iterative/jacobi.m

   1 function [x, iterations, residual_norms] = ...
   2          jacobi(A, b, x0, tolerance, max_iterations)
   3   %
   4   % Solve the system,
   5   %
   6   %   Ax = b
   7   %
   8   % iteratively using Jacobi iterations. That is, we let,
   9   %
  10   %   A = M - N = D - (L + U)
  11   %
  12   % where D is a diagonal matrix consisting of the diagonal entries of
  13   % A (the rest zeros), and N = (L + U) are the remaining upper- and
  14   % lower-triangular parts of A. Now,
  15   %
  16   %   Ax = (M - N)x = Mx - Nx = b
  17   %
  18   % has solution,
  19   %
  20   %   x = M^(-1)*(Nx + b)
  21   %
  22   % Thus, our iterations are of the form,
  23   %
  24   %   x_{k+1} = M^(-1)*(Nx_{k} + b)
  25   %
  26   % INPUT:
  27   %
  28   %   ``A`` -- The n-by-n coefficient matrix of the system.
  29   %
  30   %   ``b`` -- An n-by-1 vector; the right-hand side of the system.
  31   %
  32   %   ``x0`` -- An n-by-1 vector; an initial guess to the solution.
  33   %
  34   %   ``tolerance`` -- (optional; default: 1e-10) the stopping tolerance.
  35   %                    we stop when the relative error (the norm of the
  36   %                    residual divided by the norm of ``b``) is less
  37   %                    than ``tolerance``.
  38   %
  39   %   ``max_iterations`` -- (optional; default: intmax) the maximum
  40   %                         number of iterations we will perform.
  41   %
  42   % OUTPUT:
  43   %
  44   %   ``x`` -- An n-by-1 vector; the approximate solution to the system.
  45   %
  46   %   ``iterations`` -- The number of iterations taken.
  47   %
  48   %   ``residual_norms`` -- An n-by-iterations vector of the residual norms
  49   %                         at each iteration. If not requested, they will
  50   %                         not be computed to save space.
  51   %
  52   save_residual_norms = false;
  53   if (nargout > 2)
  54      save_residual_norms = true;
  55      residual_norms = [];
  56   end
  57
  58   if (nargin < 4)
  59     tolerance = 1e-10;
  60   end
  61
  62   if (nargin < 5)
  63     max_iterations = intmax();
  64   end
  65
  66   % This creates a diagonal matrix from the diagonal entries of A.
  67   M = diag(diag(A));
  68   N = M - A;
  69
  70   % Avoid recomputing this each iteration.
  71   b_norm = norm(b);
  72
  73   k = 0;
  74   xk = x0;
  75   rk = b - A*xk;
  76
  77   while (norm(rk) > tolerance*b_norm && k < max_iterations)
  78     x_next = M \ (N*xk + b);
  79     r_next = b - A*x_next;
  80
  81     k = k + 1;
  82     xk = x_next;
  83     rk = r_next;
  84     if (save_residual_norms)
  85       residual_norms = [ residual_norms; norm(rk) ];
  86     end
  87   end
  88
  89    iterations = k;
  90    x = xk;
  91 end