In linear algebra, an eigenvalue problem is a problem of finding the eigenvalues and eigenvectors of a matrix. Eigenvalues and eigenvectors are special values and vectors that are associated with a linear transformation, and they play an important role in many areas of mathematics and science.

Numerical eigenvalue solvers are algorithms that are used to compute the eigenvalues and eigenvectors of a matrix numerically, rather than analytically. These solvers are typically iterative methods, which means that they compute a sequence of increasingly accurate approximations to the eigenvalues and eigenvectors. Some common numerical eigenvalue solvers include the power method, the inverse power method, and the Jacobi method.

Which solver is most appropriate for a given problem will depend on the characteristics of the matrix, the desired accuracy of the solution, and the available computational resources. It is also possible to use direct solvers, which can be more efficient for certain types of matrices, but may be more computationally expensive for very large matrices.

There are several open-source eigenvalue solvers available. Some of the more popular ones include:

1. Eigen: A C++ template library for linear algebra that includes several eigenvalue solvers.
2. LAPACK: A library of linear algebra routines that includes several eigenvalue solvers, as well as solvers for other types of linear algebra problems.
3. ARPACK: A collection of subroutines for solving large-scale eigenvalue problems.
4. ELPA: A library for the solution of large eigenvalue problems that is designed to be efficient on parallel computing architectures.
5. SLEPc: A library for the solution of large eigenvalue problems that is built on top of PETSc, a library for the solution of partial differential equations.

These solvers are available under various open-source licenses and can be used in a variety of scientific and engineering applications.