A good reference for the basic algorithms of Newton-Raphson method to calculate the square root of a number, see
And the SAS codes(self-explanatory):
data root;
/question: find the square root of 4/
x=4;/first choose a rough approximation of sqrt(4);
actually, you can start with any numbers/
y0=1;count=0;/init count number/
do until (w<1e-8); /*set a small tolerance error*/
count=count+1; /accumulate count number/
y=(y0+x/y0)/2; /Newton’s formula/
w=abs(y-y0); /if close, exit;/
y0=y; /* otherwise, keep the new one*/
end;output;
run;
The outputs:
x y0 count w y
4 2 6 2.2204E-15 2
After 6 iterations, Newton-Raphson(also called divide-and-average) gets an approximated square root. See what happed during each iteration compared the output generated by SAS function,sqrt():
data root;
x=4;
y0=1;
count=0;
do until (w<1e-8);
count=count+1;
y=(y0+x/y0)/2;
w=abs(y-y0);
y0=y;
if y =sqrt(x) then is_eq_sqrt=“YES”;
else is_eq_sqrt=“NO”;
output;
end;
run;
Outputs:
x y0 count w y is_eq_sqrt
4 2.50000 1 1.50000 2.50000 NO
4 2.05000 2 0.45000 2.05000 NO
4 2.00061 3 0.04939 2.00061 NO
4 2.00000 4 0.00061 2.00000 NO
4 2.00000 5 0.00000 2.00000 NO
4 2.00000 6 0.00000 2.00000 YES
What’s the difference between count 5 and 6 since their y values look the same? We reset the tolerance value to 1e-3 rather than 1e-8, and get the outputs:
x y0 count w y is_eq_sqrt
4 2.50000 1 1.50000 2.50000 NO
4 2.05000 2 0.45000 2.05000 NO
4 2.00061 3 0.04939 2.00061 NO
4 2.00000 4 0.00061 2.00000 NO
The system get a faster convergence at an higher error rate, with an approximated value little away from sqrt(4).
We should have a deep understanding of how SAS stores numeric values, which deserves a full session to discuss, to unearth the mystery. Some basic references: