Compute
, naive version
Compute area of the unit circle
Contents
Area of the triangle is
where . The total area is
Recurrence relation
Initialization
,
,
,
.
s=sqrt(3)/2; A=3*s; n=6; % initialization z=[A-pi n A s]; % store the results while s>1e-10 % termination if s=sin(alpha) small s=sqrt((1-sqrt(1-s*s))/2); % new sin(alpha/2) value n=2*n; A=n/2*s; % A=new polygon area z=[z; A-pi n A s]; end m=length(z); disp(" n A err sine") for i=1:m fprintf('%10d %20.15f %20.15f %20.15f\n',z(i,2),z(i,3),z(i,1),z(i,4)) end
n A err sine 6 2.598076211353316 -0.543516442236477 0.866025403784439 12 3.000000000000000 -0.141592653589794 0.500000000000000 24 3.105828541230250 -0.035764112359543 0.258819045102521 48 3.132628613281237 -0.008964040308556 0.130526192220052 96 3.139350203046872 -0.002242450542921 0.065403129230143 192 3.141031950890530 -0.000560702699263 0.032719082821776 384 3.141452472285344 -0.000140181304449 0.016361731626486 768 3.141557607911622 -0.000035045678171 0.008181139603937 1536 3.141583892148936 -0.000008761440857 0.004090604026236 3072 3.141590463236762 -0.000002190353031 0.002045306291170 6144 3.141592106043048 -0.000000547546745 0.001022653680353 12288 3.141592516588155 -0.000000137001638 0.000511326906997 24576 3.141592618640789 -0.000000034949004 0.000255663461803 49152 3.141592645321216 -0.000000008268577 0.000127831731987 98304 3.141592645321216 -0.000000008268577 0.000063915865994 196608 3.141592645321216 -0.000000008268577 0.000031957932997 393216 3.141592645321216 -0.000000008268577 0.000015978966498 786432 3.141593669849427 0.000001016259634 0.000007989485855 1572864 3.141592303811738 -0.000000349778055 0.000003994741190 3145728 3.141608696224804 0.000016042635011 0.000001997381017 6291456 3.141586839655041 -0.000005813934752 0.000000998683561 12582912 3.141674265021758 0.000081611431964 0.000000499355676 25165824 3.141674265021758 0.000081611431964 0.000000249677838 50331648 3.143072740170040 0.001480086580246 0.000000124894489 100663296 3.159806164941135 0.018213511351342 0.000000062779708 201326592 3.181980515339464 0.040387861749671 0.000000031610136 402653184 3.354101966249685 0.212509312659892 0.000000016660005 805306368 4.242640687119286 1.101048033529493 0.000000010536712 1610612736 6.000000000000000 2.858407346410207 0.000000007450581 3221225472 0.000000000000000 -3.141592653589793 0.000000000000000