.

关键词： iso-spectral flow group action orbit eigenvalues sparse matrix dynamical system ordinary differential equation vector field Toda flow double bracket flow qr algorithm differential geometry projection quasi-projection parallel sum harmonic mean

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A numerical method for computing an SVD-like decomposition

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第4期26卷 1058-1082页

作者： Xu, HG Univ Kansas Dept Math Lawrence KS 66045 USA

We present a numerical method for computing the SVD-like decomposition B = QDS(-1), where Q is orthogonal, S is symplectic, and D is a permuted diagonal matrix. The method can be applied directly to compute the canonical form of the Hamiltonian matrices of the form JB(T)B, where J = [(-I 0) (0 I)]. It can also be applied to solve the related application problems such as the gyroscopic systems and linear Hamiltonian systems. Error analysis and numerical examples show that the eigenvalues of JB(T)B computed by this method are more accurate than those computed by the methods working on the explicit product JB(T)B or BJB(T)

关键词： skew-symmetric matrix Hamiltonian matrix symplectic matrix orthogonal symplectic matrix eigenvalue problem SVD SVD-like decomposition Schur form Jordan canonical form qr algorithm Jacobi algorithm

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Asymmetric healthcare biometric image encryption and watermarking technique using QZ synthesis and qrD algorithm

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MULTIMEDIA TOOLS AND APPLICATIONS 2024年第31期83卷 76223-76245页

作者： Khurana, Mehak Rakheja, Pankaj NorthCap Univ Dept CSE Gurugram 122017 India

The research paper introduces an advanced asymmetric biometric image encryption technique that integrates Quasi-Zernike synthesis (QZS), qr algorithm, and hybrid transforms for enhanced security and integrity. The encryption process incorporates qr decomposition, generating an orthogonal matrix Q that adds obfuscation and randomness to the image representation. qr is also resilient to attacks that exploit vulnerabilities in PTFT-based cryptosystems. QZS further improves security by capturing spatial information and complex features. The use of the compressed sparse row (CSR) format enables efficient storage and secure transmission. This integration provides a robust solution for confidential, secure image communication, applicable in military, surveillance, and sensitive data transfer scenarios. Moreover, watermarking ensures protection from unauthorized access, strengthening security measures. Experimental evaluations demonstrate the proposed algorithm's robustness against common attacks, including noise addition, compression, and filtering. The results showcase high-quality image reconstruction, negligible information loss, affirming the scheme's effectiveness in achieving secure image encryption.

关键词： Asymmetric Encryption Quasi-Zernike Synthesis (QZS) qr algorithm Hybrid Transforms Watermarking

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Ortho-diffusion decompositions of graph-based representation of images

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PATTERN RECOGNITION 2015年第12期48卷 4097-4115页

作者： Gudivada, Sravan Bors, Adrian G. Univ York Dept Comp Sci York YO10 5GH N Yorkshire England

This paper proposes a new feature representation methodology for graph-based data. Initially, random walks on matrices of pairwise data similarities are considered. A diffusion process is embedded into orthonormal decompositions of such matrices at various scales while enabling a data reduction mechanism as well. At each scale, the qr orthonormal decomposition algorithm, alternating with diffusions and data reduction stages is applied recursively on the given graph-based data representations. The proposed methodology is used in extracting complex feature representations from images, which are then used for image matching and in face recognition. In the face recognition application, both global appearance models and semantic representations of biometric features are considered. Both the correlation and the covariance of images of human faces are considered for the training stage when using global appearance models. The proposed data representation is shown to be robust in face recognition applications, when face images are represented in low resolution and when they are corrupted by noise. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Ortho-diffusion decompositions qr algorithm Face recognition Optical flow estimation

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High-Performance of Eigenvalue Decomposition on FPGA for the DOA Estimation

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2023年第5期72卷 5782-5797页

作者： Zhang, Xiao-Wei Yan, Di Zuo, Lei Li, Ming Guo, Jian-Xin Xidian Univ Natl Lab Radar Signal Proc Xian 710126 Peoples R China Xijing Univ Sch Informat Engn Xian 710123 Peoples R China Changan Univ Sch Informat Engn Xian 710064 Peoples R China

For the direction of arrival (DOA) in array signal processing, eigenvalue decomposition (EVD) is one key issue in hardware implementation of the multiple signal classification (MUSIC) algorithm. Therefore, we introduce the look-ahead simplified one-sided Jacobi's method to efficiently decompose those symmetric matrices in this article and prove that the new method has the best orthogonality of eigenvector and locates eigenvectors closest to the true solution in theory. Both the numerical performance and real-time are important in engineering, so we present the novel flexible hardware architecture in single floating point arithmetic for EVD on field-programmable gate arrays (FPGAs). Finally, the simulated and raw data are used to investigate the performance of some different approaches in the context of both the EVD and MUSIC algorithm. The experimental results show that our proposed method has the best performance.

关键词： Matrix decomposition Jacobian matrices Field programmable gate arrays Eigenvalues and eigenfunctions Hardware Signal processing algorithms Multiple signal classification Eigenvalue decomposition (EVD) field-programmable gate array (FPGA) MUSIC Jacobi's method matrix inversion partial sorter qr algorithm

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An atlas for tridiagonal isospectral manifolds

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LINEAR ALGEBRA AND ITS APPLICATIONS 2008年第1期429卷 387-402页

作者： Leite, Ricardo S. Saldanha, Nicolau C. Tomei, Carlos PUC Dept Matemat BR-22453900 Rio De Janeiro Brazil Univ Fed Espirito Santo Dept Matemat BR-29075910 Vitoria ES Brazil

Let T-Lambda be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix Lambda with simple spectrum. We introduce bidiagonal coordinates, charts defined on open dense domains forming an explicit atlas for T-Lambda. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in T-Lambda now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time. (C) 2008 Elsevier Inc. All rights reserved.

关键词： Jacobi matrices tridiagonal matrices norming constants Toda flows qr algorithm

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The BR eigenvalue algorithm

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1999年第4期20卷 1083-1098页

作者： Geist, GA Howell, GW Watkins, DS Oak Ridge Natl Lab Math Sci Sect Oak Ridge TN 37831 USA Florida Inst Technol Dept Appl Math Melbourne FL 32901 USA Washington State Univ Dept Pure & Appl Math Pullman WA 99164 USA

The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the qr algorithm, but, unlike the qr algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the qr algorithm by a factor of 30-60 in computing time and a factor of over 100 in matrix storage space.

关键词： eigenvalue computation qr algorithm unsymmetric Lanczos process

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DYNAMICAL ANALYSIS OF NUMERICAL SYSTEMS

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 1995年第3期2卷 297-310页

作者： BATTERSON, S Department of Mathematics and Computer Science Emory University Atlanta GA 30322 USA

For many years techniques from numerical analysis have been applied fruitfully to the study of dynamical systems. In this paper it is shown that the theory of dynamical systems may be applied to certain computational problems. In particular the question of global convergence of various qr algorithms can be reduced to the study of certain vector iterations derived from Schur forms of matrices. The technique is illustrated in determining the convergence behavior of normal Hessenberg matrices under the Francis and multishift qr iterations.

关键词： DYNAMICAL SYSTEMS EIGENVALUES qr algorithm FRANCIS SHIFT

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