Matrix Inversion Cholesky Factorization -> results not exact [closed]

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I am calculating the inverse of a square matrix using different libraries via Cholesky factorization. However my results are not as I was expecting. I am not an expert in maths, but I was expecting to get a closer result.

I am using MLK, magma and CULA libraries to calculate the inverse of a matrix in CPU and GPUs. After doing the calculation those libraries I've noticed the results always differ in one element. Say I want to calculate the inverse of A= [0.237306,0.000458;0.000458,0.238497]:

A[0] = 0.237306 
A[1] = 0.000458 
A[2] = 0.000458 
A[3] = 0.238497

The result I obtain is:

inv(A)[0] = 4.213983 
inv(A)[1] = -0.008092 
inv(A)[2] = 0.000458 
inv(A)[3] = 4.192946 

However, the correct result should be

   4.2139841  -0.0080924
  -0.0080924   4.1929404

As you can see, inv(A)[3] is different, although the rest of them are fine. Is that how Cholesky Inversion should work? Is this a correct/approximate result or am I doing something wrong here?

Answer 1

I understand what is going on. This libraries calculate the inverse of the matrix, upper or lower as Alexandre C pointed out. So depending on the argument you pass to the library (Upper or Lower), it calculates the inverse of the upper or lower side of the matrix. I thought you could calculate the complete inverse matrix, but apparently it is not possible this way.

Answer 2

It's probably due to the IEEE spec for floating point numbers and the imprecision that's built into working with them.

You don't say if you've specified 32-bit or 64-bit floating point numbers. Try increasing your precision.

Another point to make is that people don't usually calculate the inverse of a matrix because it's prone to suffer from round off errors. A better choice is LU decomposition and forward back substitution.

Where did you get the "correct" answer from? How are you judging the correctness of your result?

I know it's not the best tool for numerical methods, but if I take your matrix:

0.237306 0.000458 0.000458 0.238497

and run it through MINVERSE, I get this answer:

4.213984 -0.00809 -0.00899 4.19294

I can check its validity by multiplying the original and the inverse to see if I get an identify matrix back. Here's what I got out of MMULT:

1 2.1684e-19 -2.168e-19 1

That's what a numerical methods person would call an identity matrix.

So I say the answer I get is correct. You should check yours in the same way, and read What Every Computer Scientist Should Know About Floating Point Arithmetic.