Algorithm 641: Exact Solution of General Systems of Linear Equations

1986 ◽  
Vol 12 (2) ◽  
pp. 149
Author(s):  
Jörn Springer
Keyword(s):  
Exact Solution ◽  
Linear Equations ◽  
General Systems ◽  
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.

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