[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.
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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participants (2)
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Henry Baker -
Joerg Arndt