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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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Error analysis of the quasi-gram-schmidt algorithm

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第2期27卷 493-506页

作者： Stewart, GW Univ Maryland Dept Comp Sci College Pk MD 20742 USA Univ Maryland Inst Adv Comp Studies College Pk MD 20742 USA

Let the n x p ( n >= p) matrix X have the QR factorization X = QR, where R is an upper triangular matrix of order p and Q is orthonormal. This widely used decomposition has the drawback that Q is not generally sparse even when X is. One cure is to discard Q, retaining only X and R. Products like a = Q(T)y = R(-T)X(T)y can then be formed by computing b = X(T)y and solving the system R(T)a = b. This approach can be used to modify the gram-schmidt algorithm for computing Q and R to compute R without forming Q or altering X. Unfortunately, this quasi-gram-schmidt algorithm can produce inaccurate results. In this paper it is shown that with reorthogonalization the inaccuracies are bounded under certain natural conditions.

关键词： gram-schmidt algorithm sparse matrix QR factorization orthogonalization rounding-error analysis

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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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PARALLEL QR FACTORIZATION BY HOUSEHOLDER AND MODIFIED gram-schmidt algorithmS

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PARALLEL COMPUTING 1990年第1期16卷 99-112页

作者： OLEARY, DP WHITMAN, P UNIV MARYLAND INST ADV COMP STUDIESCOLLEGE PKMD 20742

In this paper, the parallel implementation of two algorithms for forming a QR factorization of a matrix is studied. We propose parallel algorithms for the modified gram-schmidt and the Householder algorithms on message passing systems in which the matrix is distributed by blocks or rows. The models that predict performance of the algorithms are validated by experimental results on several parallel machines.

关键词： gram-schmidt algorithm Householder algorithm Message passing systems QR factorization

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ROOT PROJECTION OF ONE-SIDED TIME-SERIES

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JOURNAL OF RESEARCH OF THE NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY 1991年第3期96卷 333-343页

作者： SIMMONS, JA National Institute of Standards and Technology Gaithersburg MD 20899.

Until recently it has been impossible to accurately determine the roots of polynomials of high degree, even for polynomials derived from the Z transform of time series where the dynamic range of the coefficients is generally less than 100 dB. In a companion paper, two new programs for solving such polynomials were discussed and applied to signature analysis of one-sided time series [1]. We present here another technique, that of root projection (RP), together with a gram-schmidt method for implementing it on vectors of large dimension. This technique utilizes the roots of the Z transform of a one-sided time series to construct a weighted least squares modification of the time series whose Z transform has an appropriately modified root distribution. Such a modification can be employed in a manner which is very useful for filtering and deconvolution applications [2]. Examples given here include the use of boundary root projection for front end noise reduction and a generalization of Prony's method.

关键词： CAUSAL BOUNDARY ROOTS gram-schmidt algorithm PRONYS METHOD ROOT PROJECTION SIGNAL PROCESSING TIME SERIES Z-TRANSFORMS

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Simple precoding algorithms using gram-schmidt orthonormalization process for multiuser relay communications with optimized power allocation

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ANNALS OF TELECOMMUNICATIONS 2013年第5-6期68卷 247-266页

作者： Meghdadi, Hamid Cances, Jean-Pierre Meghdadi, Vahid Univ Limoges ENSIL C2S2 XlimESTECNRSUMR 6172 F-87068 Limoges France

This paper proposes a new simple precoding solution based on the gram-schmidt orthonormalization to be used at the relay station of a multirelay wireless networks where the different mobile stations belong to the same network, in order to mitigate the multiuser interference at each mobile station. The strength of this method is that it only requires the knowledge of all channel impulse responses from a given relay to all the mobile stations. In other words, to compute its precoding vectors, each relay does not need to know the channel impulse responses of the channels of other relays. Unlike the centralized reference method where each mobile station benefits from the same diversity gain, using this algorithm, some mobile stations will improve their diversity gain at the cost of a loss in the diversity gain of other users. This constitutes a simple solution to supply different qualities of service in the case of a multiservices network. Furthermore, this work proposes an optimized power allocation between the relays. Analytical and accurate performance analyses for the different studied contexts are provided.

关键词： Cooperative diversity gain gram-schmidt algorithm Multiuser interference mitigation Expectation maximization Power allocation optimization

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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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When modified gram-schmidt generates a well-conditioned set of vectors

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IMA JOURNAL OF NUMERICAL ANALYSIS 2002年第4期22卷 521-528页

作者： Giraud, L Langou, J CERFACS F-31057 Toulouse 1 France

In this paper, we show why the modified gram-schmidt algorithm generates a well-conditioned set of vectors. This result holds under the assumption that the initial matrix is not 'too ill-conditioned' in a way that is quantified. As a consequence we show that if two iterations of the algorithm are performed, the resulting algorithm produces a matrix whose columns are orthogonal up to machine precision. Finally, we illustrate through a numerical experiment the sharpness of our result.

关键词： gram-schmidt algorithm orthogonalization schemes floating point arithmetic

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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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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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