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Applied Analysis
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
The Gram–Schmidt orthogonalization process is a procedure for constructing an orthonormal set of vectors from an arbitrary linearly independent set of vectors. This construction follows from the following observation. Let (V, 〈·,·〉 be an inner product vector space over the field ℂ. Also let {u1, u2,…, ur} be an orthonormal set of vectors which belong to the set V. These vectors are linearly independent. Furthermore, for any υ ∈ V, the vector w given by w=υ−〈υ,u1〉u1−〈υ,u2〉u2−⋯−〈υ,ur〉ur
Iterative Methods for Solving Linear Systems
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
Obviously, all the resulting vectors u1, u2,…,uk,… are orthonormal. If the Arnoldi process terminates at step m, then the vectors {u1, u2um} will form an orthonormal basis in Km(A, u). The process can also terminate prematurely, i.e., yield Vk = 0 at some k < m. This will indicate that the dimension of the corresponding Krylov subspace is lower than m. Note also that the classical Gram-Schmidt orthogonalization is prone to numerical instabilities. Therefore, in practice one often uses its stabilized version (see Remark 7.4 on page 219). The latter is not not completely fail proof either, yet it is more robust and somewhat more expensive computationally.
Linear Vector Spaces
Published in Sohail A. Dianat, Eli S. Saber, ®, 2017
Sohail A. Dianat, Eli S. Saber
The Gram–Schmidt orthogonalization process is used to convert a set of independent or dependent vectors in a given vector space to an orthonormal set. Given a set of n nonzero vectors S = {x1x2 … xn} in vector space V, we would like to find an orthonormal set of vectors Ŝ = {u1u2 … um} with the same span as S. It is obvious that m ≤ n. If the vectors forming set S are linearly independent, then m = n. This is accomplished by using the Gram–Schmidt orthogonalization process. The steps of the process as explained below:
Channel Contributions of EEG in Emotion Modelling Based on Multivariate Adaptive Orthogonal Signal Decomposition
Published in IETE Journal of Research, 2023
Hence, after the corresponding IMFs are acquired in the EMD process, it is clear that a method that is interested in only orthogonal modes can facilitate the process. As such, the acquired IMFs are processed by the Gram-Schmidt Orthogonalization method to estimate the number of orthogonal components [71]. [72,73] have studied with numeric examples of how all IMFs are not mutually orthogonal. Let V the be a finite-dimensional inner product space, with a basis as a linearly independent subset of V. Then the Gram-Schmidt Orthogonalization process utilize the vectors to construct new vectors , such that for and for i=1,2, … ,n. Algorithm 2 defines, and Figure 2 depicts the Gram-Schmidt Orthogonalization process.
QR decomposition for the least squares method: theory and practice
Published in International Journal of Mathematical Education in Science and Technology, 2022
The traditional algorithm of QR factorization is the Gram-Schmidt orthogonalization process. However, it won’t be discussed here because it is rarely used for the least squares problem solving due to its numerical instability. The more common and numerically stable methods that will be considered in this work are Givens rotations and Householder reflections. Both can be represented as a gradual transformation of the X design matrix to the upper triangular R matrix, i.e.: The orthogonal transformations are used to replace at least one element of X to zeros with possible change of some other elements of this matrix. If we consider X columns as vectors, then they can be interpreted as rotations or reflections for Givens and Householder methods respectively.
Parametric model order reduction based on parallel tensor compression
Published in International Journal of Systems Science, 2021
Zhen Li, Yao-Lin Jiang, Hong-liang Mu
Select parameter vectors . For every , the first r Taylor expansion coefficients are iteratively computed by (17) with , which is similar to the methods in Baur et al. (2011), Benner and Feng (2014) and Bonin et al. (2016). Then, the space spanned by the columns of the projection matrix can be obtained by The orthogonal methods, such as Gram–Schmidt orthogonalisation and QR decomposition, are used to compute the matrix V, which can make sure that the projection matrix is a column-orthogonal matrix. Taking the approximation , we can get the reduced parametric system where and . The order of the reduced system is . Furthermore, we denote and , and then we can get the corresponding reduced parametric system (3).