Calculation of simultaneous equations

In summary, there are various methods for solving simultaneous equations using a computer, such as numerically or analytically using Cramer's rule. Gaussian elimination is a common method for solving linear systems, but may not be the most efficient for larger systems. Non-linear systems require more complicated methods, such as linearizing and solving iteratively. The vortex panel method in Java involves converting a matrix to triangular form using elementary row operations. For multiple systems with the same matrix, LU decomposition can be used. Other factors to consider are round-off errors and combining iterative and direct methods.
  • #1
sid_galt
502
1
How are simultaneous equations calculated using a computer?
 
Mathematics news on Phys.org
  • #2
By a solver of some type.
Be a bit more specific, please.
 
  • #3
There are lots of different methods. For example, numerically. Here is an analytical method I wrote using Cramer's rule to invert the matrix. Delphi source code included.
http://www.geocities.com/peterbone3/LinearEquations.zip
I assume you're familiar with the matrix representation of a set of linear equations.

Peter Bone
 
  • #4
I wouldn't disagree with there being a lot of methods, peterbone; we agree on that.
I was sort of thinking what kinds of simultaneous equations OP had in mind, in particular if I needed to go into the whole mucky business of solvers involving non-linear loop structures and so on..:yuck:


At the end, of course, since we really only can solve systems of linear equations, one ought perhaps to focus on solutions techniques for linear systems, as you did.
 
Last edited:
  • #5
Massive parallel processor supercomputers are typically used to effect simultaneous processing.
Of course, the hard part is encoding correctly.
 
  • #6
sid_galt said:
How are simultaneous equations calculated using a computer?
As long as the equations are linear, you can solve them by using Gaussian elimination on the augmented matrix. It's not the fastest way to do it, especially for larger systems of equations, but it's the most intuitive technique, IMO, and pretty simple to code up. Are you looking to write something for yourself to help you learn more about linear algebra?

Or, as was stated earlier, if your systems of equations contain non-linear terms, things get a lot more complicated...
 
  • #7
There are pretty efficient sparse solvers available in many codes (and the implementations aren't that many lines of code actually), when it gets bigger in many problems iterative solvers can produce a nice performance increase (and lessen e.g. storage requirements) ... lots of stuff available about this topic.
 
  • #8
At the end, of course, since we really only can solve systems of linear equations

What do you mean by that?
 
  • #9
peterbone said:
There are lots of different methods. For example, numerically. Here is an analytical method I wrote using Cramer's rule to invert the matrix.
I hope you know the evils of cramer's rule. It is bad news.
 
  • #10
Hurkyl said:
What do you mean by that?


Probably that most nonlinear systems are "solved" by linearizing over and over and solving the linear systems.
 
  • #12
Hurkyl said:
What do you mean by that?
As LeBrad mentioned, there aren't any standard, foolproof solution procedure for non-linear equations in general in the way there are for linear systems.

I'm not saying that various types of iteration processes are "wrong".
Most definitely, they are not.
 
  • #13
Thank you for all the replies.

Actually I am looking forward to implementing the vortex panel method in Java. Thing is, methods like Gaussian elimination are fine, only I am finding it difficult how to encode the appropriate elementary row operations to convert the matrix to a triangular one.
 
  • #14
sid_galt said:
Thank you for all the replies.

Actually I am looking forward to implementing the vortex panel method in Java. Thing is, methods like Gaussian elimination are fine, only I am finding it difficult how to encode the appropriate elementary row operations to convert the matrix to a triangular one.
You just need to think of about Gaussian elimination in a systematic way. That is for an arbitrary matrix at an arbitrary stage in reduction what should be done next. A few whiles and fors and it is done. Other things to think about are if you need to solve multiple systems having the same matrix you might consider using a LU decomposition. To help with round off you might consider full or partial pivoting, or conbining iterative and direct methods.
 

1. What are simultaneous equations?

Simultaneous equations are equations that have two or more unknown variables and must be solved simultaneously in order to find the values of all the variables.

2. How do you solve simultaneous equations?

There are various methods for solving simultaneous equations, including substitution, elimination, and graphing. The most commonly used method is substitution, where one of the equations is solved for one variable and then substituted into the other equation.

3. Why is it important to solve simultaneous equations?

Solving simultaneous equations allows us to find the values of multiple variables in a system of equations, which is useful in solving real-world problems in fields such as physics, economics, and engineering.

4. Can simultaneous equations have more than two variables?

Yes, simultaneous equations can have any number of variables. However, the number of equations must be equal to the number of variables in order to find a unique solution.

5. Are there any tips for solving simultaneous equations?

One tip for solving simultaneous equations is to always check your solution by plugging it back into the original equations. Another tip is to eliminate fractions by multiplying both sides of an equation by the common denominator.

Similar threads

Replies
4
Views
710
  • General Math
Replies
2
Views
718
  • Special and General Relativity
Replies
25
Views
847
  • General Math
Replies
11
Views
1K
  • Special and General Relativity
2
Replies
51
Views
2K
Replies
1
Views
854
  • Special and General Relativity
2
Replies
38
Views
2K
Replies
13
Views
1K
  • General Math
2
Replies
44
Views
3K
  • Special and General Relativity
Replies
2
Views
211
Back
Top