Least Squares Method for Solving a System of Linear Equations Based on Multilevel Wavelet Decomposition of the Residual

2018 ◽  
Author(s):  
V. Esaulov ◽  
R. Sinetsky
Keyword(s):  
Least Squares ◽  
Wavelet Decomposition ◽  
Least Squares Method ◽  
Linear Equations ◽  
System Of Linear Equations

A Nonlinear Least-Squares Graphical Tool (‘Gaussfit’) for Educational Purposes

2010 ◽  
Vol 47 (1) ◽  
pp. 11-22 ◽  
Author(s):  
Krešimir Malarić ◽  
Roman Malarić ◽  
Hrvoje Hegeduš
Keyword(s):  
Least Squares ◽  
Input Data ◽  
Least Squares Method ◽  
Linear Equations ◽  
Computer Calculation ◽  
Nonlinear Least Squares ◽  
Undergraduate Level ◽  
Graphical Tool ◽  
Systems Of Linear Equations ◽  
Graphical Presentation

This paper describes a computer program that finds a function which closely approximates experimental data using the least-squares method. The program finds parameters of the function as well as their corresponding uncertainties. It also has a subroutine for graphical presentation of the input data and the function. The program is used for educational purposes at undergraduate level for students who are learning least-squares fitting, how to solve systems of linear equations and about computer calculation errors.

scholarly journals A Representation of Nonhomogeneous Quadratic Forms with Application to the Least Squares Solution

2011 ◽  
Vol 2011 ◽  
pp. 1-5 ◽  
Author(s):  
Czesław Stępniak
Keyword(s):  
Least Squares ◽  
Quadratic Forms ◽  
Differential Calculus ◽  
Linear Models ◽  
Linear Equations ◽  
Algebraic Approach ◽  
System Of Linear Equations ◽  
Least Squares Solution ◽  
Inconsistent System ◽  
Usual Solution

The least squares problem appears, among others, in linear models, and it refers to inconsistent system of linear equations. A crucial question is how to reduce the least squares solution in such a system to the usual solution in a consistent one. Traditionally, this is reached by differential calculus. We present a purely algebraic approach to this problem based on some identities for nonhomogeneous quadratic forms.

scholarly journals A novel interpretation of least squares solution

1992 ◽  
Vol 15 (1) ◽  
pp. 41-46
Author(s):  
Jack-Kang Chan
Keyword(s):  
Least Squares ◽  
Source Localization ◽  
Convex Combination ◽  
Linear Equations ◽  
Singular Values ◽  
Statistical Interpretation ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
The Matrix ◽  
Cramer’S Rule

We show that the well-known least squares (LS) solution of an overdetermined system of linear equations is a convex combination of all the non-trivial solutions weighed by the squares of the corresponding denominator determinants of the Cramer's rule. This Least Squares Decomposition (LSD) gives an alternate statistical interpretation of least squares, as well as another geometric meaning. Furthermore, when the singular values of the matrix of the overdetermined system are not small, the LSD may be able to provide flexible solutions. As an illustration, we apply the LSD to interpret the LS-solution in the problem of source localization.

scholarly journals Improved Computing-Efficiency Least-Squares Algorithm with Application to All-Pass Filter Design

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Lo-Chyuan Su ◽  
Yue-Dar Jou ◽  
Fu-Kun Chen
Keyword(s):  
Least Squares ◽  
Filter Design ◽  
Linear Equations ◽  
Hankel Matrix ◽  
Cholesky Decomposition ◽  
Matrix Inversion ◽  
System Of Linear Equations ◽  
Computationally Efficient ◽  
Pass Filter ◽  
Simulation Results

All-pass filter design can be generally achieved by solving a system of linear equations. The associated matrices involved in the set of linear equations can be further formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided. Consequently, the optimal filter coefficients can be solved by using computationally efficient Levinson algorithms or Cholesky decomposition technique. In this paper, based on trigonometric identities and sampling the frequency band of interest uniformly, the authors proposed closed-form expressions to compute the elements of the Toeplitz-plus-Hankel matrix required in the least-squares design of IIR all-pass filters. Simulation results confirm that the proposed method achieves good performance as well as effectiveness.

Why we should give up the sin2ψ method

2009 ◽  
Vol 24 (S1) ◽  
pp. S16-S21 ◽  
Author(s):  
Balder Ortner
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Linear Equations ◽  
Single System ◽  
Stress Determination ◽  
Method Of Least Squares ◽  
X Ray ◽  
Stress Tensors ◽  
Course Of Action ◽  
Simultaneous Linear Equations

The sin2ψ method can be formulated as a single system of simultaneous linear equations. Using this it is easy to show that the sin2ψ method is not a least-squares method. It further helps to compare the accuracies of the stress tensors obtained by the sin2ψ method and the method of least squares. Quantitative comparisons have been made for different fictitious measurements. It is shown that the unnecessary loss in accuracy by using the sin2ψ method is quite significant and by no means negligible. The same course of action has been applied to compare the so-called Dölle-Hauk method with a least-squares method; the result is similar. Some other methods for X-ray stress determination, most often similar to the sin2ψ method, and their shortcomings are also discussed briefly, together with the corresponding, more effective and more versatile least-squares method.

scholarly journals SIMULTANEOUS SPECTROPHOTOMETRIC DETERMINATION AND CLASSICAL LEAST SQUARES METHOD: A SIMPLE EXPERIMENT TO INTRODUCE THE CONCEPT OF MULTIVARIATE CALIBRATION

Química Nova ◽  
2020 ◽  
Author(s):  
Jhonatas Carvalho ◽  
Larissa Lopes ◽  
Luciano Vidal ◽  
Poliana Santos
Keyword(s):  
Least Squares ◽  
Multivariate Calibration ◽  
Least Squares Method ◽  
Linear Equations ◽  
Natural Extension ◽  
Sunset Yellow ◽  
Absorption Data ◽  
Calibration Models ◽  
Calibration Methods ◽  
Food Dyes

We herein present an experiment where the concentrations of tartrazine, sunset yellow and amaranth in samples containing these three food dyes are determined by system of equations (SE) and classical least squares (CLS) multivariate calibration methods using light absorption data. Firstly, concentrations are obtained by means of the well-known SE method, that is, by solving a set of three linear equations in which the Beer-Lambert’s proportionality coefficients are obtained from analytical curves. Then, it is shown that the CLS method is a natural extension to SE, with an arbitrarily large number of equations. Nevertheless, within the CLS method, the unknown coefficients are found using mixtures with known concentrations of each dye. In order to introduce the students to the basics of algorithms and numerical computations, data treatment is performed in a command-line fashion using a freely available software. Advantages of multivariate calibration models over univariate ones are made clear, and the performance of the CLS and SE methods is compared based on the root-mean-square error.

A Generalization of the Orthogonal Regression Technique for Life Cycle Inventory

2012 ◽  
Vol 3 (1) ◽  
pp. 51-71 ◽  
Author(s):  
Antonino Marvuglia ◽  
Maurizio Cellura ◽  
Marcello Pucci
Keyword(s):  
Life Cycle ◽  
Least Squares ◽  
Least Squares Method ◽  
Linear Equations ◽  
Mathematical Formulation ◽  
Ordinary Least Squares ◽  
Least Square ◽  
Regression Problems ◽  
Whole Life Cycle ◽  
Interesting Development

Life cycle assessment (LCA) is a method used to quantify the environmental impacts of a product, process, or service across its whole life cycle. One of the problems occurring when the system at hand involves processes delivering more than one valuable output is the apportionment of resource consumption and environmental burdens in the correct proportion amongst the products. The mathematical formulation of the problem is represented by the solution of an over-determined system of linear equations. The paper describes the application of an iterative algorithm for the implementation of least square regression to solve this over-determined system directly in its rectangular form. The applied algorithm dynamically passes from an Ordinary Least Squares (OLS) problem to the regression problems known as Total Least Squares (TLS) and Data Least Squares (DLS). The obtained results suggest further investigations. In particular, the so called constrained least squares method is identified as an interesting development of the methodology.

scholarly journals Interval Approach to Magnetotelluric Data Processing

2021 ◽  
Vol 2096 (1) ◽  
pp. 012127
Author(s):  
A A Lavrukhin ◽  
A S Tukanova
Keyword(s):  
Data Processing ◽  
Computational Algorithm ◽  
Least Squares Method ◽  
Linear Equations ◽  
Reference Method ◽  
Impedance Tensor ◽  
System Of Linear Equations ◽  
Magnetotelluric Data ◽  
New Approach ◽  
Interval Approach

Abstract The article presents a new approach to estimate the frequency characteristics of the impedance tensor for processing magnetotelluric data. The approach is based on the applying of interval analysis methods when solving a system of linear equations. As a reference method, to compare with, a combined robust algorithm is used (with discarding data by the coherence criterion, median estimating, and weighting least squares method). This algorithm is compared with the results of the proposed interval computational algorithm that is based on the method of J. Rohn, implemented in the intvalpy Python library. Computational experiments on the data processing were performed using natural magnetotelluric field data. The interval approach can be successfully applied to the processing of magnetotelluric data.

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