Changing roles in linear system for stability

Math and Physics Programming

Started by kevop December 26, 2012 12:37 PM

6 comments, last by DonTzzy 11 years, 3 months ago

kevop

111

Author

December 26, 2012 12:37 PM

I have a linear system like: AX = B

Normally to measure how stable the solution will be for such a system condition numbers are used.

With X being the only unknown the condition number will be checked for A normally.

However in this case A only defines the locations at which observations in B are made.

So checking the condition number and reducing it over multiple observations of A does not benefit me much as observed B at those locations might still be contributing to errors significantly in estimated X.

Instead if I check the condition number of B and try to reduce it over multiple observations by selecting A that reduces condition number of B I can more reliably estimate X.

But the issue is to justify the use of evaluating condition number for B which is not involved in inversion in the above linear system.

Otherwise it seems incorrect to evaluate the condition number of B instead of A.

Normally condition numbers are defined to be the measure of relative error in X divided by the relative error in B etc.

It shows they are related so there must be a way to justify the use of minimizing condition number of B instead of A in this system and I want to know of some opinion or suggestions for this.

HappyCoder

5,076

December 26, 2012 05:38 PM

Let me see if I understand what you mean. So you are trying to solve for x in Ax=b. Where A is a nxn matrix and x and b are nx1 matrices. And you want to do this without inverting A?

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kevop

111

Author

December 26, 2012 06:20 PM

I want to test the stability of system using a non-invertible matrix B of the system instead of A, when X is unknown and A, B are both known.

I am not interested in the solution directly but want to reduce error in the resulting solution by reducing condition number of B instead of A.

All of them are matrices: X = 4 x M, A = 4 x N, B = N x M.

alvaro

21,604

December 26, 2012 06:41 PM

All of them are matrices: X = 4 x M, A = 4 x N, B = N x M.

That doesn't seem right. If you can multiply A times X, the number of columns in A has to match the number of rows in X.

kevop

111

Author

December 26, 2012 06:51 PM

All of them are matrices: X = 4 x M, A = 4 x N, B = N x M.
That doesn't seem right. If you can multiply A times X, the number of columns in A has to match the number of rows in X.

The transpose can take care of such issues. They are not important but if you wish it can be rewritten as:

A = N x 4

X = 4 x M

B = N x M

HappyCoder

5,076

December 26, 2012 10:21 PM

There are an infinite number of solutions to what you are trying to solve when N < 4. For example, let N and M be 1. That leaves us with 1x4 * 4x1 = 1x1. So Ax=b would basically be b = A1 * x1 + A2 * x2 + A3 * x3 + A4 * x4. For any given value of b, there are an infinite number of solutions of x that satisfy Ax=b. This is the case for any value of N < 4. When N = 4, x can be solved simply by calculating A-1b. When N > 4 and M=1, you have an overdetermined system. You can use Least Squares for that. I am not sure if least squares will work for you if M > 1.

EDIT: Upon further thought, I think least squares is what you are looking for. It should work when M > 1.

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kevop

111

Author

December 27, 2012 06:46 AM

Well I am not interested in the solution which is trivial to obtain but in the quality of future solutions.

Let me put it this way:

There are a total of H locations at which observations can be made.

Then if I have initially N locations in A (N >=4) at a given time then which of location from set H that are not in A will be most optimal for addition to A at N+1.

That is when you use condition numbers to test the quality of possible future solutions for addition into A.

So the point is to test the quality of the future solution not using location in A but using the observation in B. By checking which one observation from H can improve the quality of future solution without ever solving the least square itself.

However checking condition number of B would be incorrect as it is not involved in inversion when finding X.

So there must a reason to relate condition number of A and B together for this linear system.

DonTzzy

487

January 09, 2013 10:58 PM

http://en.wikipedia.org/wiki/Preconditioner

Changing roles in linear system for stability

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