The square root problem Suppose a is a positive real number and that we want to compute its square root geometrically this task is equivalent to constructing a square whose area is A. x
The square root problem Suppose A is a positive real number and that we want to compute its square root. Geometrically, this task is equivalent to constructing a square whose area is A. c c x A x x 2 1
XC 6000/Xc 60.00000000000000100.00000000000000 80.000000000000007500000000000000 77.500000000000007741935483870968 77459677419354857745965642894325 77459666924149057745966692414763 77459666924148347745966692414834 77459666924148347745966692414834 7745966692414834
xc 6000/xc ------------------------------------------- ------- ------ 60.00000000000000 100.00000000000000 80.00000000000000 75.00000000000000 77.50000000000000 77.41935483870968 77.45967741935485 77.45965642894325 77.45966692414905 77.45966692414763 77.45966692414834 77.45966692414834 77.45966692414834 77.45966692414834 77.45966692414834
To improve an approximate square root A=m米A,4 <m<1 √A=√m*2,m∈[.25,1 a good initial guess for the reduced range problem can be obtained by linear interpolation with L(m)=(1+2m)/3 L(m)-√m|≤0.05
To improve an approximate square root 1 4 1 A m*4 , m e A m *2 , m [.25,1] e a good initial guess for the reduced range problem can be obtained by linear interpolation with L(m) (1 2m)/ 3 | L(m) m | 0.05
sqrt(m) 0.8 04 0.2 0.5 15
0 0.5 1 1.5 0 0.2 0.4 0.6 0.81 1.2 m sqrt(m)
x+ C -L(m), ek=xk 2x k It can be shown that the iterates x are al ways in the interval [5, 1, and so k+1 e4≤e3≤e2≤e8≤6≤(05)6
2 2 1 2 1 c c c c x x m m x m x m x x0 L(m),ek xk m 2 1 2 1 k k k e x e It can be shown that the iterates xk are always in the interval [.5,1], and so 2 k 1 k e e 16 16 0 8 1 4 2 2 4 3 e e e e e (.05)