For the five-point discretization of on the unit square, the weighted iteration eigenvalues are . A fixed parameter converges for every mesh when , but poorly damps the highest frequencies.
For the centered second-difference matrix with diagonal and off-diagonals , the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when .
For a three-by-three matrix with diagonal entries one and every off-diagonal entry , the Jacobi method iteration matrix has eigenvalue on the all-ones vector and eigenvalue with multiplicity two on its orthogonal complement. Its spectral radius is , so Jacobi iteration converges exactly when .
If is symmetric positive definite and tridiagonal, set . Thenis positive definite. Applying the Householder-John theorem to and proves that the Jacobi iteration converges.
Articles by others on the same topic
The Jacobi method is an iterative algorithm used to solve systems of linear equations. It is particularly useful for large sparse systems, where the matrix involved has a significant number of zero elements. The method is named after the German mathematician Carl Gustav Jacob Jacobi.