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# Square roots close range approximation

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### Square roots close range approximation

1. 1. A procedure for finding square roots of numbers close to familiar square roots. Dave Coulson, 2014
2. 2. This procedure is based on the expansion for square roots close to 1.                   .... x1C x1C x1C x1Cx1x1 3 3 2 2 1 1 0 0 2 5 2 1 2 3 2 1 2 1 2 1 2 1 2 1 2 1        
3. 3. This procedure is based on the expansion for square roots close to 1.            ...xxx1 x1x1 3 2 3 2 1 2 12 2 1 2 1 2 1 2 1  
4. 4. This procedure is based on the expansion for square roots close to 1.   ...xxx1 x1x1 3 8 32 4 1 2 1 2 1  
5. 5. This procedure is based on the expansion for square roots close to 1.   x1 x1x1 2 1 2 1  
6. 6. This procedure is based on the expansion for square roots close to 1. Similarly   x1 x1x1 2 1 2 1     x1 x1x1 2 1 2 1  
7. 7. Error < 0.01%
8. 8. Error ~ 0.01%
9. 9. In general, for k between 0 and 0.5, This should be equal to N+k for a perfect square root 
10. 10. k can never be bigger than 0.5.  Therefore error decreases with N. Relative error decreases (roughly) with N2. Therefore....
11. 11. 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.
12. 12. Greater accuracy can be achieved by referencing the squares of halves. Error 0.2%
13. 13. [END] dtcoulson@gmail.com