What are the methods used for computing eigenvalues and eigenvectors numerically?

There are several methods used for computing eigenvalues and eigenvectors numerically in the field of numerical analysis. Some of the commonly used methods include:

1. Power Iteration Method: This method is used to find the dominant eigenvalue and its corresponding eigenvector. It involves iteratively multiplying a vector by a matrix and normalizing the result until convergence is achieved.

2. Inverse Iteration Method: This method is used to find eigenvalues close to a given value. It involves iteratively solving a linear system of equations using the matrix and the shifted eigenvalue, and then normalizing the resulting eigenvector.

3. QR Algorithm: This method is an iterative algorithm that computes all eigenvalues and eigenvectors of a matrix. It involves decomposing the matrix into a product of an orthogonal matrix and an upper triangular matrix, and then repeating the process until convergence is achieved.

4. Jacobi Method: This method is used to find all eigenvalues and eigenvectors of a symmetric matrix. It involves iteratively applying orthogonal transformations to the matrix to diagonalize it.

5. Lanczos Algorithm: This method is used to find a few eigenvalues and eigenvectors of a large sparse matrix. It involves iteratively constructing a tridiagonal matrix that is similar to the original matrix, and then applying the QR algorithm to find the desired eigenvalues and eigenvectors.

6. Arnoldi Iteration: This method is used to find a few eigenvalues and eigenvectors of a large sparse matrix. It involves iteratively constructing an orthogonal basis for the Krylov subspace of the matrix, and then applying the QR algorithm to find the desired eigenvalues and eigenvectors.

These methods vary in terms of their efficiency, accuracy, and applicability to different types of matrices. The choice of method depends on the specific problem at hand and the characteristics of the matrix being analyzed.