Derivation of a low multiplicative complexity algorithm for multiplying hyperbolic octonions

Transcription

1 rivtion of low multiplitiv omplit lgorithm for multipling hproli otonion lkndr Criow Glin Criow Jrołw Knpińki ult of Computr Sin nd nformtion Thnologi Żołnirk - Szzin olnd {triov gtriov jknpinki}@wi.zut.du.pl tl. 9 9 trt. prnt n ffiint lgorithm to multipl two hproli ountrompl otonion. Th dirt multiplition of two hproli otonion rquir rl multiplition nd rl ddition. or fftiv olution till do not it. how how to omput produt of th hproli otonion with rl multiplition nd 9 rl ddition. uring nthi of th diud lgorithm w u th ft tht produt of two hproli otonion m rprntd mtri vtor produt. Th mtri multiplind tht prtiipt in th produt lulting h uniqu truturl proprti tht llow prforming it dvntgou ftoriztion. Nml thi ftoriztion ld to ignifint rduing of th omputtionl omplit of hproli otonion multiplition. Kword: hproli otonion multiplition of hprompl numr ft lgorithm.. ntrodution Th dvlopmnt of thor nd prti of dt proing wll nit of olving mor nd mor ompl prolm of thortil nd pplid omputr in rquir uing dvnd mthmtil mthod nd formlim. t prnt hprompl numr [] r ing inrd pplition in vriou fild of digitl ignl nd img proing [-] omputr grphi nd mhin viion [ ] tlommunition [- ] nd in puli k rptogrph []. mong othr rithmtil oprtion in th hprompl lgr multiplition i th mot tim onuming on. Th ron for thi i u th uul multiplition of th numr rquir N N rl ddition nd N rl multiplition. t i to tht th inring of dimnion of hprnumr inr th omputtionl omplit of it multiplition. Thrfor rduing th omputtionl omplit of th multiplition of hprompl numr i n importnt thortil nd prtil tk. ffiint lgorithm for th multiplition of vriou hprompl numr lrd it [-]. No uh lgorithm for th multiplition of th hproli otonion hv n propod. n thi ppr n ffiint lgorithm for thi purpo i uggtd.. rliminr mrk hproli otonion n dfind follow [ ]: o ˆ whr { i } i... r rl numr r qutrnion imginr unit i ountrimginr unit nd th of hproli otonion r dfind follow: []. Th of hproli otonion hv multiplition rul in Tl :

2 Tl. ul for multiplition of hproli otonion um w wnt to omput th produt of two hproli otonion ˆ ˆ ˆ o o o : ˆ o ˆ o ˆ o Uing pn nd ppr mthod w n writ: ˆ o. Thn w hv: p. n tht th hoolook mthod of multiplition of two hproli otonion rquir rl multiplition nd rl ddition. n mtri nottion th ov rltion n writtn mor omptl : X Y whr Τ ]. [ X Τ ]. [ Y nd

3 Th dirt rliztion of rquir rl multiplition nd rl ddition too. hll prnt th lgorithm whih rdu rithmtil omplit to rl multiplition nd 9 rl ddition.. Snthi of rtionlizd lgorithm for multipling two hproli otonion t firt w multipl - th ith vnth nd ighth row of th mtri. Thn w intrhng th firt nd th fifth olumn of thi mtri nd ll th rulting mtri.. Thn w n writ X Y whr. Thi trnformtion i don in ordr to prnt modifid in thi mnnr mtri n lgri um of th lok-mmtri Toplitz-tp mtri nd om pr mtri i.. mtri ontining onl mll numr of nonzro lmnt. Now th mtri n rprntd n lgri um of mmtri Toplitz-tp mtri nd nothr mtri whih h mn zro lmnt :

4 Tking into ount propod dompoition th omputtionl produr for multiplition hproli otonion n rwrittn follow: X Σ Y whr ign dnot th dirt um of two mtri [] dig Σ. t i to tht h th following trutur:. t i il to vrif [-] tht th mtri with thi trutur n fftivl ftorizd: ] [

5 whr i th ordr dmrd mtri N i th ordr N idntit mtri nd ign dnot th Kronkr produt of two mtri rptivl []. Thn th omputtionl produr for multiplition of th hproli otonion t thi tp of th lgorithm dign n rprntd follow: X Σ Y whr. ig. how dt flow digrm of th rtionlizd lgorithm for omputtion of produt of hproli otonion. n thi ppr dt flow digrm r orintd from lft to right. Stright lin in th figur dnot th oprtion of dt trnfr. oint whr lin onvrg dnot ummtion. Th dhd lin indit th ign hng oprtion. dlirtl u th uul lin without rrow on purpo o not to luttr th pitur. Th rtngl indit th mtri vtor multiplition with th mtri inrid inid rtngl.

6 ig.. t flow digrm for rtionlizd hproli otonion multiplition lgorithm in ordn with th produr. Lt u now onidr th trutur of th mtri nd. irt w multipl - vr lmnt of th firt row of mtri nd ll th rulting mtri :. Th mtri n dompod n lgri um of mmtri Toplitz mtri nd nothr mtri whih h mn zro lmnt :

7 ] [. Lt u rturn now to th trutur of th mtri. t to thn th mtri n lo rprntd n lgri um of mmtri Toplitz mtri nd nothr mtri whih h mn zro lmnt : C C C t i il to vrif [-] tht th mtri n ftorizd in th m w: ] [ C C

8 K C. L C Sutituting nd in w n writ: X Σ Σ Y 9 whr Σ. L K L K.

9 9

10 ig. how dt flow digrm of th rtionlizd lgorithm for multipling of two hproli otonion t th ond tg of nthi. Conidr now th mtri K nd L. n n th mtri lo hv "good" trutur lding to dr in th numr of rl multiplition during lultion of th hproli otonion produt. ] [ ] [ d d d d ] [ K f f f f ] [ L h g h g g h h g. ntrodu th following nottion: d d f f h g h g. nd. Uing th ov nottion nd omining prtil dompoition in ingl omputtionl produr w finll n writ following: X Σ Y whr.

11 ig. how dt flow digrm of th rtionlizd lgorithm for multipling of two hproli otonion t th finl tg of th lgorithm drivtion. Th irl in thi figur how th oprtion of multiplition vril or ontnt inrid inid irl. n tht th ordinr pproh to lultion of lmnt... } { k k rquir ddition. t i to tht th rltion for lultion of } { k ontin rptd lgri um. Thrfor th numr of ddition nr to lult th lmnt n rdud. So it i to vrif tht th lmnt } { k... k n lultd uing th following rtionlizd mtri vtor produr: S Τ ] [ S Τ ] [ dig.. ig. how dt flow digrm of th pro for lulting th vtor S lmnt.

12 K L ig.. t flow digrm for rtionlizd hproli otonion multiplition lgorithm in ordn with th produr 9.

13 ig.. t flow digrm for rtionlizd hproli otonion multiplition lgorithm in ordn with th produr.

14 ig.. t flow digrm driing th pro of lulting lmnt of th vtor S in ordn with th produr.. timtion of omputtionl omplit lult how mn rl multiplition luding multiplition powr of two nd rl ddition r rquird for rliztion of th propod lgorithm nd ompr it with th numr of oprtion rquird for dirt vlution of mtri-vtor produt in q.. Lt u look to th dt flow digrm in igur. t i to vrif tht ll th rl multiplition whih to prformd to omputing th produt of two hproli otonion r rlizd onl during multipling vtor of dt th qui-digonl mtri. t n rgud tht th multiplition of vtor th mtri rquir rl multiplition nd onl fw trivil multiplition th powr of two. ultiplition powr of two m implmntd uing onvntion rithmti hift oprtion whih hv impl rliztion nd hn m ngltd during omputtionl omplit timtion. So th numr of rl multiplition rquird uing th propod lgorithm i. Thu uing th propod lgorithm th numr of rl multiplition to lult th hproli otonion produt i ignifintl rdud. Now w lult th numr of ddition rquird in th implmnttion of th lgorithm. t i il to vrif tht th numr of rl ddition rquird uing our lgorithm i 9. Thrfor th totl numr of rithmti oprtion i till lightl l thn th totl numr of rithmti oprtion in th niv lgorithm.. Conluion n thi ppr w hv prntd n originl lgorithm tht llow u to omput th produt of two hproli otonion with rdud multiplitiv omplit. Th propod lgorithm v rl multiplition omprd to th hoolook lgorithm. Unfortuntl th numr of rl ddition in th propod lgorithm i omwht grtr thn in th dirt lgorithm ut th totl numr of rithmtil oprtion i till l. or pplition whr th ot of rl multiplition i grtr thn tht of rl ddition th nw lgorithm i gnrll mor ffiint thn dirt mthod.. frn. Kntor. nd Solodovnikov. prompl numr Springr-Vrlg Nw York. 99. ülow T. nd Sommr G. prompl ignl - novl tnion of th nlti ignl to th multidimnionl Trn. Sign. ro. vol. S-9 No. -.. lfmnn. On fmili of N -dimnionl hprompl lgr uitl for digitl ignl proing in ro. uropn Signl roing Conf. USCO lorn tl.

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