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I am a computer science student. But I start writing here because my problem is not code related. It is purely computational related.

I am trying to calculated inverse of a large (e.g., $2000 \times 2000$) square matrix. This matrix is band diagonal matrix.

The non zero elements are very small numbers (e.g: $1--6$). They never exceed 6.

Now when I invert this matrix the result of inversion are all same number displayed.

Output:

9.711E013  9.711E013  9.711E013  9.711E013....................
9.711E013  9.711E013  9.711E013  9.711E013......................

9.711E013  9.711E013  9.711E013  9.711E013.....................
..............................................................

 9.711E013  9.711E013  9.711E013  9.711E013

My question is that why this type of output I get? What is the mathematical reason behind this?

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    $\begingroup$ what are you using to invert it? Matlab, for example, should be easily inverting this. $\endgroup$ – dezdichado Aug 4 '18 at 20:28
  • $\begingroup$ I am using universal java matrix package for matrix inversion. I write down this code in java. $\endgroup$ – Encipher Aug 4 '18 at 21:36
  • $\begingroup$ You should post your code, I don't think we have enough information to see what's going wrong. $\endgroup$ – littleO Aug 5 '18 at 2:26
  • $\begingroup$ math.stackexchange.com/questions/2872771/… Please go through this link. $\endgroup$ – Encipher Aug 5 '18 at 9:23
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EDITED: thanks to comment of copper.hat.

The mathematical/numerical reason is that the condition number defined as $$ \varkappa(A) = \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)} $$ is too large, say of order $10^{10}$ or even more. $\sigma_{\min, \max}(A)$ is the minimal and maximal singular values of matrix $A$.

The case of small determinant is a particular case and in some cases implied from the argument above.

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  • $\begingroup$ This doesn't explain why all the entries in the output are the same, though. $\endgroup$ – littleO Aug 4 '18 at 20:56
  • $\begingroup$ My guess would be that this number is the “infinity” of used software. $\endgroup$ – pointguard0 Aug 4 '18 at 21:00
  • $\begingroup$ I used intellj idea to write down java code and use universal java matrix package to invert this matrix. $\endgroup$ – Encipher Aug 4 '18 at 21:38
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    $\begingroup$ A small determinant does not necessarily translate into ill conditioned. Generally the issue is because the condition number is large, which is related to the singular values, not the eigenvalues. $\endgroup$ – copper.hat Aug 4 '18 at 22:08
  • $\begingroup$ math.stackexchange.com/questions/2872771/… Please go through this link where I explain my code. $\endgroup$ – Encipher Aug 5 '18 at 9:23

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