Scripts for generating positive definiton inputs

[hcq9102]

Start Clean up:

Clean-up1:

Issue :The original input matrix is pascal matrix greatly limited the input matrix size(max matrix size will be 18*18).
Solution: This py script is for generating Positive Definition Matrix as input which will be used in benchmark part in each cholesky cpp files.

In this way, we can definitely get positive-definite matrix after the verification: Line30

[mhaseeb123]

Thank you for the contribution. I have a few comments that I feel like should be addressed before merging into main:

1. Instead of generating np.rand, consider methods that generate positive definitive matrices with high probability such as: https://math.stackexchange.com/questions/357980/how-to-generate-random-symmetric-positive-definite-matrices-using-matlab

2. Consider using a computationally inexpensive method such as checking eigen values to verify positive definitive instead of cholesky itself to avoid heavy penalty for larger matrices

3. Please ensure the script doesn't end with an error and instead loops until a good matrix is generated.

4. Please use argparse module to take matrixsize as input to the script instead of hardcoding,

[mhaseeb123]

Hardcoding this is not a good idea. Consider using argparse module and use that to take input from the user on the matrix size to generate.

[mhaseeb123]

print(A) for large matrixSize would just flood the console

[mhaseeb123]

Same comment for large matrixSize for B

[mhaseeb123]

It looks like the script will either generate or not generate a positive definitive matrix when run so that should be in a loop until we do get a correct solution.

Also using Cholesky transformation itself to check if the generated matrix is positive definitive doesn't seem to be the correct solution especially for larger matrices. I would recommend using a cheaper (eigen value based or Gershgorin circle theorem to verify this). You can refer to Gilbert Strang's book/videos or online resources for this.

[mhaseeb123]

I would recommend not generating matrices using rand here as the probability of generating a wrong matrix may get increasingly higher for larger matrices. Instead I would recommend looking into methods to generate positive definitive matrices with high probability such as translating this matlab code to python: https://math.stackexchange.com/questions/357980/how-to-generate-random-symmetric-positive-definite-matrices-using-matlab

[hcq9102]

agree, I have another helper function in tiled_cholesky implementation which can generate the pd matrix using blas lib.
I suggest to leave this PR alone at this moment(I usually use this when i did performance test before).
And we can close this pr when get the better method.