Jacobi and Gauss-Seidel Iteration

Warning.  

    Iteration does not always converge.  A sufficient condition for iteration to Jacobi iteration to converge is that A is strictly diagonally  dominant. The following subroutine will check to see if a matrix is strictly diagonally dominant.  It should be used before any call to Jacobi iteration or Gauss-Seidel iteration is made.  There exists a counter-example for which Jacobi iteration converges and Gauss-Seidel iteration does not converge.  The "true" sufficient condition for Jacobi iteration to converge is that the "spectral radius" of     is less than 1, where   is the diagonal of  .  That is, the magnitude of the largest eigenvalue of M must be less than 1.  This condition seems harsh because numerical computation of eigenvalues is an advanced topic compared to solution of a linear system.  

 

More efficient subroutines

    A tolerance can be supplied to either the Jacobi or Gauss-Seidel method which will permit it to exit the loop if convergence has been achieved.  

 

Mathematica Subroutine (Jacobi Iteration).

Mathematica Subroutine (Gauss-Seidel Iteration).

 

Subroutines using matrix commands

    In the Jacobi subroutine we can use fix point iteration as suggested by the theory.  

Mathematica Subroutine (Jacobi Iteration).