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Talk:Arithmetic-geometric mean: Difference between revisions
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const ε = 1e-14</lang>
==rapidity==
From this Rosetta Code task's prologue:
:: <big><big>Since the limit of <math>a_n-g_n</math> tends (rapidly) to zero with iterations, this is an efficient method.</big></big>
<br>With this in mind, I modified the REXX's entry (and suppressed the normal output being displayed), and
<br>added a display of the iteration count along with the number of decimal digits being used.
The modified REXX program is:
<lang rexx>/*REXX program calculates the AGM (arithmetic─geometric mean) of two numbers.*/
parse arg a b digs . /*obtain optional numbers from the C.L.*/
if digs=='' | digs==',' then digs=100 /*No DIGS specified? Then use default.*/
numeric digits digs /*REXX will use lots of decimal digits.*/
if a=='' | a==',' then a=1 /*No A specified? Then use default.*/
if b=='' | b==',' then b=1/sqrt(2) /*No B specified? " " " */
call AGM a,b /*invoke AGM & don't show A,B,result.*/
exit /*stick a fork in it, we're all done. */
/*────────────────────────────────────────────────────────────────────────────*/
agm: procedure: parse arg x,y; if x=y then return x /*equality case?*/
if y=0 then return 0 /*is Y zero? */
if x=0 then return y/2 /* " X " */
d=digits(); numeric digits d+5 /*add 5 more digs to ensure convergence*/
tiny='1e-' || (digits()-1); /*construct a pretty tiny REXX number. */
ox=x+1
do #=1 while ox\=x & abs(ox)>tiny; ox=x; oy=y
x=(ox+oy)/2; y=sqrt(ox*oy)
end /*while ··· */
numeric digits d /*restore numeric digits to original.*/
/*this is the only output displayed ►─┐*/
say 'digits='right(d,7)", iterations=" right(#,3) /* ◄───────────────┘*/
return x/1 /*normalize X to the new digits. */
/*────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); i=; m. =9
numeric digits 9; if x<0 then do; x=-x; i='i'; end; numeric form; h=d+2
parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'E'_%2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=.5*(g+x/g); end /*k*/
numeric digits d; return (g/1)i</lang>
'''outputs''' with a specified number of digits, and with the various outputs being combined/reduced/edited:
<pre>
digits= 100, iterations= 9
digits= 200, iterations= 10
digits= 400, iterations= 11
digits= 800, iterations= 12
digits= 1600, iterations= 13
digits= 3200, iterations= 14
digits= 6400, iterations= 15
digits= 12800, iterations= 16
digits= 25600, iterations= 17
digits= 51200, iterations= 18
</pre>
Needless to say, the CPU time increased everytime the digits were doubled.
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