An Efficient Algorithm for Ill-Conditioned Separable Nonlinear Least Squares

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.

A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

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¨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.

Generation of Tri-Directional Spectra-Compatible Time Histories Coupling the Influence Matrix Method and Gram–Schmidt Orthogonalization

This study presents a new approach for obtaining a set of tri-directional time histories compatible with target design spectra by modifying real recorded earthquake ground motions. The influence matrix method (IMM) based on eigenfunction expansion is improved for typical design response spectra with different shapes and employed in order to achieve accurate matching with the target design spectra. By applying the Gram–Schmidt orthogonalization in each iteration of the IMM procedure, the correlation coefficient between any two components can be guaranteed to be strictly zero. Hence, the generated three components in the orthogonal directions are statistically independent. The generated time histories satisfy the requirements of current codes and standards. Two examples are presented to illustrate the procedure and the superiority of the proposed method, with the maximum relative error between the generated time histories and target design spectra being less than 0.2% in [0.6, 100] Hz, and the code requirements being satisfied strictly.

Comparative study of loudspeaker position optimization techniques for multizone sound field reproduction

Mutlizone sound field reproduction aims to generate personal sound zones in a shared space with multiple loudspeakers. Conventionally, loudspeakers are placed to form a regular pattern such as circular, arc or linear array, which are empirical rather than optimal mainly for the convenience of physical placement. Recently, several algorithms have been proposed to select a fixed number of loudspeaker locations from a large set of candidate positions, such as the sparse regularization (i.e. Lasso and Elastic Net) methods, the Constrained Match Pursuit (CMP) method, the Gram-Schmidt Orthogonalization (GSO) method etc. Most of these methods were investigated for single-zone rather than mulit-zone sound field reproduction based on the pressure matching techniques. This paper compares the performance of the state-of-the-art techniques for loudspeaker position optimization in a multizone sound field reproduction system in terms of reproduction error, acoustic contrast and array effort. Simulation results demonstrate that the CMP-LS method shows the best performance in terms of lower MSE and higher AC while the Lasso method needs the lowest AE.

Orthogonal Signals and the Orthogonalization Procedure

Chapter 2 is dedicated to the principle of signal orthogonalization, because orthogonal signals are widely used in telecommunication theory and practice, like the carriers of baseband signals, subcarriers in orthogonal frequency division multiplexing systems, and the spreading sequences in spread-spectrum and code division multiple access (CDMA) systems. The orthonormal basis functions are defined and the procedure of the vector representation of signals is demonstrated. The Gram–Schmidt orthogonalization procedure and construction of the space diagram are presented in detail. Using orthonormal signals, signal synthesizers and analysers that can be used to form discrete-time transmitters and receivers are theoretically founded. Understanding of this chapter is a prerequisite for understanding Chapters 4–10, because the orthonormal signals defined in this chapter will be used throughout the book. The basis harmonic orthonormal functions will define the carriers in the discrete and digital communication systems.

Analysis of Feature Influence on Covid-19 Death Rate Per Country Using a Novel Orthogonalization Technique

We have developed a new technique of Feature Importance, a topic of machine learning, to analyze the possible causes of the Covid-19 pandemic based on country data. This new approach works well even when there are many more features than countries and is not affected by high correlation of features. It is inspired by the Gram-Schmidt orthogonalization procedure from linear algebra. We study the number of deaths, which is more reliable than the number of cases at the onset of the pandemic, during Apr/May 2020. This is while countries started taking measures, so more light will be shed on the root causes of the pandemic rather than on its handling. The analysis is done against a comprehensive list of roughly 3,200 features. We find that globalization is the main contributing cause, followed by calcium intake, economic factors, environmental factors, preventative measures, and others. This analysis was done for 20 different dates and shows that some factors, like calcium, phase in or out over time. We also compute row explainability, i.e. for every country, how much each feature explains the death rate. Finally we also study a series of conditions, e.g. comorbidities, immunization, etc. which have been proposed to explain the pandemic and place them in their proper context. While there are many caveats to this analysis, we believe it sheds light on the possible causes of the Covid-19 pandemic.

An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

Our aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

A Fast Adaptive Beamforming Algorithm Based on Gram-Schmidt Orthogonalization

Fast implementation is one of the important indexes of the ADBF algorithm. The advantages of the Gram-Schmidt (GS) orthogonalization algorithm are that it can reconstruct the interference subspace well under the high signal-to-noise ratio and has fast convergence speed and low computational complexity. This paper studies the RGS algorithm for GS orthogonalization of sampling covariance matrix. To estimate the interference subspace more accurately, this paper modifies the orthogonal adaptive threshold of covariance matrix, and extends the proposed GS orthogonal algorithm of covariance matrix based on data preprocessing to the adaptive beamforming processing at subarray level.