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An approach of orthogonalization within the gram-schmidt algorithm

COMPUTATIONAL & APPLIED MATHEMATICS 2018年第2期37卷 1250-1262页

作者： Rivaz, A. Moghadam, M. Mohseni Sadeghi, D. Kermani, H. Momenaee Shahid Bahonar Univ Kerman Fac Math & Comp Dept Appl Math Kerman *** Iran

In this paper we consider the variants of gram-schmidt such as Classical gram-schmidt and Modified Grain-schmidt algorithms. It is shown that for problems of dimension more than two the round-off error of operation q(1)(T)q(2) has more propagation in both of algorithms. To cure this difficulty we will present an algorithm, namely Optimized Modified gram-schmidt algorithm. Numerical examples indicate the accuracy of this algorithm. We show that this method can improve the loss of orthogonality of the orthogonalization in some ill-conditioned cases.

关键词： gram-schmidt algorithm Loss of orthogonality Ill-conditioned matrix Optimized gram-schmidt algorithm Reorthogonalization

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Evaluation of soil fertility using combination of Landsat 8 and Sentinel-2 data in agricultural lands

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ENVIRONMENTAL MONITORING AND ASSESSMENT 2024年第2期196卷 131-131页

作者： Zhang, Ming Aqdam, Mohammad Khosravi Fadel, Hassan Abbas Wang, Lei Waheeb, Khlood Kadhim, Angham Hekmati, Jamal Anhui Vocat & Technol Coll Forestry Dept Resources & Environm Hefei 230031 Anhui Peoples R China Adm Educ Saqqez Kurdistan Iran Natl Univ Sci & Technol Dhi Qar Iraq Minist Ecol & Environm Nanjing Inst Environm Sci Nanjing 210042 Peoples R China Al Farahidi Univ Med Tech Coll Baghdad Iraq Al Zahrawi Univ Coll Dept Opt Tech Karbala Iraq Univ Campus 2 Rasht Iran

Today, remote sensing is widely used to estimate soil properties. Because it is an easy and accessible way to estimate soil properties that are difficult to estimate in the field. Based on this, to evaluate the soil fertility (SF), soil sampling was performed irregularly from the surface depth of 0-30 cm in 216 points, 11 soil properties were measured, and the soil fertility index (SFI) was calculated by soil properties. Simultaneously, we combined satellite images of Landsat 8 and Sentinel-2 using the gram-schmidt algorithm. Finally, multiple linear regression SFI was calculated using satellite data, as well as the spatial distribution of SFI was obtained in very low, low, moderate, high, and very high classes. Our findings showed that the combination of Landsat 8 and Sentinel-2 data using the gram-schmidt algorithm has a higher correlation with SFI than when these data are individually. Therefore, combined Landsat 8 and Sentinel 2 data were used for SFI modeling. Using model selection procedure indices (including Cp, AIC, and rho c criteria), the visible range bands, notably blue (r = 0.65), green (r = 0.63), and red (r = 0.61), provide the best model for estimating SFI (R2 = 0.43, Cp = 3.34, AIC = -277.4, and rho c = 0.44). Therefore, these bands were used to estimate the SFI index. Also, the spatial distribution of the SIF index showed that the most significant area was related to the low class, and the lowest area belonged to the high and very high fertility classes. According to these results, it can be concluded that using the combination of Landsat 8 and Sentinel 2 bands to estimate soil fertility index in agricultural lands can increase the accuracy of soil fertility estimation.

关键词： gram-schmidt algorithm Remote sensing Soil fertility index Soil fertility

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Numerical solution of the fractional model representing the HIV infection of CD4⁺T cells

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INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2025年 1-17页

作者： Boroghani, Reza Nouri, Kazem Torkzadeh, Leila Semnan Univ Fac Math Stat & Comp Sci Dept Math POB 35195-363 Semnan Iran

This study introduces a novel numerical approach that utilizes orthonormal Bell polynomials (OBPs) and block-pulse functions (BPFs) to address the fractional model of HIV infection in CD4(+)T cells. The model consists of three fractional nonlinear ordinary differential equations. The initial phase of this approach involves orthonormalizing the Bell polynomials using the gram-schmidt algorithm. Following this, BPFs are used to create an operational matrix related to fractional integration for OBPs. By utilizing this matrix, the problem is transformed into a series of nonlinear algebraic equations. These equations can be effectively solved using an appropriate numerical technique, such as Newton's method. Ultimately, the numerical results illustrate the method's applicability, efficiency, and accuracy, and these findings are compared with existing methods documented in the literature.

关键词： Orthonormal Bell polynomials (OBPs) gram-schmidt algorithm Block-pulse functions (BPFs) HIV infection CD4(+)T cells

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Singular Value Decomposition Optimizes Multi-Level Orthogonal Code Index Modulation DCSK System

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IEEE COMMUNICATIONS LETTERS 2024年第12期28卷 2874-2878页

作者： Zhang, Gang Chen, Xibiao Hu, Yanan Chongqing Univ Posts & Telecommun Sch Commun & Informat Engn Chongqing 400065 Peoples R China

To improve the bit error rate (BER) performance of the differential chaotic shift keying (DCSK) system, this letter proposes a multi-level orthogonal code index modulation DCSK (MOCIM-DCSK) system. A set of mutually orthogonal reference signals is generated using the gram-schmidt algorithm, with each reference signal carrying code index bits and modulation bits for the transmission of information signals. Additionally, to further enhance the system's reliability, singular value decomposition (SVD)-aided detection is introduced, successfully suppressing additive white Gaussian noise (AWGN). The BER formulas of MOCIM-DCSK system over AWGN channel and multipath Rayleigh fading channels is derived, and the complexity of MOCIM-DCSK system is compared with other systems. Simulation results validate the correctness of the theoretical derivations and demonstrate the outstanding BER performance of the system. Furthermore, the application of SVD to the MC-DCSK system provides additional evidence of its advantages and versatility in enhancing BER performance.

关键词： Indexes Codes Modulation Matched filters Bit error rate Chaotic communication Receivers Filtering algorithms Information filters Spectral efficiency Differential chaos shift keying (DCSK) gram-schmidt algorithm code index modulation singular value decomposition

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Direct Sequential Generation of Optimal Eigenvectors of the Symmetric Kernel of the Discrete Hankel Transform 67

Direct Sequential Generation of Optimal Eigenvectors of the ...

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67th IEEE International Midwest Symposium on Circuits and Systems (MWSCAS)

作者： Hanna, Magdy Tawfik Fayoum Univ Dept Engn Math & Phys Al Fayyum 63514 Egypt

ISBN: (纸本)9798350387186;9798350387179

Generating orthonormal Laguerre- Gaussian-power-like eigenvectors of the symmetric kernel matrix T of the discrete Hankel transform ( DHT) is the cornerstone in developing the discrete fractional Hankel transform (DFRHT). The direct sequential generation of the columns of each partition of the modal matrix of matrix T pertaining to one distinct eigenvalue is considered. The focus of the research is the assessment of the direct sequential algorithms: the gram-schmidt algorithm (GSA), modified gram-schmidt algorithm (MGSA), sequential orthogonal procrustes algorithm (SOPA) and direct sequential evaluation by constrained optimization algorithm (DSEOA). Although the four algorithms are theoretically equivalent, they are quite numerically distinct. The difference in performance becomes more pronounced as the order of the square matrix T becomes larger. The assessment shows that the MGSA outperforms both the GSA and the SOPA and that the DSEOA is by far the best in terms of computation accuracy.

关键词： discrete Hankel transform fractional transform modal matrix gram-schmidt algorithm modified gram-schmidt algorithm constrained minimization

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Consideration of a robust watermarking algorithm for color image using improved QR decomposition

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SOFT COMPUTING 2022年第11期26卷 5069-5093页

作者： Phuong Thi Nha Ta Minh Thanh Nguyen Tuan Phong Le Quy Don Tech Univ 236 Hoang Quoc Viet Hanoi Vietnam

In order to protect the digital image copyright, it is necessary to design a robust watermarking algorithm. To achieve this purpose, a novel color image watermarking scheme based on an improved QR decomposition is proposed in this paper. The proposed method gives a new algorithm to find elements of Q and R matrices instead of using the gram-schmidt algorithm for QR factorization. First, the R matrix is performed by solving a set of linear equations where diagonal elements of R are checked and modified if they are zero or negative. After that, the Q matrix is computed based on the R matrix. In addition, a novel formula is proposed to improve the extracting time where the first element R(1, 1) of the R matrix is found instead of computing QR decomposition as the previous proposals. Experimental results show that the proposed method outperforms other considered methods in this paper in terms of the quality of the watermarked images. Furthermore, the execution time is significantly improved, and the extracted watermark is more robust against almost tested attacks.

关键词： Digital watermarking Copyright protection gram-schmidt algorithm QR factorization Quantization index modulation (QIM)

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A Receding Horizon-type Solution of the Inverse Kinematic Task of Redundant Robots 15

A Receding Horizon-type Solution of the Inverse Kinematic Ta...

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IEEE 15th International Symposium on Applied Computational Intelligence and Informatics (SACI)

作者： Issa, Hazem Varga, Bence Tar, Jozsef K. Obuda Univ Doctoral Sch Appl Informat & Appl Math Budapest Hungary Obuda Univ Inst Mechatron & Vehicle Engn Donat Banki Fac Mech & Safety Engn Budapest Hungary Obuda Univ Antal Bejczy Ctr Intelligent Robot Univ Res & Innovat Ctr Budapest Hungary Obuda Univ John von Neumann Fac Informat Inst Appl Math Budapest Hungary

ISBN: (纸本)9781728195438

With the exception of specially designed kinematic structures, the inverse kinematic task of redundant robots cannot be solved in "closed form". Redundancy is often needed to make the arm "dexterous". In general numerical techniques have to be applied. A typical approach is finding the "differential solution" by formulating the Jacobian of the problem in the first step, and using some generalized inverse that selects a definite solution of the ambiguous set of the possible ones. Each of these solutions suffers from huge joint coordinate time-derivatives in the vicinity of the kinematic singularities that are evaded by "deforming" the original task. A quasi-differential approach was recently elaborated that, though calculates the Jacobian, instead "inverting" it, applies a fixed point iteration that automatically evades the singularities. However, its solution depends on the eigenvalues of the Jacobian, and it was found not "flexible enough" for all practical purposes. The Moore-Penrose solution can be generalized as an optimization task under constraints for quite complex cost functions. The hard constraint in this approach forces the exact solution of the inverse kinematic task when it is possible. Due to it the intent of "minimizing the costs" is "overridden" in the vicinity of the singularities. In the present paper, based on the formal structure of the Receding Horizon Controllers, an alternative solution is suggested that allows flexibility for relaxing the costs and the "hard constraints", too. Its computational complexity to some extent is reduced via evading the technique of gradient reduction. Simulation results are presented for a 7 DoF robot arm.

关键词： Inverse Kinematic Task Gradient Descent Method Reduced Gradient Method Moore-Penrose Pseudoinverse gram-schmidt algorithm Receding Horizon Control

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Speeding up the Reduced Gradient Method for Constrained Optimization 19

Speeding up the Reduced Gradient Method for Constrained Opti...

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19th IEEE World Symposium on Applied Machine Intelligence and Informatics (SAMI)

作者： Issa, Hazem Tar, Jozsef K. Obuda Univ Doctoral Sch Appl Informat & Appl Math Budapest Hungary Obuda Univ Res & Innovat Ctr Antal Bejczy Ctr Intelligent Robot Budapest Hungary

ISBN: (纸本)9781728180533

In various technical applications the local minimum of a differentiable cost function must be found under constraints that are interpreted as embedded hypersurfaces in the whole space of search. Generally Lagrange's "Reduced Gradient Method" can be applied for solving such problems in which the Lagrange multipliers associated with the individual constraint equations have important physical interpretation, therefore it is desirable to compute them. Though in special cases this algorithm can be replaced by closed form calculations via considering the "Auxiliary Function", in other cases the algorithmic realization cannot be avoided. In this paper it is shown that via avoiding the calculation of the individual Lagrange multipliers the algorithm can be made considerably faster especially if the constraint equations are appropriately handled. Simulation investigations are presented to substantiate the suggested method.

关键词： Reduced Gradient Auxiliary Function gram-schmidt algorithm Fixed Point Iteration Optimal Control

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New Learning Approach for Unsupervised Neural Networks Model with Application to Agriculture Field

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INTERNATIONAL JOURNAL OF ADVANCED COMPUTER SCIENCE AND APPLICATIONS 2020年第5期11卷 360-369页

作者： Sara, Belattar Otman, Abdoun Haimoudi, El Khatir Univ Abdelmalek Essaadi UAE Comp Sci Dept Adv Sci & Technol Lab Polydisciplinary Fac Larache Morocco

An accurate and lower cost hybrid machine learning algorithm based on a combination of Kohonen-Self Organizing Map (SOM) and gram-schmidt (GSHM) algorithm was proposed, to enhance the crop yield prediction and to increase the agricultural production. The combination of GSHM and SOM allows to withdraw the most informative components about our data, by overcoming correlation issues between input data prior to the training process. The improved hybrid algorithm was trained firstly on data that have a correlation problem, and it was compared with another hybrid model based on SOM and Principal Component Analysis (PCA), secondly, it was trained using selected soil parameters related to the atmosphere (e.g. pH, nitrogen, phosphate, potassium, depth, temperature, and rainfall). A comparative study with the standard SOM was conducted. The improved Kohonen-Self Organizing Map when applied to correlated data, demonstrated better results in terms of classification accuracy (8/8), and rapidity = 0.015s compared to a classification accuracy (7/8) and a rapidity = 97,828 s using SOM combined with PCA. Moreover, the proposed algorithm resulted in better results for crop prediction in terms of maximum iteration number of 675, mean error <= 0.00022, and rapidity = 18.422s versus an iteration number of 729, mean error <= 0.000916 and rapidity= 23.707s with the standard SOM. The proposed algorithm allowed us to overcome correlation issues, and to improve the classification, learning process, and rapidity, with the potential to apply for predicting crop yield in the agricultural field.

关键词： Kohonen-self organizing map gram-schmidt algorithm principal component analysis agriculture field crop yield prediction

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Cost Function-Free Optimization in Inverse Kinematics of Open Kinematic Chains 24th

Cost Function-Free Optimization in Inverse Kinematics of Ope...

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24th International Conference on Robotics in Alpe-Adria-Danube Region (RAAD)

作者： Tar, Jozsef K. Nadai, Laszlo Felde, Imre Rudas, Imre J. Obuda Univ Antal Bejczy Ctr Intelligent Robot Becsi Ut 96-B H-1034 Budapest Hungary Obuda Univ Smartlab Knowledge Ctr H-1034 Budapest Hungary

ISBN: (纸本)9783319212906;9783319212890

The traditional ways of solving various tasks "optimally" in control technology and robotics normally are based on the minimization of some cost function (or functional). On the basis of function minimization various "generalized inverse matrices" can be introduced that have special significance in the inverse kinematic tasks of redundant manipulators, where the possible solutions are ambiguous-therefore various choices are available. The solution suggested here tackles the question of optimality by the geometric interpretation of the simple and computationally efficient gram-schmidt algorithm. The method is presented via simulations using a redundant arm structure.

关键词： Open kinematic chain gram-schmidt algorithm Redundant robot arms Generalized inverse Inverse kinematic task

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