Goldschmidt's algorithm comes to mind. See section 29.6 of my book (pp.581ff, but also the sections up to p.586). The matrix product hints to continued fractions. Best regards, jj * Henry Baker <hbaker1@pipeline.com> [Nov 26. 2017 15:45]:
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.
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