# Sparse matrix computational difficulties

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?

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

## 1 Answer

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.

• This doesn't explain why all the entries in the output are the same, though. – littleO Aug 4 '18 at 20:56
• My guess would be that this number is the “infinity” of used software. – pointguard0 Aug 4 '18 at 21:00
• I used intellj idea to write down java code and use universal java matrix package to invert this matrix. – Encipher Aug 4 '18 at 21:38
• 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. – copper.hat Aug 4 '18 at 22:08
• math.stackexchange.com/questions/2872771/… Please go through this link where I explain my code. – Encipher Aug 5 '18 at 9:23