README.md

Fast solver to find the extreme eigenvalue

I present a simple code that allows one to compute the maximum and minimum eigenvalues of a positive definite matrix in Python. The code is faster than scipy.sparse.linalg.eigsh, providing results that converge while scipy.sparse.linalg.eigsh doesn't.

Example code can be seen in the Example.ipynb file. A description of the method used can be seen in Eigenvalues.pdf. To ensure accuracy up to some error $\varepsilon$ see Guaranteed Accuracy.ipynb.

The following pictures are the results when trying to approximate the maximum and minimum eigenvalues of a positive definite $1000\times 1000$ matrix with eigenvalues logarithmically from $10^{-4}$ to $10^4$. We remark that in this example, scipy.sparse.linalg.eigsh fails to converge for the minimum eigenvalue. The code for this can be found in utils.py and Tests.ipynb.