Numerical linear solvers are algorithms that are used to solve systems of linear equations numerically, rather than analytically. These systems can be represented in the form Ax = b, where A is a matrix of coefficients, x is a vector of unknown variables, and b is a vector of constant terms. Some common numerical linear solvers include:

1. Gaussian elimination
2. LU decomposition
3. QR decomposition
4. Conjugate gradient
5. Jacobi method
6. Gauss-Seidel method
7. Successive over-relaxation (SOR)

Which solver is most appropriate for a given problem will depend on a variety of factors, including the size and structure of the matrix A, the desired accuracy of the solution, and the available computational resources. It is also possible to use iterative methods, which can be more efficient for very large systems, but may require more iterations to converge to the solution.

A direct solver is an algorithm that computes the exact solution to a system of linear equations in a finite number of steps. These methods typically work by constructing a decomposition of the matrix A (such as an LU decomposition) and using it to efficiently solve the system Ax = b. Direct solvers are generally more accurate and reliable than iterative solvers, but they can be computationally expensive for very large systems, as the time complexity is typically O(n^3) or higher.

Iterative solvers, on the other hand, do not compute the exact solution to the system, but rather compute a sequence of increasingly accurate approximations. These methods work by starting with an initial guess for the solution and repeatedly refining it until the desired accuracy is reached. Iterative solvers are generally faster than direct solvers for large systems, but they may require a large number of iterations to converge, and the accuracy of the solution may be less predictable. Some common iterative solvers include the conjugate gradient method and the Gauss-Seidel method.

MUMPS (MUltifrontal Massively Parallel Sparse direct Solver) is a software package for solving large, sparse systems of linear equations. It is a direct solver, meaning that it computes the exact solution to the system in a finite number of steps. MUMPS is designed to be highly efficient and scalable, and can take advantage of parallel computing architectures to solve very large systems quickly. MUMPS is widely used in a variety of scientific and engineering applications, and has been incorporated into many software packages and libraries.

PARDISO (Parallel Direct Sparse Solver) is another software package for solving large, sparse systems of linear equations. Like MUMPS, it is a direct solver that is designed to be highly efficient and scalable, and can take advantage of parallel computing architectures. PARDISO is also widely used in scientific and engineering applications, and is available for a variety of platforms.

SuiteSparse is a suite of libraries for numerical linear algebra that includes several solvers for sparse systems of linear equations. One of the solvers included in SuiteSparse is CHOLMOD, a direct solver based on the Cholesky decomposition. CHOLMOD is designed to be efficient and scalable, and can take advantage of parallel computing architectures.

Eigen is a C++ template library for linear algebra, which includes several numerical linear algebra functions, including solvers for systems of linear equations. One of the solvers included in Eigen is the LDLT decomposition, which is a direct solver based on the LDLT decomposition of the matrix A. Eigen is designed to be fast and efficient, and is widely used in scientific and engineering applications.