function newx = jacobiStep(x, A, b)
% Arguments
%   x = an n-vector, either in column or row format;
%       This is the current iterate
%   A = an n-by-n matrix
%   b = an n-vector, either in column or row format
%
% Return value: an n-vector (column format);
%       This is the next Jacobi iterate for the system Ax = b

  % Check that every input is of the same dimension.
  [m, n] = size(A);
  if (m ~= n || length(x) ~= n || length(b) ~= n)
    error('Matrix/vector dimensions not appropriate');
    return;
  end
  
  % The following loop implements equation (2.6.3) from p. 71
  for i = 1:n
    indices = [1:(i-1) (i+1):n];       % Leave out the one undesirable index
    newx(i) = (b(i) - dot(A(i,indices),x(indices)))/A(i,i);
  end

  % At this point, newx is a row vector.
  % The next command takes its transpose (the ' does this) and makes
  %     newx this transpose.  That is, it makes newx a column vector.
  newx = newx';