The Lanczos algorithm helps make finding eigenvalues for large sparse matrices much easier. Here’s how it works:

1. Less Work to Calculate:

   • The algorithm only needs about $O(n^2)$ calculations every time it runs. Here, $n$ is the size of the matrix. This makes it faster, especially when $n$ is big.

2. Saves Memory:

   • It only has to keep track of a few vectors. This means it uses much less memory than other methods.

3. Fast Results:

   • The eigenvalues show up quickly. Most of the time, it only takes around $k \approx 10$ runs to get really accurate results.

4. Works Well with Sparse Matrices:

   • It is designed to take advantage of sparse matrices, which means lots of zeros in the data. This can make calculations up to 90% faster in real-world use!