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Useful Computational Methods

Cube roots using mediants

Letting N denote the coefficient matrix, we see that N^n is a matrix that is symmetrical about the trailing diagonal[1]. For instance, with Q = 2, we get the following.

This means that after six iterations, we get
c' = 1081c + 1362a + 1716b
a' = 858c + 1081a + 1362 b and
b'= 681c + 858a+ 1081 b.
We can choose a,b,c to be any positive intgers to start with. So if c is chosen to be a very large integer compared to a and b,
c' will be approximately equal to 1081c
a' will be approximately equal to 858c and
b' will be approximately equal to 681c
after six iterations. But c'/a', a'/b', and 2b'/c' approach the cube root of 2 after a large number of iterations. This means the entries in the matrix after a few iterations will increase by a factor of the cube root of 2 vertically. Making a and b large, in turn, compared to the remaining initial values, we also conclude that the entries of the matrix after a few iterations increase by the cube root of 2 horizontally as well. In other words, the approximation of the cube root of 2 is the quotient of any two consecutive entires vertically or horizontally.

Analagous results apply to roots of higher orders, such as generating the 4th roots of Q by computing the powers of

[1] Brown, Kevin. "Generalized Mediant," webpage: http://www.mathpages.com/home/kmath055.htm
[2] Gómez Morín, Domingo. “La Quinta Operación Aritmética, Media Aritmónica”, 2nd. Edition, 2006. ISBN:980-12-1671-9
Webpage: http://domingogomez.web.officelive.com/rm.aspx




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