I worked out by hand the formulas for the iterates of the 2x2 QR
algorithm.  Note that the square roots needed to compute Q and R
disappear when the iterate RQ is computed.

The following c++ program shows the iteration process. The starting
symmetric matrix used was {{1,1},{1,2}}, which has eigenvalues
(3 +/- sqrt(5))/2. For a matrix this small the eigenvalues are
easily computed by hand.

/* *** C++ program for 2x2 QR eigenvalues *** */
#include <iostream>

int main(int argc, char** argv)
{
  double a00 = 1.0;
  double a11 = 2.0;
  double a01 = 1.0;
  char* sp = "  ";

  double d, b00, b11, b01;
  for (int i = 0; i < 25; i++)
  {
    cout<<i<<sp<<a00<<sp<<a11<<sp<<a01<<sp<< (a00*a11 -a01*a01) << endl;
    d = a00*a00 + a01*a01;
    b00 = a00*a00*a00 + 2.*a01*a01*a00 + a01*a01*a11;
    b01 = a00*a01*a11 - a01*a01*a01;
    b11 = a00*a00*a11 - a00*a01*a01;

    a00 = b00/d;
    a01 = b01/d;
    a11 = b11/d;
  }
}

The output of this program, showing the iterates and the determinant is:

0  1  2  1  1
1  2.5  0.5  0.5  1
2  2.61538  0.384615  0.0769231  1
3  2.61798  0.382022  0.011236  1
4  2.61803  0.381967  0.00163934  1
5  2.61803  0.381966  0.000239177  1
6  2.61803  0.381966  3.48955e-05  1
7  2.61803  0.381966  5.09118e-06  1
8  2.61803  0.381966  7.42794e-07  1
9  2.61803  0.381966  1.08372e-07  1
10  2.61803  0.381966  1.58113e-08  1
11  2.61803  0.381966  2.30683e-09  1
12  2.61803  0.381966  3.36563e-10  1
13  2.61803  0.381966  4.91038e-11  1
14  2.61803  0.381966  7.16415e-12  1
15  2.61803  0.381966  1.04524e-12  1
16  2.61803  0.381966  1.52498e-13  1
17  2.61803  0.381966  2.22491e-14  1
18  2.61803  0.381966  3.2461e-15  1
19  2.61803  0.381966  4.736e-16  1
20  2.61803  0.381966  6.90974e-17  1
21  2.61803  0.381966  1.00812e-17  1
22  2.61803  0.381966  1.47082e-18  1
23  2.61803  0.381966  2.1459e-19  1
24  2.61803  0.381966  3.13083e-20  1

The QR algorithm depends on the original matrix having a reasonable
mixture of its eigenvectors in the proper order.  For example here
is the output of the 2x2 QR iterations when starting with the
ill-conditioned matrix {{1.0, .0001},(.0001,3.0}}

0  1  3  0.0001  3
1  1  3  0.0003  3
2  1  3  0.0009  3
3  1  3  0.0027  3
4  1.00003  2.99997  0.00809987  3
5  1.0003  2.9997  0.0242964  3
6  1.00265  2.99735  0.0728033  3
7  1.02363  2.97637  0.216116  3
8  1.19432  2.80568  0.592353  3
9  1.98402  2.01598  0.999872  3
10  2.79417  1.20583  0.60769  3
11  2.97483  1.02517  0.222959  3
12  2.99717  1.00283  0.0751606  3
13  2.99969  1.00031  0.0250851  3
14  2.99997  1.00003  0.00836286  3
15  3  1  0.00278766  3
16  3  1  0.000929223  3
17  3  1  0.000309741  3
18  3  1  0.000103247  3
19  3  1  3.44157e-05  3
20  3  1  1.14719e-05  3
21  3  1  3.82396e-06  3
22  3  1  1.27465e-06  3
23  3  1  4.24885e-07  3
24  3  1  1.41628e-07  3 

Notice how the off-diagnal portion of the matrix increases
until a suitable mixture is attained.