Solving System of 2n Equations

Ultimately, it's up to you and your specific needs and preferences. In summary, the conversation discusses the question of whether to leave a system of 2n equations intact or simplify it into n equations through back substitution. The decision depends on the size of n and the condition number of the resulting matrix, with the 2n case requiring more computational resources but potentially having more favorable properties. The final decision should be based on individual needs and preferences.
  • #1
defunc
55
0
Hi there,

I have a system of 2n equations. n of these unknowns can be expressed explicitly in terms of the remaining n. The resulting n x n matrix is more dense than the original 2n x 2n and the original 2n x 2n system has some desired properties like diagonal dominance etc. So, should I leave the 2n equations intact or perform the back substitutions and solve the more complex n equations?

Thanks!
 
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  • #2
Sounds like you are concerned with computational efficiency and/or accuracy. I think it depends on (1) how big n is and (2) what the condition number is of the matrix in each case. The 2n case will require roughly 8 times the computational resources versus the n case, but if the condition number is more favorable, it might be worth it.
 

What is a system of 2n equations?

A system of 2n equations is a set of n equations that are all related to each other and must be solved simultaneously. Each equation contains 2 variables, and the solution to the system is the values of the variables that make all n equations true.

Why is solving a system of 2n equations important?

Solving a system of 2n equations is important because it allows us to find the values of multiple variables at once, rather than solving for each variable individually. This can be useful in many applications, such as engineering, economics, and physics.

What methods can be used to solve a system of 2n equations?

The most common methods for solving a system of 2n equations are substitution, elimination, and graphing. Other methods, such as Gaussian elimination and Cramer's rule, can also be used for more complex systems.

What is the difference between consistent and inconsistent systems of 2n equations?

A consistent system of 2n equations has at least one solution that satisfies all n equations, while an inconsistent system has no solution that satisfies all equations. This can be determined by looking at the number of solutions and the number of equations in the system.

How can I check if my solution to a system of 2n equations is correct?

To check if a solution is correct, you can plug the values of the variables into each equation and see if they make the equation true. If all n equations are true with the given values, then the solution is correct.

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