Linear Algebraic System Solution for Known and constrained variables

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Discussion Overview

The discussion revolves around solving a linear algebraic system represented as K*X=F, particularly focusing on scenarios where some variables in X are known or constrained relative to each other. Participants explore methods for incorporating these known values and constraints into the solution process.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that a brute force approach could involve adding more equations to the system, thereby enlarging the matrices K and F, but expresses uncertainty about the efficiency of standard software routines in handling such modifications.
  • Another participant mentions that known values of X can be used to modify the matrices K and F without adding additional equations, allowing for the use of elimination or other solution methods.
  • A participant raises a concern about the effectiveness of Gaussian elimination in ensuring that specific known values, such as x_{13} = 25, are achieved, citing potential roundoff errors with pseudoinverse calculations.
  • One participant expresses a desire for a matrix-oriented method to implement substitutions for known values and constraints but admits to not knowing one offhand.
  • Another participant introduces linear programming as a possible avenue for addressing other types of constraints within the system.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the best method for solving the system with known or constrained variables, as multiple approaches and uncertainties are discussed.

Contextual Notes

There are limitations regarding the assumptions about the capabilities of software for solving linear systems, as well as the potential for roundoff errors in numerical methods. The discussion also reflects a lack of clarity on contemporary methods versus older techniques.

Ronankeating
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Dear All,

If I have the linear algebraic system where its composed as of matrices in that form K*X=F, what column/row operations should I perform if I want to solve it where some of the X variables are known (targeted values) or if I want to solve when variables are constrained relative to each other.

For example, say that I want to solve linear algebraic equations(K*X=F) when its constrained (e.g. X5=X12=X125=X128) or when it has targeted values such as (X13=25; X23=5.4; X33=13 etc...)

Your help will be appreciated!

Regards,
 
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That's a good question! Off the top of my head, the brute force approach would be to add more equations and enlarge the matrix K and the vector F so they have more columns. That would work if you have software that solve systems where there are more equations than unknowns. For example x_5 = x_{12} is expressed by the equation x_5 - x_{12} = 0. Of course the "target" x_{13} = 25 is itself an equation.

However, I don't know that the average software routine for solving linear equations can handle the problem efficiently. For example, if it uses Gaussian elimination, I don't know that it would perform the operations in an order that would make the answer for x_{13} exactly equal to 25. If the software computes the answer by finding the pseudoinverse of a non-square matrix then roundoff errors might be a problem.

Obviously the proper way to do things would be to use substitutions and rewrite your original system of equations so its has fewere variables. You seem to be asking for a matrix oriented method of implementing the substitutions. Off hand, I don't know one. I'll have to think about it.
 
Your question about how to apply known values of X when solving the matrix equation K*X = F has been discussed recently in this thread:

https://www.physicsforums.com/showthread.php?t=691663

In short, using the known values of X, the matrices K and F can be modified without adding additional equations such that the original matrix equation can be solved using elimination or whatever solution method you choose. The attached pdf in the thread will illustrate the procedure.
 
SteamKing said:
Your question about how to apply known values of X when solving the matrix equation K*X = F has been discussed recently in this thread:

https://www.physicsforums.com/showthread.php?t=691663

Yes SteamKing you've answered this question in the shown thread and thank you for your help, but if you remember I was looking additionally for the constrained type solution also where I couldn't find entry point.

What inspires me actually to ask those question again, I still do have belief deep down inside my brain that I'm following very old methods for this, instead of contemporary and robust ones.
 
For other types of constraint, one possible source of a solution would be to study a branch of mathematics called 'linear programming'.
 

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