1. A procedure for finding square roots of numbers
close to familiar square roots.
Dave Coulson, 2014
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. 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. 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. This procedure is based on the expansion for square roots close to 1.
 
x1
x1x1
2
1
2
1


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. Error < 0.01%
8. Error ~ 0.01%
9. In general, for k between 0 and 0.5,
This should be equal to N+k for a perfect square root 
10. k can never be bigger than 0.5. 
Therefore error decreases with N.
Relative error decreases (roughly) with N2.
Therefore....
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. Greater accuracy can be achieved by referencing the squares of halves.
Error 0.2%
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