function x = pagerank2(U,G,p)
% PAGERANK  Google's PageRank
% pagerank(U,G,p) uses the URLs and adjacency matrix produced by SURFER,
% together with a damping factory p, (default is .85), to compute and plot
% a bar graph of page rank, and print the dominant URLs in page rank order.
% x = pagerank(U,G,p) returns the page ranks instead of printing.
% See also SURFER, SPY.

if nargin < 3, p = .85; end

% Eliminate any self-referential links
%G = G - diag(diag(G));

% c = out-degree, r = in-degree
[~,n] = size(G);
c = sum(G,1);
r = sum(G,2);

% Scale column sums to be 1 (or 0 where there are no out links).
k = find(c~=0);
D = sparse(k,k,1./c(k),n,n);

e = ones(n,1);
I = speye(n,n);

% ---------------------------- INVERSE ITERATION --------------------------
disp('Using inverse iteration implementation\n');

z = ((1 - p) * (c ~= 0) + (c == 0)) / n;
A = p * G * D + e * z;  
x = e/n;

% Check if B will be a singular matrix. If so, change it
alpha = 0.8;
while rcond(A - alpha * I) < eps
   alpha = alpha + 0.01;
end
B = A - alpha * I;

old_x = zeros(n, 1);
old_norm = -1;
no = +Inf;

it = 0;
while (old_norm == -1 || old_norm > no) && no > 1e-8
   old_norm = no;
   old_x = x;
   x = B \ x;
   x = x/sum(x);
   it = it + 1;
   no = norm(x - old_x, 1);
end

x = abs(old_x);
display(it); 

% -------------------------------------------------------------------------

% Normalize so that sum(x) == 1.
x = x/sum(x);

% Bar graph of page rank.
shg
bar(x)
title('Page Rank')

% Print URLs in page rank order.

if nargout < 1
   [~,q] = sort(-x);
   disp('     page-rank  in  out  url')
   k = 1;
   maxN = length(U);
   while  (k <= maxN) && (x(q(k)) >= .005)
       disp(k)
      j = q(k);
      temp1  = r(j);
      temp2  = c(j);
      disp(fprintf(' %3.0f %8.4f %4.0f %4.0f  %s', j,x(j),full(temp1),full(temp2),U{j}))
      k = k+1;
   end
end