It is reported that Groebner basis is a powerful method for new applications in engineering. It would be exciting to learn this method in a matrix formulation and whether it is advantageous computationally.
I don't know of a matrix formulation and I don't think there is one because the problems it solves are not always linear. It's better to think of it as a constraint satisfaction algorithm. At its core, you give it a set of polynomials the algorithm gives out something that you can think of as the "least common denominator" (much generalized since the polynomials are multi-variable).
To use the algorithm, you have to encode your problem as a constraint-satisfaction problem. For example, you can encode everything that needs to be true as polynomials, and ask the question of whether all of the polynomials can be satisfied at the same time. For example, with Sedoku you make polynomials for each of the squares, and specify the values sum to 9. Asking the question of whether all of these can be satisfied is then equivalent to whether the Groebner basis has a 1 in it. I included a link to a detailed (at the undergraduate level, I wrote this many years ago) example of using Groebner basis algorithms to do automated geometric theorem proving. With these examples in mind, you should be able to map the algorithm to what you need it for.
Note that the problem of solving for Groebner bases is NP-complete, and so the worst case scenario for the run times is very poor. However, "semi-fast" algorithms in specific cases exist (and are implemented in most computational algebra programs) because the ubiquity of the technique makes this a major area of research.
http://chrisrackauckas.com/assets/Papers/ChrisRackauckas-AutomatedGeometricTheoremProving.pdf
@Chris Rackauckas Thank you very much for your valuable reply..Will examine how we can use it in circuit analysis.
Cheers
Groebner basis arise in the study of polynomial representation of partial differential equations (PDE). I'm not sure about where you could use PDE in electrical circuits unless you do not assume lumped parameters. If you assume distributed parameters (L,C,R) as functions of space and time , i.e, L=L(x,t), or L=L(x,,y,,z,t) you colud use PDE.
@Ricardo Zavala-Yoe thank you for a different view. Yes, PDE occurs in transmission line voltage/ current waves. However i was thinking of proving trivial cases of lumped circuits. For instance a balanced bridge circuit seems to have some connection with the bisector of a parallelogram (Ricardo Zavala-Yoe ·) above. a conjecture then, is if Groebner basis can be used to prove conrollability /observability in control systems.
More interesting would be to see if it helps in Power system transient stability equation (Differential algebraic equation) nonlinear circuits ( ferro resonance ) etc.
Cheers
You have to dive in Prof Jan Willems' page (RIP) who was an enormous researcher in these fields (he accepted me originally as his PhD student and later he got retired but I kept constant contact). See: http://homes.esat.kuleuven.be/~jwillems/Lectures/2002/lln.pdf That may help.
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