Exact solution of general integer systems of linear equations

1986 ◽  
Vol 12 (1) ◽  
pp. 51-61 ◽  
Author(s):  
Jörn Springer
Keyword(s):  
Exact Solution ◽  
Linear Equations ◽  
Systems Of Linear Equations

The use of negative penalty functions in linear systems of equations

2006 ◽  
Vol 462 (2074) ◽  
pp. 2965-2975 ◽  
Author(s):  
Harm Askes ◽  
Sinniah Ilanko
Keyword(s):  
Exact Solution ◽  
Linear Systems ◽  
Vibration Analysis ◽  
Linear Equations ◽  
Penalty Functions ◽  
Penalty Parameter ◽  
Systems Of Equations ◽  
Mathematical Proofs ◽  
Systems Of Linear Equations ◽  
Linear Systems Of Equations

Contrary to what is commonly thought, it is possible to obtain convergent results with negative (rather than positive) penalty functions. This has been shown and proven on various occasions for vibration analysis, but in this contribution it will also be shown and proven for systems of linear equations subjected to one or more constraints. As a key ingredient in the developed arguments, a pseudo-force is identified as the derivative of the constrained degree of freedom with respect to the inverse of the penalty parameter. Since this pseudo-force can be proven to be constant for large absolute values of the penalty parameter, it follows that the exact solution is bounded by the results obtained with negative and positive penalty parameters. The mathematical proofs are presented and two examples are shown to illustrate the principles.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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