function [R,err,lambda_n] = explQR_shift(A,m)
% explicit QR-algorithm with Rayleigh shift
%   Input:  A - matrix
%           m - number of iterations
%   Output: R - upper triangular matrix
%           err - velikosti prvku A(n,n-1)
%           lambda_n - aproximace nejmensiho vl.c.

n = size(A,1);
err = [abs(A(n,n-1))];
TOL=1e-10;

lambda_n = A(n,n);                      % Reyl. shift

for i=1:1,                              % inicial iterations without shift
    [Q,R] = qr(A);
    A = R*Q;    
    err = [err,abs(A(n,n-1))];
    lambda_n = [lambda_n, A(n,n)];
end

while ((i < m) && (err(i+1) > TOL))     % stopping criterium 
    i=i+1;
    [Q,R] = qr(A - lambda_n(i)*eye(n));
    A = R*Q + lambda_n(i)*eye(n);    
    err = [err,abs(A(n,n-1))];
    lambda_n = [lambda_n, A(n,n)];    
end

if (n > 2)                              % recursive dimensional reduction
    [A(1:n-1,1:n-1),err_sub,lambda_sub] = explQR_shift(A(1:n-1,1:n-1),m);
    err = [err,err_sub];
    lambda_n = [lambda_n, lambda_sub];   
end    

R=A;