Solving a positive definite system of linear equations via the matrix exponential

2011 ◽  
Author(s):  
Ammar Hasan ◽  
Eric C. Kerrigan ◽  
George A. Constantinides
Keyword(s):  
Linear Equations ◽  
Positive Definite ◽  
Matrix Exponential ◽  
System Of Linear Equations ◽  
The Matrix ◽  
Positive Definite System

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 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 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 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.

scholarly journals A New Version of the Accelerated Overrelaxation Iterative Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Shi-Liang Wu ◽  
Yu-Jun Liu
Keyword(s):  
Iterative Method ◽  
Linear Equations ◽  
Positive Definite ◽  
System Of Linear Equations ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Convergence Results ◽  
Aor Method ◽  
Symmetric Positive Definite Matrices ◽  
Overrelaxation Iterative Method

Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant,L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant,H-matrices, symmetric positive definite matrices, andL-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method.

scholarly journals Analysis of the Production Task Model With Fuzzy Information About Direct Cost Factors and the Final Product Demand

2019 ◽  
Vol 09 (4) ◽  
pp. 32-45 ◽  
Author(s):  
A.V. Panteleev ◽  
V.S. Saveleva
Keyword(s):  
Strong Solution ◽  
Direct Cost ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Production Task ◽  
Fuzzy Information ◽  
System Of Linear Equations ◽  
Fuzzy Matrix ◽  
Linear System Of Equations ◽  
The Matrix

The article discusses the study of a mathematical model of execution of the production task in the presence of fuzzy information about the matrixes of direct costs and final demand. By solving a problem with fuzzy information we mean the solution of a linear system of equations with a fuzzy matrix and a fuzzy right-hand side described by fuzzy triangular numbers in a form of deviations from the mean. In this task of search of inter-sectoral balance the LU-decomposition method for the matrix of direct cost which is further used for solving the system of linear equations is applied. A software implementation of a numerical method for finding a strong solution of a fuzzy system of linear equations consisting of two successive stages is described. At the first stage, the necessary and sufficient conditions for the existence of a strong solution are verified. At the second stage, the solution of the system is found, which is written in the form of a fuzzy matrix. The influence of the fuzzy numbers parameters on the final result was studied.

scholarly journals Matrix Diophantine equations over quadratic rings and their solutions

2020 ◽  
Vol 12 (2) ◽  
pp. 368-375
Author(s):  
N.B. Ladzoryshyn ◽  
V.M. Petrychkovych ◽  
H.V. Zelisko
Keyword(s):  
Matrix Equation ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Standard Form ◽  
Matrix Equations ◽  
System Of Linear Equations ◽  
Structure Of Solutions ◽  
Linear Diophantine Equations ◽  
The Matrix ◽  
Solution Methods

The method for solving the matrix Diophantine equations over quadratic rings is developed. On the basic of the standard form of matrices over quadratic rings with respect to $(z,k)$-equivalence previously established by the authors, the matrix Diophantine equation is reduced to equivalent matrix equation of same type with triangle coefficients. Solving this matrix equation is reduced to solving a system of linear equations that contains linear Diophantine equations with two variables, their solution methods are well-known. The structure of solutions of matrix equations is also investigated. In particular, solutions with bounded Euclidean norms are established. It is shown that there exists a finite number of such solutions of matrix equations over Euclidean imaginary quadratic rings. An effective method of constructing of such solutions is suggested.

Dissertation on “The Rank Technique” and its Application

1965 ◽  
Vol 69 (652) ◽  
pp. 280-283 ◽  
Author(s):  
John Robinson
Keyword(s):  
Linear Equations ◽  
Three Dimensional ◽  
Attractive Feature ◽  
System Of Linear Equations ◽  
Initial Development ◽  
The Matrix ◽  
Plane Frames ◽  
General Plane ◽  
Basic Characteristics ◽  
Selection Of

Summary“The Rank Technique” is a method for automatic selection of redundancies in the Matrix Force Method. The method was developed for the complete linear analysis of general plane frames, but is equally applicable to other forms of two- and three-dimensional configurations whose state can be expressed as a system of linear equations. An attractive feature of the method is that the structure is systematically and automatically investigated to determine its basic characteristics. The first point considered is whether the structure is stable or unstable for the prescribed load conditions; if stable, whether determinate or redundant and if redundant, the degree of redundancy. A consistent set of redundants is automatically isolated. For general structures the technique automatically generates the basic and redundant load systems in an indirect manner which can be made readily available, if required. The initial development of “The Rank Technique” was carried out in collaboration with Robert R. Regl and is given in reference 1.

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