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 Matrix and vector sequence transformations revisited

1995 ◽  
Vol 38 (3) ◽  
pp. 495-510 ◽  
Author(s):  
C. Brezinski ◽  
A. Salam
Keyword(s):  
Linear Equations ◽  
Convergence Acceleration ◽  
Difference Operators ◽  
Scalar Case ◽  
Extrapolation Methods ◽  
System Of Linear Equations ◽  
Vector Sequence ◽  
The Matrix ◽  
Sequence Transformations

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two different approaches which are equivalent. The first one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation difference operators. In this paper, these two approaches are generalized to the matrix and the vector cases.

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 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.

scholarly journals MATHEMATICAL MODELING OF THE 3(Н)-QUINAZOLIN-4-ONЕ SYNTHESIS PROCESS

2021 ◽  
pp. 51-57
Keyword(s):  
Mathematical Modeling ◽  
Reaction Rate ◽  
Linear Equations ◽  
Product Yield ◽  
Farm Animals ◽  
Synthesis Process ◽  
Optimal Conditions ◽  
System Of Linear Equations ◽  
Molar Ratios ◽  
The Matrix

The aim is to optimize the conditions for the synthesis of 3(H)-quinazolin-4-one by the method of mathematical modeling to develop a technology for producing the substance of a new domestic drug used in the treatment of farm animals from helminths. In mathematical modeling, the method of a small number of squares was used. Analytical dependences of the product yield on temperature, reaction time, and molar ratios of the starting materials were determined. A system of linear equations has been compiled. The system of linear equations was performed by the matrix method (A, B, C).The average reaction rate was determined. Based on the results obtained, a 3(H)-quinazolin-4-one diagram using the Maple 18 program and an icon diagram of the reaction duration, temperature, and reaction rate are shown. Based on the results of mathematical modeling, a highly efficient technological scheme for obtaining 3(H)-quinazolin-4-one has been developed. Based on this technology, compound 3(H)-quinazolin-4-one was synthesized in quantitative products at the Institute of Plant Chemistry, at a pilot production plant.The results obtained confirmed the found optimal conditions

scholarly journals HERMITE POLYNOMIAL APPROACH FOR SOLVING SINGULAR PERTURBATED DELAY DIFFERENTIAL EQUATIONS

2020 ◽  
Vol 20 (4) ◽  
pp. 845-854
Author(s):  
SUAYIP YUZBASI ◽  
NURCAN BAYKUS SAVASANERIL
Keyword(s):  
Algebraic System ◽  
Hermite Polynomial ◽  
Linear Equations ◽  
Hermite Polynomials ◽  
System Of Linear Equations ◽  
Collocation Points ◽  
The Matrix ◽  
Collocation Approach ◽  
Delay Differential ◽  
Derivatives Of

In this study, a collocation approach based on the Hermite polyomials is applied to solve the singularly perturbated delay differential eqautions by boundary conditions. By means of the matix relations of the Hermite polynomials and the derivatives of them, main problem is reduced to a matrix equation. And then, collocation points are placed in equation of the matrix. Hence, the singular perturbed problem is transformed into an algebraic system of linear equations. This system is solved and thus the coefficients of the assumed approximate solution are determined. Numerical applications are made for various values of N.

scholarly journals L1 Norm Solution of Overdetermined System of Linear Equations

2019 ◽  
Vol 1 (1) ◽  
pp. 19-30
Author(s):  
Bijan Bidabad
Keyword(s):  
Linear Equations ◽  
Computer Programs ◽  
System Of Linear Equations ◽  
Overdetermined System ◽  
L1 Norm ◽  
Weighted Median

In this paper, three algorithms for weighted median, simple linear, and multiple m parameters L1 norm regressions are introduced. The corresponding computer programs are also included.   

scholarly journals Improved Heuristics for Short Linear Programs

2019 ◽  
pp. 203-230
Author(s):  
Quan Quan Tan ◽  
Thomas Peyrin
Keyword(s):  
Random Matrices ◽  
Selection Criteria ◽  
Block Cipher ◽  
State Of The Art ◽  
Linear Equations ◽  
Linear Programs ◽  
System Of Linear Equations ◽  
New Methods ◽  
The Matrix

In this article, we propose new heuristics for minimising the amount of XOR gates required to compute a system of linear equations in GF(2). We first revisit the well known Boyar-Peralta strategy and argue that a proper randomisation process during the selection phases can lead to great improvements. We then propose new selection criteria and explain their rationale. Our new methods outperform state-of-the-art algorithms such as Paar or Boyar-Peralta (or open synthesis tools such as Yosys) when tested on random matrices with various densities. They can be applied to matrices of reasonable sizes (up to about 32 × 32). Notably, we provide a new implementation record for the matrix underlying the MixColumns function of the AES block cipher, requiring only 94 XORs.

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