square roots close range approximation
TRANSCRIPT
A procedure for finding square roots of numbers close to familiar square roots.
Dave Coulson, 2014
This procedure is based on the expansion for square roots close to 1.
....
x 1 C
x 1 C
x 1 C
x 1 C x1 x1
3 3
2 2
1 1
0 0
25
21
23
21
21
21
21
21
21
This procedure is based on the expansion for square roots close to 1.
... x x x 1
x1 x13
23
21
212
21
21
21
21
This procedure is based on the expansion for square roots close to 1.
... x x x 1
x1 x13
832
41
21
21
This procedure is based on the expansion for square roots close to 1.
x 1
x1 x1
21
21
This procedure is based on the expansion for square roots close to 1.
Similarly
x 1
x1 x1
21
21
x 1
x1 x1
21
21
Error < 0.01%
Error ~ 0.01%
In general, for k between 0 and 0.5,
This should be equal to N+k for a perfect square root
k can never be bigger than 0.5.
Therefore error decreases with N.
Relative error decreases (roughly) with N2.
Therefore....
Blue is absolute error, Red is relative error.
Relative error is never higher than 6% and very quickly reduces to less than 1%.
The worst estimates occur when estimating square roots of numbers less than 4.
Greater accuracy can be achieved by referencing the squares of halves.
Error 0.2%
