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Journal articles on the topic 'Gram-Schmidt Process'

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Published: 3 June 2025

Last updated: 19 July 2025

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1

Okazaki, Hiroyuki. "On the Formalization of Gram-Schmidt Process for Orthonormalizing a Set of Vectors." Formalized Mathematics 31, no. 1 (2023): 53–57. http://dx.doi.org/10.2478/forma-2023-0005.

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Summary In this article, we formalize the Gram-Schmidt process in the Mizar system [2], [3] (compare another formalization using Isabelle/HOL proof assistant [1]). This process is one of the most famous methods for orthonormalizing a set of vectors. The method is named after Jørgen Pedersen Gram and Erhard Schmidt [4]. There are many applications of the Gram-Schmidt process in the field of computer science, e.g., error correcting codes or cryptology [8]. First, we prove some preliminary theorems about real unitary space. Next, we formalize the definition of the Gram-Schmidt process that finds orthonormal basis. We followed [5] in the formalization, continuing work developed in [7], [6].

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2

Wardani, Irma Budi, and Hartanto Sunardi. "ORTONORMALISASI VEKTOR BASIS DENGAN PROSES GRAM SCHMIDT." Buana Matematika : Jurnal Ilmiah Matematika dan Pendidikan Matematika 5, no. 2: (2016): 1–8. http://dx.doi.org/10.36456/buanamatematika.v5i2:.391.

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Gram schmidt process is one of linear algebra roles that associated by basis vector. This thesis aims to determine theoretically step by step in the process of gram schmidt. Gram schmidt process is a method that used to convert an arbitrary basis vector into an orthogonal basis vector. After orthogonal basis vector had been obtained, the orthogonal basis vector was compiled into an orthonormal basis through step by step. A vector on will be expressed as a basis vector if the vector if the vectors in are linear independently and spinning against . And a basis vector can be expressed as a set of orthonormal vectors, then the vector is an orthogonalvector and has norm = 1. If the basis vector has norm 1, to normalize the basis vector by using gram schmidt process. Keywords: vector base, ortonormalisasi, gram schmidt proses.

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3

Amoozad, Taher, Saeid Abbasbandy, Tofigh Allahviranloo, and Mohsen Rostamy Malkhalifeh. "A new approach for Solving a nonlinear system of second-order BVPs." Mathematical Modelling and Analysis 29, no. 4 (2024): 669–83. http://dx.doi.org/10.3846/mma.2024.19217.

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In this paper, we introduce a new approach based on the Reproducing Kernel Method (RKM) for solving a nonlinear system of second-order Boundary Value Problems (BVPs) without the Gram-Schmidt orthogonalization process. What motivates us to use the RKM without the Gram-Schmidt orthogonalization process is its easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy. Finally, the compatibility of numerical results and theorems demonstrates that the Present method is effective.

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4

Balabanov, Oleg, and Laura Grigori. "Randomized Gram--Schmidt Process with Application to GMRES." SIAM Journal on Scientific Computing 44, no. 3 (2022): A1450—A1474. http://dx.doi.org/10.1137/20m138870x.

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5

Lin, Ying-Hsiung. "GRAM-SCHMIDT PROCESS OF ORTHONORMALIZATION IN BANACH SPACES." Taiwanese Journal of Mathematics 1, no. 4 (1997): 417–31. http://dx.doi.org/10.11650/twjm/1500406120.

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6

Sreedharan, V. P. "A note on the modified gram-schmidt process." International Journal of Computer Mathematics 24, no. 3-4 (1988): 277–90. http://dx.doi.org/10.1080/00207168808803649.

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7

Zhao, Yong-Ping, Zhi-Qiang Li, Peng-Peng Xi, Dong Liang, Liguo Sun, and Ting-Hao Chen. "Gram–Schmidt process based incremental extreme learning machine." Neurocomputing 241 (June 2017): 1–17. http://dx.doi.org/10.1016/j.neucom.2017.01.049.

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8

Havlicek, Hans, and Karl Svozil. "Dimensional Lifting through the Generalized Gram–Schmidt Process." Entropy 20, no. 4 (2018): 284. http://dx.doi.org/10.3390/e20040284.

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9

Robinson, P. John, and A. Saranraj. "Intuitionistic Fuzzy Gram-Schmidt Orthogonalized Artificial Neural Network for Solving MAGDM Problems." Indian Journal Of Science And Technology 17, no. 24 (2024): 2529–37. http://dx.doi.org/10.17485/ijst/v17i24.1386.

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Objectives: To propose a suitable decision-making model based on Intuitionistic Fuzzy sets (IFSs) and Gram-Schmidt orthogonalization process for Artificial Neural Network (ANN). Methods: The IFS data sets appearing in the form of matrices are aggregated using the available aggregation operators in the literature and then the collective aggregated information is processed through Gram-Schmidt orthogonalization for the revised input vectors which is then fed into the ANN algorithm following Delta Learning Rule for the next phase. The weight updation is performed through the ANN and the output is improvised. Findings: The proposed Gram-Scmidt Orthogonalization process is utilized in Intuitionistic Fuzzy Artificial Neural Network model. The Delta learning rule is utilized in the process of the Neural Network, where the Intuitionistic Fuzzy nature of the input data is transformed into a fuzzy data and then the ranking of the alternatives is done based on the weights updation through the learning phase of the ANN. Once the vector is trained out of the learning phase, it is then processed through the activation function for the final selection of the best alternative required of the Multiple Attribute Group Decision Making (MAGDM) problem posed in this work. To demonstrate the usefulness and applicability of this new method with the Gram-Schmidt process, the numerical example also adds more insight to the proposed methodology of ANN with the application of some Linear Space techniques. Novelty: Most of the research done on Intuitionistic Fuzzy Artificial Neural Network model are based on learning rules or using some other calculations. The proposed Gram-Schmidt Orthogonalization process is used to find the orthogonal basis that are used as input training vectors in the Delta learning rule for ANN. Keywords: MAGDM, ANN, Aggregation operators, Learning Rules, Intuitionistic Fuzzy sets, Gram-Scmidt Orthogonalization

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10

Zink. "SCHAUDER BASES, SCHAUDER FUNCTIONS, AND THE GRAM-SCHMIDT PROCESS." Real Analysis Exchange 19, no. 1 (1993): 301. http://dx.doi.org/10.2307/44153843.

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11

Giraud, L., J. Langou, and M. Rozloznik. "The loss of orthogonality in the Gram-Schmidt orthogonalization process." Computers & Mathematics with Applications 50, no. 7 (2005): 1069–75. http://dx.doi.org/10.1016/j.camwa.2005.08.009.

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12

Giraud, Luc, Julien Langou, Miroslav Rozložník, and Jasper van den Eshof. "Rounding error analysis of the classical Gram-Schmidt orthogonalization process." Numerische Mathematik 101, no. 1 (2005): 87–100. http://dx.doi.org/10.1007/s00211-005-0615-4.

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13

P, John Robinson, and Saranraj A. "Intuitionistic Fuzzy Gram-Schmidt Orthogonalized Artificial Neural Network for Solving MAGDM Problems." Indian Journal of Science and Technology 17, no. 24 (2024): 2529–37. https://doi.org/10.17485/IJST/v17i24.1386.

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Abstract <strong>Objectives:</strong> To propose a suitable decision-making model based on Intuitionistic Fuzzy sets (IFSs) and Gram-Schmidt orthogonalization process for Artificial Neural Network (ANN). <strong>Methods:</strong> The IFS data sets appearing in the form of matrices are aggregated using the available aggregation operators in the literature and then the collective aggregated information is processed through Gram-Schmidt orthogonalization for the revised input vectors which is then fed into the ANN algorithm following Delta Learning Rule for the next phase. The weight updation is performed through the ANN and the output is improvised. <strong>Findings:</strong> The proposed Gram-Scmidt Orthogonalization process is utilized in Intuitionistic Fuzzy Artificial Neural Network model. The Delta learning rule is utilized in the process of the Neural Network, where the Intuitionistic Fuzzy nature of the input data is transformed into a fuzzy data and then the ranking of the alternatives is done based on the weights updation through the learning phase of the ANN. Once the vector is trained out of the learning phase, it is then processed through the activation function for the final selection of the best alternative required of the Multiple Attribute Group Decision Making (MAGDM) problem posed in this work. To demonstrate the usefulness and applicability of this new method with the Gram-Schmidt process, the numerical example also adds more insight to the proposed methodology of ANN with the application of some Linear Space techniques.<strong> Novelty:</strong> Most of the research done on Intuitionistic Fuzzy Artificial Neural Network model are based on learning rules or using some other calculations. The proposed Gram-Schmidt Orthogonalization process is used to find the orthogonal basis that are used as input training vectors in the Delta learning rule for ANN. <strong>Keywords:</strong> MAGDM, ANN, Aggregation operators, Learning Rules, Intuitionistic Fuzzy sets, Gram-Scmidt Orthogonalization

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14

Oduro, Francis T., and Amos Odoom. "A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator." Journal of Mathematics Research 13, no. 6 (2021): 20. http://dx.doi.org/10.5539/jmr.v13n6p20.

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This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr&uml;odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

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15

Rashid, Kawa M. Jamal. "Determining the approximate value of MD-Mahalanobis-Taguchi-Gram-Schmidt process by (K-MTGS) process." International Journal of Statistics and Applied Mathematics 9, no. 6 (2024): 195–204. https://doi.org/10.22271/maths.2024.v9.i6c.1918.

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16

LONDJIRINGA, Emilien LORANU, Camile LIKOTELO BINENE, Grace NKWESE MAZONI, Jonathan OPFOINTSHI ENGOMBANGI, Anyta MUKAWA LUKENZU та Clara PALUKU KASOKI. "Design and Implementation of the Gram-Schmidt Orthogonalization Process in the Sobolev Space H^1 (Ω)". Journal of Advances in Mathematics and Computer Science 40, № 6 (2025): 107–22. https://doi.org/10.9734/jamcs/2025/v40i62016.

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This article presents the design and implementation of the Gram-Schmidt orthogonalization method in the Sobolev space \(H^1\)(\(\Omega\)). However, through computational processes, it becomes clear that the Gram-Schmidt orthogonalization algorithm in \(H^1\)(\(\Omega\)) aims to transform a set of functions into an orthogonal set. By considering an arbitrary basis of a subspace of functions in the \(H^1\)(\(\Omega\)) space, we can construct a new orthogonal basis. However, this method presents certain complexities and Streamlined to improve flow due to the tedious calculation of inner products and norms of the functions. This complexity can lead to an accumulation of errors during the orthogonalization process, thereby compromising the accuracy of the results obtained. The motivation behind the development of the new implementation method is based on the need to reduce the maximum computation time and optimize precision while minimizing the risk of errors during critical steps. Consequently, the improved new approach can not only facilitate the use of this method but also ensure reliable and efficient results in practical applications.

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17

Rebollo-Neira, Laura. "Constructive updating/downdating of oblique projectors: a generalization of the Gram–Schmidt process." Journal of Physics A: Mathematical and Theoretical 40, no. 24 (2007): 6381–94. http://dx.doi.org/10.1088/1751-8113/40/24/007.

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18

Bar-On, Ilan. "Efficient logarithmic time parallel algorithms for the cholesky decomposition and gram-schmidt process." Parallel Computing 17, no. 4-5 (1991): 409–17. http://dx.doi.org/10.1016/s0167-8191(05)80144-9.

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19

Jakobs, Thomas, Billy Naumann, and Gudula Rünger. "Performance and energy consumption of the SIMD Gram–Schmidt process for vector orthogonalization." Journal of Supercomputing 76, no. 3 (2019): 1999–2021. http://dx.doi.org/10.1007/s11227-019-02839-0.

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20

Chen, Ray-Ming. "A Non-Self-Referential Characterization of the Gram–Schmidt Process via Computational Induction." Mathematics 13, no. 5 (2025): 768. https://doi.org/10.3390/math13050768.

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The Gram–Schmidt process (GSP) plays an important role in algebra. It provides a theoretical and practical approach for generating an orthonormal basis, QR decomposition, unitary matrices, etc. It also facilitates some applications in the fields of communication, machine learning, feature extraction, etc. The typical GSP is self-referential, while the non-self-referential GSP is based on the Gram determinant, which has exponential complexity. The motivation for this article is to find a way that could convert a set of linearly independent vectors {u→i}j=1n into a set of orthogonal vectors {v→}j=1n via a non-self-referential GSP (NsrGSP). The approach we use is to derive a method that utilizes the recursive property of the standard GSP to retrieve a NsrGSP. The individual orthogonal vector form we obtain is v→k=∑j=1kβ[k→j]u→j, and the collective orthogonal vectors, in a matrix form, are Vk=Uk(BΔk+). This approach could reduce the exponential computational complexity to a polynomial one. It also has a neat representation. To this end, we also apply our approach on a classification problem based on real data. Our method shows the experimental results are much more persuasive than other familiar methods.

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21

Balabanov, Oleg, and Laura Grigori. "Randomized Block Gram–Schmidt Process for the Solution of Linear Systems and Eigenvalue Problems." SIAM Journal on Scientific Computing 47, no. 1 (2025): A553—A585. https://doi.org/10.1137/22m1518396.

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22

Rezghi, Mansoor, and S. Mohammad Hosseini. "An ILU preconditioner for nonsymmetric positive definite matrices by using the conjugate Gram–Schmidt process." Journal of Computational and Applied Mathematics 188, no. 1 (2006): 150–64. http://dx.doi.org/10.1016/j.cam.2005.04.002.

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23

Batael, Hugo O., and Elso Drigo Filho. "Variational eigenfunctions for excited states inspired by supersymmetric quantum mechanics and the Gram–Schmidt process." Journal of Physics A: Mathematical and Theoretical 56, no. 28 (2023): 285201. http://dx.doi.org/10.1088/1751-8121/acde23.

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Abstract Factorization methods such as the Hamiltonian hierarchy have been useful to find eigenfunctions for Schrödinger equations, in particular, for potentials that are partially or approximately solvable. In this paper, an alternative approach is proposed to study excited states via the variational method. The trial functions are built from the exact or approximate superpotential for the ground state combined with the Gram–Schmidt process to ensure orthogonalization between the functions. The results found variationally for one dimensional potentials are compared with previous results from the literature. The energy eigenvalues obtained agree with previous ones and, for most of the results, the percentage difference between the proposed approach and others in the literature is less than 0.1%. The method introduced is an effective and intuitive approach to determine trial wave functions for the excited states. This approach can be useful in studying the Schrödinger equation and related problems which can be mapped onto a Schrödinger type-equation as, for example, the Fokker–Planck equation.

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24

Hidayati, Iswari Nur, Eni Susanti, and Westi Utami. "Analisis Pan-Sharpening untuk Meningkatkan Kualitas Spasial Citra Penginderaan Jauh dalam Klasifikasi Tata Guna Tanah." BHUMI: Jurnal Agraria dan Pertanahan 3, no. 1 (2017): 122. http://dx.doi.org/10.31292/jb.v3i1.95.

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Pan-sharpened transformation methods improve the quality of spatial resolution remote sensing imagery. This study used pan-sharpened analysis to improve the quality of image. Pan-sharpening method was used to increase spatial quality of each research object. The aims of reserach were to study image sharpening using Quickbird imagery (multispectral band and pancromatic band) and to calculate overall accuracy of land use classification base on pan-sharpened imagery classification. This study used Brovey transformation and Gram-Schmidt transformation for pan-sharpened process. The classification system used Suharyadi Classification scheme (2001) for urban areas. The results showed that Brovey transformation better than gram-schmidt transformation for the elements of texture, shape, pattern, height, and shading. Gram-Schmidt method was more suitable for the analysis concerned to its original color combination associated with the color or hue of the elements of visual interpretation. The accuracy of the study is 90.70%.Sebagian besar proses citra pan-sharpened yang diperoleh dengan formulasi berbagai algoritma yang sudah ditentukan merupakan representasi antara hubungan karakteristik dari resolusi spektral untuk meningkatkan kualitas secara visual dari citra itu sendiri. Hasil dari beberapa citra pan-sharpened tersebut menjadikan salah satu alternative untuk analisis visual citra penginderaan jauh. Penelitian ini mencoba melakukan analisis pan-sharpened untuk mendapatkan hasil yang lebih maksimal dalam berbagai kenampakan untuk setiap tata guna tanah perkotaan. Pemanfaatan pan-sharpening untuk meningkatkan kualitas spasial dari tiap objek penelitian akan dikaji agar mendapatkan masukan dalam pengembangan metode pan-sharpening untuk klasifikasi tata guna tanah di perkotaan. Penelitian ini menggunakan metode transformasi Brovey dan Gram-Schimdt untuk proses pan-sharpened. Sistem klasifikasi yang digunakan adalah sistem Klasifikasi Suharyadi (2001) untuk daerah perkotaan. Hasil penelitian ini menunjukkan bahwa metode Brovey lebih baik dalam penyajian untuk tekstur, bentuk, pola, tinggi, dan bayangan. Metode Gram-Schimdt lebih cocok untuk analisis yang lebih mementingkan perpaduan warna (komposit) aslinya terkait dengan warna ataupun rona dalam unsur interpretasi visual. Hasil akurasi penelitian penggunaan tanah yang digunakan dalam penelitian ini sebesar 90,70%.

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25

Zhang, Shuang, Yifei Han, Hua Wang, and Daishuang Hou. "Gram-Schmidt Remote Sensing Image Fusion Algorithm Based on Matrix Elementary Transformation." Journal of Physics: Conference Series 2410, no. 1 (2022): 012013. http://dx.doi.org/10.1088/1742-6596/2410/1/012013.

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Abstract In the application of the Internet of vehicles, it is difficult for a single remote sensing image to have accurate spatial information and spectral information at the same time, so engineers must use image fusion to improve the utilization of remote sensing image information. Aiming at the shortcomings of low spectral resolution and relatively complex programming of traditional fusion algorithms, this paper proposes a Gram-Schmidt remote sensing image fusion algorithm based on matrix elementary transformation, which aims to improve the spatial resolution of multispectral images by using panchromatic images with high spatial resolution. The orthogonalization process of the algorithm in this paper uses the elementary matrix transformation, which is simple and easy to perform. The experimental results show that it can effectively improve the spatial resolution of the original image and has good spectral retention.

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26

Liang, C. G., and George M. Lance. "A Differentiable Null Space Method for Constrained Dynamic Analysis." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 3 (1987): 405–11. http://dx.doi.org/10.1115/1.3258810.

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A geometric approach to the solution of the dynamic response of constrained mechanical systems is proposed. A continuous and differentiable basis of the constraint null space is automatically generated using the Gram-Schmidt process. The independent coordinates are obtained by transforming the physical velocity coordinates to the tangent hyperplane of the constraint surface. As a result the independent coordinates lie on the constraint surface and no constraint violation control is necessary.

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27

Kilela Makiese, Michael, Lucien Adam Sita Butsana, Antoine Lohadje Djombo, Christian Ndaga-Jekha, Oscar Mamona Lungiambudila, and Michel-charmy Onya Omalohoso Emongo. "SUR L’IMPLEMENTION SOUS PYTHON DU PROCEDE D’ORTHOGONALISATION DE GRAM SCHIMDT." Journal Africain des Sciences 2, no. 1 (2025): 197–205. https://doi.org/10.70237/jafrisci.2025.v2.i1.16.

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This work presents a Python implementation of the Gram-Schmidt orthogonalization process applied to sets of vectors in 2D and 3D Euclidean spaces. The goal is to transform an arbitrary basis into an orthonormal one while preserving the vector space spanned. The approach is illustrated through practical examples, highlighting the usefulness of this method in simplifying the resolution of associated linear systems. Additionally, algorithms are developed to automate the orthogonalization process and solve the systems using substitution methods or triangular matrix techniques. This study demonstrates the efficiency and accuracy of numerical approaches in handling low-dimensional algebraic problems.

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28

Mohammed Noori, Abbas, Sumaya Falih Hasan, Qayssar Mahmood Ajaj, Mustafa Ridha Mezaal, Helmi Z. M. Shafri, and Muntadher Aidi Shareef. "Fusion of Airborne Hyperspectral and WorldView2 Multispectral Images for Detailed Urban Land Cover Classification A case Study of Kuala Lumpur, Malaysia." International Journal of Engineering & Technology 7, no. 4.37 (2018): 202. http://dx.doi.org/10.14419/ijet.v7i4.37.24102.

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Detecting the features of urban areas in detail requires very high spatial and spectral resolution in images. Hyperspectral sensors usually offer high spectral resolution images with a low spatial resolution. By contrast, multispectral sensors produce high spatial resolution images with a poor spectral resolution. Therefore, numerous fusion algorithms and techniques have been proposed in recent years to obtain high-quality images with improved spatial and spectral resolutions by sensibly combining the data acquired for the same scene. This work aims to exploit the extracted information from images in an effective way. To achieve this objective, a new algorithm based on transformation was developed. This algorithm primarily depends on the Gram–Schmidt process for fusing images, removing distortions, and improving the appearance of images. Images are first fused by using the Gram–Schmidt pansharpening method. The obtained fused image is utilized in the classification process in different areas by using support vector machine (SVM). The classification result is evaluated using a matrix of errors. The overall accuracy produced from the hyperspectral, multispectral and fused images was 72.33%, 82.83%, and 89.34%, respectively. Results showed that the developed algorithm improved the image enhancement and image fusion. Moreover, the developed algorithm has the ability to produce an imaging product with high spatial resolution and high-quality spectral data.

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Hou, Zhi Song, Zhou Yu, Xian Gang Zuo, and Wei Zheng. "An Improved Blind Multiuser Detector Algorithm Based on Subspace Tracking." Advanced Materials Research 301-303 (July 2011): 737–41. http://dx.doi.org/10.4028/www.scientific.net/amr.301-303.737.

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Adaptive OPAST algorithm can ensure the orthogonality of weighted matrix in every iterations, and accelerate the convergence rate, improve the performance of error stably. However, it also enhance the complexity of algorithm. In this paper, an improved OPAST algorithm is proposed. The algorithm reduce cumulative error in the process of every iterations by inducting the dual Gram-Schmidt orthogonalization, ensure the algorithm has a better stability. Simulation results show this algorithm has stability convergence.

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30

Meghdadi, Hamid, Jean-Pierre Cances, and Vahid Meghdadi. "Simple precoding algorithms using Gram–Schmidt orthonormalization process for multiuser relay communications with optimized power allocation." annals of telecommunications - annales des télécommunications 68, no. 5-6 (2012): 247–66. http://dx.doi.org/10.1007/s12243-012-0304-0.

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31

Trigiante, D., and K. N. Murty. "On best least square solutions of two-point boundary value problems via modified Gram-Schmidt process." Applied Mathematics and Computation 51, no. 2-3 (1992): 105–18. http://dx.doi.org/10.1016/0096-3003(92)90067-b.

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32

Farzaneh Javan, Sedigheh, Saeid Abbasbandy, and M. Ali Fariborzi Araghi. "Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations." Mathematical Problems in Engineering 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/7498136.

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A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

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33

Kopal, Jiří, Miroslav Rozložník, and Miroslav Tůma. "A note on adaptivity in factorized approximate inverse preconditioning." Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, no. 2 (2020): 149–59. http://dx.doi.org/10.2478/auom-2020-0024.

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AbstractThe problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram–Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve di cult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram– Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.

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Labecca, William, Osvaldo Guimarães, and José Roberto C. Piqueira. "Coadjoint Formalism: Nonorthogonal Basis Problems." Mathematical Problems in Engineering 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/9718962.

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Using nonorthogonal bases in spectral methods demands considerable effort, because applying the Gram-Schmidt process is a fundamental condition for calculations. However, operational matrices numerical methods are being used in an increasing way and extensions for nonorthogonal bases appear, requiring simplified procedures. Here, extending previous work, an efficient tensorial method is presented, in order to simplify the calculations related to the use of nonorthogonal bases in spectral numerical problems. The method is calledcoadjoint formalismand is based onbracketDirac’s formulation of quantum mechanics. Some examples are presented, showing how simple it is to use the method.

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35

Yuan, L., and G. Zhu. "RESEARCH ON REMOTE SENSING IMAGE CLASSIFICATION BASED ON FEATURE LEVEL FUSION." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-3 (April 30, 2018): 2185–89. http://dx.doi.org/10.5194/isprs-archives-xlii-3-2185-2018.

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Remote sensing image classification, as an important direction of remote sensing image processing and application, has been widely studied. However, in the process of existing classification algorithms, there still exists the phenomenon of misclassification and missing points, which leads to the final classification accuracy is not high. In this paper, we selected Sentinel-1A and Landsat8 OLI images as data sources, and propose a classification method based on feature level fusion. Compare three kind of feature level fusion algorithms (i.e., Gram-Schmidt spectral sharpening, Principal Component Analysis transform and Brovey transform), and then select the best fused image for the classification experimental. In the classification process, we choose four kinds of image classification algorithms (i.e. Minimum distance, Mahalanobis distance, Support Vector Machine and ISODATA) to do contrast experiment. We use overall classification precision and Kappa coefficient as the classification accuracy evaluation criteria, and the four classification results of fused image are analysed. The experimental results show that the fusion effect of Gram-Schmidt spectral sharpening is better than other methods. In four kinds of classification algorithms, the fused image has the best applicability to Support Vector Machine classification, the overall classification precision is 94.01&amp;thinsp;% and the Kappa coefficients is 0.91. The fused image with Sentinel-1A and Landsat8 OLI is not only have more spatial information and spectral texture characteristics, but also enhances the distinguishing features of the images. The proposed method is beneficial to improve the accuracy and stability of remote sensing image classification.

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Zhao, Di. "Quick finite volume solver for incompressible Navier-Stokes equation by parallel Gram-Schmidt process based GMRES and HSS." Engineering Computations 32, no. 5 (2015): 1460–76. http://dx.doi.org/10.1108/ec-02-2014-0032.

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Purpose – The purpose of this paper is to develop Triple Finite Volume Method (tFVM), the author discretizes incompressible Navier-Stokes equation by tFVM, which leads to a special linear system of saddle point problem, and most computational efforts for solving the linear system are invested on the linear solver GMRES. Design/methodology/approach – In this paper, by recently developed preconditioner Hermitian/Skew-Hermitian Separation (HSS) and the parallel implementation of GMRES, the author develops a quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation. Findings – Computational results show that, the quick solver HSS-pGMRES-tFVM significantly increases the solution speed for saddle point problem from incompressible Navier-Stokes equation than the conventional solvers. Originality/value – Altogether, the contribution of this paper is that the author developed the quick solver, HSS-pGMRES-tFVM, for fast solving incompressible Navier-Stokes equation.

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Brum, R. C., M. C. S. De Castro, and C. O. Faria. "The Impact of Compilation Flags and Choosing Single- or Double-Precision Variables in Linear Systems Solvers." Trends in Computational and Applied Mathematics 24, no. 2 (2023): 319–35. http://dx.doi.org/10.5540/tcam.2023.024.02.00319.

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This paper intends to show the impact of compiler optimization flags and the variable's precision on direct methods to solve linear systems. The chosen six methods are simple direct methods, so our work could be a study for new researchers in this field. The methods are LU decomposition, LDU decomposition, Gaussian Elimination, Gauss-Jordan Elimination, Cholesky decomposition, and QR decomposition using the Gram-Schmidt orthogonalization process. Our study showed a huge difference in time between single- and double-precision in all methods, but the error encountered in the single-precision was not so high. Also, the best flags to these methods were the `-O3' and the `-Ofast' ones.

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Xu, Xiaoyong, and Fengying Zhou. "Orthonormal Euler wavelets method for time-fractional Cattaneo equation with Caputo-Fabrizio derivative." AIMS Mathematics 8, no. 2 (2022): 2736–62. http://dx.doi.org/10.3934/math.2023144.

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<abstract><p>In this paper, a new orthonormal wavelets based on the orthonormal Euler polynomials (OEPs) is constructed to approximate the numerical solution of time-fractional Cattaneo equation with Caputo-Fabrizio derivative. By applying the Gram-Schmidt orthonormalization process on sets of Euler polynomials of various degrees, an explicit representation of OEPs is obtained. The convergence analysis and error estimate of the orthonormal Euler wavelets expansion are studied. The exact formula of Caputo-Fabrizio fractional integral of orthonormal Euler wavelets are derived using Laplace transform. The applicability and validity of the proposed method are verified by some numerical examples.</p></abstract>

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Bian, Yiwen, and Sijie Li. "Ranking decision making units with large set of highly correlated performance indicators: A method based on Gram–Schmidt process." Expert Systems with Applications 38, no. 8 (2011): 10518–23. http://dx.doi.org/10.1016/j.eswa.2011.02.091.

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Bian, Yiwen. "A Gram–Schmidt process based approach for improving DEA discrimination in the presence of large dimensionality of data set." Expert Systems with Applications 39, no. 3 (2012): 3793–99. http://dx.doi.org/10.1016/j.eswa.2011.09.080.

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Zhuk, Serhii, Viacheslav Chmelov, and Oleksandr Tereshchenko. "Development of an adaptive multi-channel correlation sidelobe canceller for active noise interference based on the Gram-Schmidt orthogonalization procedure." Eastern-European Journal of Enterprise Technologies 6, no. 5 (132) (2024): 33–40. https://doi.org/10.15587/1729-4061.2024.319253.

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The object of the study is the process of adapting a multi-channel correlation sidelobe canceller to work in conditions of radio-electronic countermeasures based on the Gram-Schmidt orthogonalization procedure. The proposed approach allows developing a fast, recursive algorithm for searching for optimal values of weight coefficients. Such an algorithm will ensure fast adaptation of the sidelobe canceller to a complex interference-target situation, which can change rapidly. The obtained result of the coefficient of suppression of active noise interference in a constant value approaches the optimal value determined by the Wiener-Hopf equation, which indicates the effectiveness of the proposed approach. By using the Gram-Schmidt orthogonalization procedure, it was possible to obtain high stability of the procedures for calculating the optimal values of weight coefficients, in contrast to other considered approaches. The proposed approach can be practically implemented in existing radar systems for suppressing active noise interference. During the study, it was found that the adaptive multi-channel correlation sidelobe canceller in a steady-state mode works similarly to adaptive phased antenna arrays - it has the same efficiency in spatial signal selection and in compensation of active noise interference. When suppressing four active noise interference, the suppression coefficient is –23.35 dB. With an increase in the number of interferences at the input of the four-channel sidelobe canceller, the suppression level deteriorates rapidly. With five, six, seven, eight interferences – –22.90 dB, –21.54 dB, –20 dB, –17.02 dB, respectively. Such changes are due to the number of active interferences, which is greater than the number of compensation channels

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Gumah, Ghaleb. "Reproducing kernel Hilbert space method to solve fuzzy partial Volterra integro-differential equations." Filomat 38, no. 24 (2024): 8553–64. https://doi.org/10.2298/fil2424553g.

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In this paper, a reproducing kernel Hilbert space method for the numerical solution of fuzzy partial Volterra integro-differential equations has been presented. The reproducing Hilbert space, kernel function properties, Gram-Schmidt orthogonalization process and the bounded linear operator in the same space have been developed, which helps this method to demonstrate the convergence analysis. Moreover, we present some lemmas and theorems to prove the convergence of the reproducing kernel Hilbert space method. In this method, we give the approximate solution of the fuzzy partial Volterra integro-differential equation as a Fourier series in the Hilbert space. In order to clear the efficiency of the proposed method, some numerical examples have been solved.

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Su, Wei-Chih, Chiung-Shiann Huang, Ho-Cheng Lien, and Quang-Tuyen Le. "Identifying the stiffness parameters of a structure using a subspace approach and the Gram–Schmidt process in a wavelet domain." Advances in Mechanical Engineering 9, no. 7 (2017): 168781401770764. http://dx.doi.org/10.1177/1687814017707649.

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This article presents a procedure to improve the accuracy of calculated stiffness matrix of a structure based on the identified modal parameters from its measured responses. First, a continuous wavelet transform is applied to the measured responses of a structure, and the state–space model can be reconstructed by the wavelet coefficients of acceleration that can be obtained from the measured noisy responses. The modal parameters are identified using the subspace approach. Second, the identified mode shapes are corrected via Gram–Schmidt process. Finally, the identified natural frequencies and the corrected mode shapes in previous steps are utilized to build the stiffness matrix of structure. The accuracy of the proposed approach is numerically confirmed, and the noise effects on the ability to precisely identify the stiffness matrix are also investigated. The measured data of two eight-story steel frames in a shaking table test are analyzed to demonstrate the applicability of the procedure to real structures.

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Agapiou, Athos. "Evaluation of Landsat 8 OLI/TIRS Level-2 and Sentinel 2 Level-1C Fusion Techniques Intended for Image Segmentation of Archaeological Landscapes and Proxies." Remote Sensing 12, no. 3 (2020): 579. http://dx.doi.org/10.3390/rs12030579.

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The use of medium resolution, open access, and freely distributed satellite images, such as those of Landsat, is still understudied in the domain of archaeological research, mainly due to restrictions of spatial resolution. This investigation aims to showcase how the synergistic use of Landsat and Sentinel optical sensors can efficiently support archaeological research through object-based image analysis (OBIA), a relatively new scientific trend, as highlighted in the relevant literature, in the domain of remote sensing archaeology. Initially, the fusion of a 30 m spatial resolution Landsat 8 OLI/TIRS Level-2 and a 10 m spatial resolution Sentinel 2 Level-1C optical images, over the archaeological site of “Nea Paphos” in Cyprus, are evaluated in order to improve the spatial resolution of the Landsat image. At this step, various known fusion models are implemented and evaluated, namely Gram–Schmidt, Brovey, principal component analysis (PCA), and hue-saturation-value (HSV) algorithms. In addition, all four 10 m available spectral bands of the Sentinel 2 sensor, namely the blue, green, red, and near-infrared bands (Bands 2 to 4 and Band 8, respectively) were assessed for each of the different fusion models. On the basis of these findings, the next step of the study, focused on the image segmentation process, through the evaluation of different scale factors. The segmentation process is an important step moving from pixel-based to object-based image analysis. The overall results show that the Gram–Schmidt fusion method based on the near-infrared band of the Sentinel 2 (Band 8) at a range of scale factor segmentation to 70 are the optimum parameters for the detection of standing visible monuments, monitoring excavated areas, and detecting buried archaeological remains, without any significant spectral distortion of the original Landsat image. The new 10 m fused Landsat 8 image provides further spatial details of the archaeological site and depicts, through the segmentation process, important details within the landscape under examination.

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Mustafa Hüsrevoğlu and Ahmet Emin Karkınlı. "Enhanced Pansharpening Using Curvelet Transform Optimized by Multi Population Based Differential Evolution." International Journal of Scientific Research in Computer Science, Engineering and Information Technology 10, no. 4 (2024): 139–49. http://dx.doi.org/10.32628/cseit24104116.

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In this study, a pansharpening process was conducted to merge the color information of low-resolution RGB images with the details of high-resolution panchromatic images to obtain higher quality images. During this process, weight optimization was performed using the Curvelet Transform method and the Multi Population Based Differential Evolution (MDE) algorithm. The proposed method was tested on Landsat ETM satellite image. For Landsat ETM data, the RGB images have a resolution of 30m, while the panchromatic images have a resolution of 15m. To evaluate the performance of the study, the proposed MDE-optimized Curvelet Transform-based pansharpening method was compared with classical IHS, Brovey, PCA, Gram-Schmidt and Simple Mean methods. The comparison process employed metrics such as RMSE, SAM, COC, RASE, QAVE, SID, and ERGAS. The results indicate that the proposed method outperforms classical methods in terms of both visual quality and numerical accuracy.

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Siwi, Sukentyas Estuti. "EVALUATION OF SPOT-5 IMAGE FUSION USING MODIFIED PAN-SHARPENING METHODS." International Journal of Remote Sensing and Earth Sciences (IJReSES) 11, no. 2 (2017): 117. http://dx.doi.org/10.30536/j.ijreses.2014.v11.a2609.

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Image fusion, commonly known as pan-sharpening, is a method that combines two data: a panchromatic image that has geometric detail information with the highest spatial resolution and multi-spectral image that has the highest color information but with the lowest resolution. Pan-sharpeningis very important for various remote sensing applications, such as to improve the image classification, to change the detection using temporal data, to increase the geometric, image segmentation, and to improve the visibility of certain object that does not appear on certain data. This study aims to compare the existing pan-sharpening methods such as Brovey, Brovey modification using green and red band, Gram-Schmidt, HPF, Multiplicative, and SFIM.The quality of the pan-sharpening result should be evaluated, this study used Universal Image Quality Index (UIQI/Q index); this evaluation methodgives the opportunity to choose which method is best to provide the most similar spectral information with the original multispectral image. A pan-sharpening qualitative analysis shows that there has been a sharpening process on all pan-sharpening images. Based on spectral visualization (color display), several pan-sharpening methods such as HPF multiplicative method provides brighter colorsand Brovey transformation method displays dark colors. Gram-Schmidt method also provides a different color from the original multispectral image. A pan-sharpening quantitative analysis shows that the best pan-sharpening method with UIQI value> 0.9 is Brovey modification using green and red band. This is due to the green band (500-590 nm) and the red band(610-680 nm) wavelength are in the panchromatic band (480-710 nm) of the SPOT-5 Data.

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Wang, Jiayan, Lanlan Guo, Zongmin Li, Xueqin Wang, and Zhengqing Fu. "An Efficient Algorithm for Ill-Conditioned Separable Nonlinear Least Squares." Journal of Mathematics 2021 (November 10, 2021): 1–8. http://dx.doi.org/10.1155/2021/7625175.

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For separable nonlinear least squares models, a variable projection algorithm based on matrix factorization is studied, and the ill-conditioning of the model parameters is considered in the specific solution process of the model. When the linear parameters are estimated, the Tikhonov regularization method is used to solve the ill-conditioned problems. When the nonlinear parameters are estimated, the QR decomposition, Gram–Schmidt orthogonalization decomposition, and SVD are applied in the Jacobian matrix. These methods are then compared with the method in which the variables are not separated. Numerical experiments are performed using RBF neural network data, and the experimental results are analyzed in terms of both qualitative and quantitative indicators. The results show that the proposed algorithms are effective and robust.

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Gourary, Mark, Sergey Rusakov, Mikhail Zharov, and Sergey Ulyanov. "Computational Algorithms for Reducing Rational Transfer Functions’ Order." SPIIRAS Proceedings 19, no. 2 (2020): 330–56. http://dx.doi.org/10.15622/sp.2020.19.2.4.

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A problem of reducing a linear time-invariant dynamic system is considered as a problem of approximating its initial rational transfer function with a similar function of a lower order. The initial transfer function is also assumed to be rational. The approximation error is defined as the standard integral deviation of the transient characteristics of the initial and reduced transfer function in the time domain. The formulations of two main types of approximation problems are considered: a) the traditional problem of minimizing the approximation error at a given order of the reduced model; b) the proposed problem of minimizing the order of the model at a given tolerance on the approximation error. Algorithms for solving approximation problems based on the Gauss-Newton iterative process are developed. At the iteration step, the current deviation of the transient characteristics is linearized with respect to the coefficients of the denominator of the reduced transfer function. Linearized deviations are used to obtain new values of the transfer function coefficients using the least-squares method in a functional space based on Gram-Schmidt orthogonalization. The general form of expressions representing linearized deviations of transient characteristics is obtained. To solve the problem of minimizing the order of the transfer function in the framework of the least squares algorithm, the Gram-Schmidt process is also used. The completion criterion of the process is to achieve a given error tolerance. It is shown that the sequence of process steps corresponding to the alternation of coefficients of polynomials of the numerator and denominator of the transfer function provides the minimum order of transfer function. The paper presents an extension of the developed algorithms to the case of a vector transfer function with a common denominator. An algorithm is presented with the approximation error defined in the form of a geometric sum of scalar errors. The use of the minimax form for error estimation and the possibility of extending the proposed approach to the problem of reducing the irrational initial transfer function are discussed. Experimental code implementing the proposed algorithms is developed, and the results of numerical evaluations of test examples of various types are obtained.

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Elbouyahyaoui, Lakhdar, and Mohammed Heyouni. "On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems." Boletim da Sociedade Paranaense de Matemática 36, no. 3 (2018): 155–72. http://dx.doi.org/10.5269/bspm.v36i3.32109.

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In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments that are done with different matrices coming from the Matrix Market repository or from the university of Florida sparse matrix collection are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.

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Ider, S. K., and F. M. L. Amirouche. "Coordinate Reduction in the Dynamics of Constrained Multibody Systems—A New Approach." Journal of Applied Mechanics 55, no. 4 (1988): 899–904. http://dx.doi.org/10.1115/1.3173739.

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In this paper a new theorem for the generation of a basis for the null space of a rectangular matrix, with m linearly independent rows and n (n > m) columns is presented. The method is based on Gaussian row operations to transform the constraint Jacobian matrix to an uptriangular matrix. The Gram-Schmidt process is then utilized to identify basis vectors orthogonal to the uptriangular matrix. A complement orthogonal array which forms the basis for the null space for which the algebraic constraint equations are satisfied is then formulated. An illustration of the theorem application to constrained dynamical systems for both Lagrange and Kane’s equations is given. A numerical computer algorithm based on Kane’s equations with embedded constraints is also presented. The method proposed is well conditioned and computationally efficient and inexpensive.

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