761d764b-7912-4288-b776-00e2dcde7fe2 6 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

"Heron" sqrt(x)= √x as function x ε SQRT k 0 : r x 2 / : r 2^x - abs ε ≥ r r 2 / x 2 r * / + : k k 1 + : 21 line while r k 21 mat 41 line :

objective demand x 9999 : ε 1013 -^: 12 mat

x sqrt x ε SQRT 21 mat 99.9949998749938 99.9949998749938 2221 mat 21 mat

An iterative method solves for an objective demand. It solves recursively guessing next result from previous one. RootDichotomy, Golden ratio ...

x 1.23456789 :

U x sqrt : 1.11111110605556

x Heron r x 2 / : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : r r 2 / x 2 r * / + : 81 sys :

j 18 range : j Xcollect x Heron j el : Xj j Xcollect sys2mat : 31 sys

Xj 0.617283945 1.3086419725 1.12601909617746 1.11120979362233 1.11111111043782 1.11111110605556 1.11111110605555 1.11111110605555 81 mat