Computations with large numbers

Transcription

1 Comutatos wth large umbers Wehu Hog, Det. of Math, Clayto State Uversty, 2 Clayto State lvd, Morrow, G 326, Mgshe Wu, Det. of Mathematcs, Statstcs, ad Comuter Scece, Uversty of Wscos-Stout, Meomoe, WI 5475 bstract. Sce ay ersoal comuter has a lmted rage of teger values, therefore, t wll result a teger overflow whe a rogram tres to comute a value larger tha mache s maxmum value. We wll dscuss a worable algorthm that wll be able to deal wth ay large umbers wthout gettg a teger overflow. Keywords: Iteger overflow.. Itroducto y rogrammer ows that whe a varable s declared to be of teger tye, t has a maxmum value. It wll result a teger overflow whe a rogram s to comute a teger value larger tha your mache s maxmum value. Some mache gves you a error message whe overflow occurs, but others do t. Therefore, t s ecessary to revet t from occurrg. It would be ce to fd a algorthm that wll be able to deal wth ay large umber wthout gettg a teger overflow o ay mache. I the followg, we wll rovde a algorthm that wll fulfll such goal. 2. lgorthm Our aroach s smle ad easy for mlemetato. The dea s to tae ay teger ut as a strg, whch ca be as log as you wsh. For stace, f you have to calculate you ca vew both umbers as strg, the maulate them ether dgt by dgt or bloc by bloc. The followg algorthm dd (, ) s a algorthm of dgt by dgt. lgorthm dd (, ) : = ( f = where the C C 2 2 Italze Carry = ; For ( = ; < ; ) Outut : C = C, where s the ut dgt, s the ut dgt, +, ot of the same legth, fll f + + Carry > 9, f Carry >, the C = + + Carry, Carry = + C C C C s the teth dgt, ad so o s the teth dgt, ad so o = + + Carry, Carry = = Carry C t out by zeros to the left) + = "";

2 Cosderg the effcecy for large umbers, we mght dvde each umber to blocs of legth that ca be determed by the mache s maxmum teger, ad the use regular oerato of addto to carry out the sum of each bloc, ad the cocateate the sums of blocs to get the sum. The followg algorthm demostrates the dea of bloc by bloc. lgorthm dd2 (, ) ssume that s a ostve whole umber; = = where ( + + ( ) ( ) ) ad, + + ( 2 2 ( 2) ( 2) + ; + ; ) are whole umbers of or fewer dgts; Each sum + + ( =,, 2,,m(, s less tha the mache'max; s () Use regular addto to fd each sum S If S. legth = +, the Carry =, otherwse, Carry = ; Set S = ( f > ) or ( f = + ( =,,,m(, ; ) for = m(, ) + to max(, ); (2) If ( Carry = ) the ( S C = ""& S ); = S ; ach ( S. legth) s to the left of S ; (3) For ( =, 2,, max(, S = S f ( S Carry = ; S C = S + Carry;. legth < + ) the f ( < max(, the ach ( S C = S & C; Carry = ; ach ( S & C; = S ;. legth) s to the left of S. legth) s to the left of ;) S ; (4) If ( Carry = ) the C = ""& C; (5) Outut C as the sum +.

3 Lewse, we ca do subtracto a smlar way. The followg s a algorthm for subtracto dgt by dgt. We ca also do subtracto bloc by bloc as show lgorthm Subtract2 (, ). lgorthm Subtract2 (, ) : ssume that s a ostve whole umber; = = where ( =,, 2, 3,, ) ad, whole umbers of ad ad lgorthm Subtract (, ) : = = ( f C + + () f ( = ) the ot of the C Outut : f 2 2 sg = " ";, Italze orrow = ; For ( = ; < ; ) where s the ut dgt, are less tha the mache'max; s outut D = ; Ext; ( ) ( ) dgts ( ach s f are whole umbers of f ( > ) the sg = ' + ;' sg = ' ;' swa ad ;, where s the ut dgt, If the sg = " + " f orrow <, ( sg == " ") the C C s the teth dgt, ad so o s the teth dgt, ad so o the same legth, fll t out by zeros to the left = orrow +, orrow = = orrow, orrow = C C C 2 ( 2) ( 2) C ( =,, 2, 3,, ) are ecessary); or fewer dgts; ) + ; + ;

4 (2) ssume > ; Italze orrow = ; Use regular subtracto to fd each dfferece : For ( = to max(, f ( m(, the f ( f ( (3) Let m = max(, ); (4) For( = to m ), ach ( D. legth) s (5) D = ""; for( = to m) D = D (6) f ( sg = ' )' the D = sg & D; (7) Outut orrow > ) the ( D ( D orrow > ) the ( D ( D = + = + orrow; orrow = ;) & D; D as the dfferece. = orrow; orrow = ;) orrow; orrow = ;) = orrow; orrow = ;) to the left of D ; For multlcato, we ca do ether dgt wse or bloc wse. The followg s a algorthm for multlcato bloc wse. lgorthm Multly (, ) : ssume that = = where, () Use regular multlcato to (2) ttchg ( + ) (3) Use the dd(, ) or dd 2(, ) to sum u all roducts; (4) Outut + + Each roduct ( ) ( ) ( ; s less the sum as the ) are ( 2) ( 2) tha the mache'max; s zeros to the ed whole umbers of fd each roduct ; roduct of ad. of the roduct ; + ; + ; or fewer dgts, I order to chec dvsblty ad rmalty of tegers, we eed a algorthm of dvso. We ca roceed as follow: lgorthm Dvde (, ) : = where s the ut dgt, s the teth dgt, ad so o = m where m 2 m 2,, s the ut dgt, s the teth dgt, ad so o

5 (). If <, the Outut uotet =, ad remader =. (2). If =, the Outut uotet =, ad remader =. (3). If >, the Dvded =, dvsor =, K = the legth of ; Whle (Dvded > dvsor) D = the frst dgts of Dvded from the left. D2 = the substrg of Dvded startg from the (+) to the ed If (D >= dvsor) the Fd the greatest o-egatve teger such that (dvsor)<= D; cout = ; Udate Dvded = Subtract (D, Multly (, dvsor)) & D2; Else = ; cout D = the frst + dgts of Dvded D2 = the substrg of Dvded startg from the (+2) to the ed Fd the greatest o-egatve teger such that (dvsor) <= D; cout = ; Udate Dvded = Subtract (D, Multly (, dvsor)) & D2; If (Dvded = dvsor) the = ; Remader = ; cout Else Remader = Dvded; Outut : uotet = 2cout remader = Remader Oce we mlemet Dvde (, ), we ca defe modulo fucto Mod (, ), whch returs the remader whe s dvded by. Therefore, we ca chec dvsblty ad rmalty of tegers. 3. lcatos It s very commo that wheever we teach Number Theory, we eed dslay some large umbers such as Fboacc rmes ad Mersee umbers by usg comuter. If we are ot careful, we wll ru to a teger overflow. We ca use the algorthms develoed secto 2 above to elmate such errors. s a demostrato, we gve two examles the followg. The frst examle s a comarso of dslays of large umbers o comuters wth or wthout usg our algorthms.

6 Examle. It s well ow [ 2] that the 3 rd, 4 th,7 th, th, 3 th,7 th,23 rd, 29 th,43 rd,47 th,83 rd,3 st, 37 th, 359 th,43 st, 433 rd,449 th,59 th,569 th,57 st,297 st,4723 rd,5387 th,93 th,9677 th,443 st, 2556 st, 3757 th, 3757 th, th, ad 8839 th terms of Fboacc seuece are rmes. It s easy to dslay ay of the frst few small oes usg a comuter. However, t s mossble to dslay a large oe wthout gettg a teger overflow. I the table, t was show that the dslay va JavaScrt wthout usg our algorthm wll ether loss recsos or get a scetfc otato that ufortuately results a real umber stead of a teger. lve demostrato s avalable o our webste at htt://cms.clayto.edu/whog/tools/demo4gnumber.htm. The Dslay wthout usg our algorthm Table Fboacc Prmes. The frst 22 rmes the Fboacc Seuece The Dslay usg our algorthm e e e e e e e e e Ifty

7 Examle 2. It s well ow [ ] that the 257 th Mersee umber s a rme. To dslay the 257 th Mersee umber, we ca mae a smle loo as follows: Cout = ; X = ; Whle (Cout Not Eual to 257) X = Multly(X,"2"); Z = Subtract(X,""); Cout++; Outut Z; The 257 th Mersee umber s Cocluso It s very commo that a error of teger overflow occurs scetfc comutato. To elmate such a error, t s a good dea to mlemet algorthms as we dd ths ote ether dgt by dgt or bloc by bloc so that rograms wll be able to hadle stuatos tellgetly for lager tegers. The algorthm gve ths ote ca be easly aled to may scetfc comutatos such as RS [3] ecryto ad searchg for large rmes. cowledgmets: We tha Dr. ust ad Dr. Mg-Ju La for ther may suggestos that mroved ths ote. Refereces [] Davd M. urto, 27, Elemetary Number Theory, the 6 th Ed., McGraw Hll. [2] Chrs K. Caldwell, Fboacc Prmes, htt://rmes.utm.edu [3]. ruce Scheer, 996, led Crytograhy, Protocols, lgorthms, ad Source Code C, the 2 d Ed., Wley.