square roots close range approximation
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A procedure for finding square roots of numbers close to familiar square roots.
Dave Coulson, 2014
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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
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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
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This procedure is based on the expansion for square roots close to 1.
... x x x 1
x1 x13
832
41
21
21
![Page 5: Square roots close range approximation](https://reader035.vdocuments.us/reader035/viewer/2022062418/555a0249d8b42aa8098b4f81/html5/thumbnails/5.jpg)
This procedure is based on the expansion for square roots close to 1.
x 1
x1 x1
21
21
![Page 6: Square roots close range approximation](https://reader035.vdocuments.us/reader035/viewer/2022062418/555a0249d8b42aa8098b4f81/html5/thumbnails/6.jpg)
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
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Error < 0.01%
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Error ~ 0.01%
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In general, for k between 0 and 0.5,
This should be equal to N+k for a perfect square root
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k can never be bigger than 0.5.
Therefore error decreases with N.
Relative error decreases (roughly) with N2.
Therefore....
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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.
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Greater accuracy can be achieved by referencing the squares of halves.
Error 0.2%