ARPACK

ARPACK, the ARnoldi PACKage, is a numerical software library written in FORTRAN 77 for solving large scale eigenvalue problems in the matrix-free fashion.

The package is designed to compute a few eigenvalues and corresponding eigenvectors of large sparse or structured matrices, using the Implicitly Restarted Arnoldi Method (IRAM) or, in the case of symmetric matrices, the corresponding variant of the Lanczos algorithm. It is used by many popular numerical computing environments such as SciPy, Mathematica, GNU Octave and MATLAB to provide this functionality.

Reverse Communication Interface
A powerful matrix-free feature of ARPACK is its ability to use any matrix storage format. This is possible because it doesn't operate on the matrices directly, but instead when a matrix operation is required it returns control to the calling program with a flag indicating what operation is required. The calling program must then perform the operation and call the ARPACK routine again to continue. The operations are typically matrix-vector products, and solving linear systems.

Fork
Due to stalled upstream development, ARPAСK has been forked into ARPACK-NG, as a form of a collaborative effort of the various groups that rely on ARPACK.