Computing a matrix which transforms a quadrangle to another quadrangle in 2D

In the figure below the goal is to compute the homography matrix H which transforms the points a1 a2 a3 a4 to their counterparts b1 b2 b3 b4. That is:

[b1 b2 b3 b4] = H * [a1 a2 a3 a4]

What way would you suggest to be the best way to calculate H(3x3). a1...b4 are points in 2D which are represented in homogeneous coordinate systems ( that is [a1_x a1_y 1]', ...). EDIT: For these types of problems we use SVD, So i would like to see how this can be simply done in Matlab.

EDIT:

Here is how I initially tried to solve it using svd (H=Q/P) in Maltlab. Cosider the following code for the given example

px=[0 1 1 0];  % a square
py=[1 1 0 0];

qx=[18 18 80 80];    % a random quadrangle
qy=[-20 20 60 -60];
if (DEBUG)
  fill(px,py,'r');
  fill(qx,qy,'r');
end

Q=[qx;qy;ones(size(qx))];
P=[px;py;ones(size(px))];
H=Q/P;
H*P-Q
answer:
   -0.0000         0         0         0         0
  -20.0000   20.0000  -20.0000   20.0000    0.0000
   -0.0000         0         0         0   -0.0000

I am expecting the answer to be a null matrix but it is not!... and that's why I asked this question in StackOverflow. Now, we all know it is a projective transformation not obviously Euclidean. However, it is good to know if in general care calculating such matrix using only 4 points is possible.