Square roots   close range approximation
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Square roots close range approximation

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Square roots   close range approximation Square roots close range approximation Presentation 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.                   .... 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        
  • 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  
  • This procedure is based on the expansion for square roots close to 1.   ...xxx1 x1x1 3 8 32 4 1 2 1 2 1  
  • This procedure is based on the expansion for square roots close to 1.   x1 x1x1 2 1 2 1  
  • 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  
  • 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%
  • [END] dtcoulson@gmail.com