[math-fun] square root approximating sequence ?

I'm looking for a simple approximation to sqrt(abs(b*c)) that goes something like this: b*c=b_1*c_1=b_2*c_2=...=(sign(b)*sqrt(b*c))*(sign(c)*sqrt(b*c)) where b_(i+1),c_(i+1) are rationally computed from b_i,c_i. In other words, we preserve the product while approximating the square root. What I'm really after is a sequence: [a b] [a b_1] [a b_2] [c d], [c_1 d], [c_2 d], ... such that the determinant is preserved, but each matrix is computed from the previous one by X.M.Y, where X,Y are 2x2 matrices rationally computed from the entries of M.