
qr algoritam

numerical leanier algibrayil, qr algoritam allengil qr ittareshan oru eejan value algoritam aannu athaayath, oru maatriksinte eejan vaalyookalum egenvectorukalum kanakkakkunnathinulla oru nadapadikramam. 1950kalude avasaanathil joonn jii aanu qr algoritam vikasippichath. eph. francis vera enn. kublanovskaya, swathanthramaayi pravartthikkunnu. oru oorthoogoonal matrixinteyum mukalile thrikonaakrithiyilulla matrixinteyum ulppannamaayi matricks ezhuthukayum vipareeta kramatthil gadakangale gunikkukayum aavarthikkukayum cheyyuka ennathaanu adisthaana ashayam.
praayogika qr algoritam.
aupachaarikamaayi, "a" ennathu oru yadhaartha matricks ayirikkatte, athil ninnu eejan moolyangal kanakkakkaan nammal aagrahikkunnu, koodaathe "a" 0: = "a" ayirikkatte. k"-aamath ghattathil ("k" = 0 ll aarambikkunnu), "q" k "=" q "k" r "k" enna qr vighadanam nammal kanakkakkunnu, ivide "q" k "oru oorthoogoonal matricks aanu (i.iii.q" t "=" q "−1)," r "k" enniva oru mukalile trikona matricks aanu. appol nammal "a" "k" + 1 = "r" "k" "q" "k" roopappeduthunnu. athu shradhikkuka
soothravaakyam _ 1
athinaal ellah "a" k "umm samaanamaan, athinaal avaykku ore eejan moolyangalundu. oorthoogoonal" samanatha parivartthanangaliloode munnott pookunnathinaal algoritam samkhyaaparamaayi sthirathayullathaanu.
chila saahacharyangalil, "a" k "metriksukal" a "yude shharr roopamaaya oru thrikonaakrithiyilulla metrixilekku othucherunnu. oru trikona maatriksinte eejan moolyangal diagonalil pattikappedutthukayum eejan moolya prashnam pariharikkappedukayum cheiyunnu. kanverjans parisodhanayil krithyamaaya poojyangal aavashyamaayi varunnath apraayogikamaanu, ennaal gershgorin sarkkil sidhaantham pishakin oru paridhi nalkunnu.
eee asamskritha roopathil aavarthanangal thaarathamyena chelaveriyathaanu. aadyam matricks "a" neey upper hezenberg roopathilekku konduvannu (homeholder reduction adisthaanamaakkiyulla oru saankethikavidya upayogichu formula _ 2 ganitha pravarthanangalkku chelavu varum), oru paridhivare dvimukha qr vighadanam pole oorthoogoonal saamyam parivartthanangalude parimithamaaya kramam upayogichu ithu laghookarikkaam. (qr vighadanathinu, homeholder reflectorukal idathuvashathu maathrame gunikkoo, ennaal hessenberg kesil avaa idathuvashathum valathuvasathum gunikkoo.) oru upper hezenberg maatriksinte qr vighadanam nirnnayikkaan formula _ 3 ganitha pravartthanangal chelavaakum. koodaathe, hezenberg roopam ithinakam thanne eathaandu mukalile thrikonaakrithiyilullathinal (oro diagonalinum thaazhe poojyamallaatha oru entry maathrame ulloo), ithu oru prarambha pooyintaayi upayogikkunnathu qr algoritam othucherunnathinu aavashyamaaya ghattangalude ennam kuraykkunnu.
yadhaartha matricks samamithiyaanengil, mukalile hessenberg matrix samamithiyaan, athinaal trikonamitheeyamaanu, athupole thanne ellah "a" k umm. eee prakriyaykku hoome housing reduction adisthaanamaakkiyulla oru saankethikavidya upayogichu formula 4 ganitha pravartthanangal chelavaakum. oru samamita tribhuja maatriksinte qr vighadanam nirnnayikkunnathin formula 5 pravartthanangal chelavaakum.
othucheralinte nirakk eejan moolyangal thammilulla verthirivine aasrayichirikkunnu, athinaal oru praayogika algoritam verthirikkal vardhippikkunnathinum othucheral thwarithappeduthunnathinum vyakthamaayatho avyakthamo aaya shifttukal upayogikkum. oru saadhaarana cemetric qr algoritam oro eejan moolyatheyum (pinneedu maatriksinte valuppam kuraykkunnu) onno rando aavarthanangal upayogichu verthirikkunnu, ithu kaaryakshamavum karuthuttathumaakkunnu.
drishyavalkkaranam.
a" oru positteev-dephinit cemetric matricks aaya saahacharyathil adisthaana qr algoritam drishyavalkkarikkaam. anganeyaanengil, "a" yeei 2 alavukalulla deerghavrithamaayi chithreekarikkaam allengil uyarnna alavukalulla deerghavrithamaayi chithreekarikkaam. algoritam inputum orotta aavarthanavum thammilulla bandham chithram 1-ll chithreekarikkaam (oru aanimeshan kaanaan klikkucheyyuka). qr algorithathinoppam lr algoritam chithreekarichirikkunnu shradhikkuka.
orotta aavarthanam deerghavritham x-achuthandilekku charinjo veezhukayo cheiyunnu. deerghavrithathinte valia ardha-aksham x-akshathinu samaantharamaanengil, qr-nte oru aavarthanam onnum cheiyunnilla. valia ardha-aksham x-akshathinu pakaram y-akshathinu samaantharamaakumbozhaanu algoritam "onnum cheyyaatha" mattoru sahacharyam. aaa saahacharyathil, deerghavritham rund dishakalilekkum veezhaathe apakadakaramaayi santhulithamaakkunnathaayi kanakkakkaam. rund saahacharyangalilum, matricks diagonal aanu. algoritam "onnum cheyyaatha" oru aavarthanathe oru nischitha pooyint ennu vilikkunnu. oru nischitha binduvilekkulla aavarthanamaanu algoritam upayogikkunna thanthram. oru nischitha pooyint sthiramaanennum mattonn asthiramaanennum nireekshikkuka. deerghavritham asthiramaaya nischitha binduvil ninnu valare cheria alavil charichaal, kyuaarinte oru aavarthanam deerghavritham nischitha binduvil ninnu nere chaayunnathin pakaram charivinu kaaranamakum. ennirunnaalum, oduvil, algoritam mattoru nischitha poyintilekku othucherumenkilum athinu valareyadhikam samayamedukkum.
eejan moolyangal kandethunnathum egenvectorukal kandethunnathum.
oru cemetric maatriksinte orotta eejanvektar polum kandethunnathu kanakkukoottaanaakaathaannathaa choondikkaanikkendathund (kanakkukoottaavunna vishakalanathile nirvachanangal anusarichu krithyamaaya yadhaartha ganithathil). oru maatriksinte eejan moolyangalude gunithangal ariyappedaathappozhellaam eee budhimuttu nilanilkkunnu. maruvashathu, eejan moolyangal kandethunnathinu ithe prashnam nilavililla. oru maatriksinte eejan moolyangal ellaayppozhum kanakkukoottavunnavayaanu.
adisthaana qr algorithathil eee budhimuttukal engane prakadamaakumennu nammal ippol charcha cheyyum. ithu chithram 2 ll chithreekarichirikkunnu. (laghuchithrathil klikkucheyyan oorkkuka). deerghavrithangal positteev-dephinit cemetric maatriksukale prathinidheekarikkunnuvennu oorkkuka. inputt maatriksinte rund eejan moolyangal parasparam sameepikkumbol, inputt deerghavritham oru vrithamaayi maarunnu. oru vritham aidantitti maatriksinte onniladhikam bhagangalumaayi poruthappedunnu. oru vrithathinaduthulla vritham aidantitti maatriksinte oru multiplumaayi poruthappedunnu, athinte eejan moolyangal maatriksinte diagonal entrikalkku ekadesham thulyamaanu. athinaal ekadesham eejan moolyangal kandethunnathinulla prashnam aaa saahacharyathil eluppamaanennu kaanikkunnu. ennaal deerghavrithangalude ardha-achuthandinu enthu sambhavikkunnuvennu shradhikkuka. inputt deerghavritham oru vrithamaakunnathinu aduthethumbol qr (allengil lr)-nte aavarthanam ardha-achuthandukale kurachukoodi charippikkunnu. semi aaksukal x-axis, y-axis ennivaykku samaantharamaakumbol maathrame egenvectorukal arian kazhiyoo. inputt deerghavritham kooduthal vrithaakrithiyilullathaakum samaantharatha kaivarikkunnathinu aavashyamaaya aavarthanangalude ennam bandhamillathe vardhikkunnu.
oru ekapaksheeyamaaya samamiti maatriksinte igendecomposition kanakkaakkunnath asaadhyamaanengilum, oru aniyanthrithamaaya cheria alavil maatriksine thadassappeduthaanum thathfalamaayundaakunna maatriksinte igendecomposition kanakkakkaanum ellaayppozhum kazhiyum. matricks oru vrithathinaduthaayi chithreekarikkappedumbol, matricks oru thikanja vritham ulla oru maathruka upayogichu maattisthaapikkaam. anganeyaanengil, matricks aidantitti maatriksinte oru multiple aanu, athinte igendecomposition udanadi aanu. thathfalamaayundaakunna eejanbasis yadhaartha eejanbaasisil ninnu valare akaleyayirikkumennu arinjirikkuka.
antharleenamaaya qr algoritam.
aadhunika computational parisheelanatthil, qr algoritam oru avyaktha pathippil nadathappedunnu, ithu onniladhikam shifttukalude upayogam avatharippikkan eluppamaakkunnu. matricks aadyam upper hezenberg phoom formula _ 6 lake konduvarunnu; thudarnnu, oro ghattathilum, formula _ 7-nte aadya nira oru cheria valippathilulla homeholder saamyam formula _ 8 (allengil formula _ 9)-nte aadya nirayilekku parivartthanam cheiyunnu, avide digri formula _ 11-nte formula _ 8, shifting straattajiye nirvachikkunna polynomialaanu (palappozhum formula _ 12, ivide formula _ 13, formula _ 14 enniva formula _ 7-nte trailing formula _ 15 principal submatricsinte rund eejan vaalyookalaanu, "implisit double shift" ennu vilikkappedunnu). thudarnnu 7 enna pravartthana matricks formula mukalile hessenberg roopathilekku thirike nalkunnathinaayi 17 enna valuppa formulayude thudarchayaaya veettamma parivartthanangal nadathunnu. algorithathinte gattangalilulla maatriksinte poojyamallaatha entrikalude savisheshamaaya aakruthi kaaranam eee pravartthanatthe "balj chasing" ennu vilikkunnu. aadya pathippilennapole, formula _ 7-nte sub diagonal entrikalil onnu mathiyaya cheruthaanenkil thanne deflation nadathunnu.
punarnaamakaranam nirddesham.
nadapadikramatthinte aadhunika antharleenamaaya pathippil qr vighadanangalonnum vyakthamaayi nadathaathathinaal, chila rachayithakkal, udaaharanathinu vaatkins, athinte peru "francis algoritam" ennu maattaan nirddeshichu. golab, vaan loonn enniva "francis qr steppu" enna padam upayogikkunnu.
vyaakhyaanavum samyojanavum.
adisthaana "power" eejan value algorithathinte kooduthal sankeernamaaya vyathiyaanamaayi qr algorithathe kaanaan kazhiyum. power algoritam oro aavarthanathinum sesham saadhaaranavalkkarikkappedunna "a" oru vektarinte eratti aavarthikkunnundenn oorkkuka. vekttar ettavum valia eejan moolyathinte oru eejanvektaraayi othucherunnu. pakaram, qr algoritam vektarukalude poornnamaaya adisthaanathil pravartthikkunnu, qr vighadanam upayogichu punarnaamakaranam cheiyunnu (oorthoganalais cheiyunnu). oru samamiti maatriksil "a", kanverjansil, "aq" = "qλ", ivide "a" kanverju cheytha eejan moolyangalude diagonal matricks aanu, koodaathe "q" avide ethaan aavashyamaaya ellah oorthoogoonal samaanathakaludeyum samyojanamaanu. athinaal "q"-nte nirakal eejanvektarukalaanu.
charithram.
qr algorithathinu munbulla "elr algoritam", qr vighadanathinu pakaram luu vighadanam upayogikkunnu. qr algoritam kooduthal sthirathayullathinaal elr algoritam ippol apoorvamaayi maathrame upayogikkunnulloo. ennirunnaalum, ithu qr algoritam vikasippikkunnathinulla oru pradhana ghattathe prathinidheekarikkunnu.
1950 kalude thudakkathil ethu soorichile edward steefalinte gaveshana sahayiyaayi pravarthicha hains ruttishousaraanu elr algoritam vikasippichath. y" 0t "a" "k" "x" 0 ", k" = 0,1, ennee nimishangalude kramam roottishousar upayogikkanamennu steefel nirddeshichu... (ivide "x" 0 umm "y" 0 umm ekapaksheeyamaaya vektarukalaanu) "a" yude eejan moolyangal kandethuka. eee jolikkayi alexander aitkente oru algoritam edutha roottishousar athine "kwaasant-deference algoritam" allengil "kyudi algoritam" aayi vikasippichu. kanakkukoottal anuyojyamaaya aakruthiyil krameekaricha sesham, qd algoritam yadhaarthathil "a" k "=" l "k" u "k" (luu vighadanam), "a" k "+ 1 =" u "k" l "k" enna aavarthanamaanennu addeham kandethi, ithu oru trikonamitheeya maatriksil prayogikkunnu, athil ninnu lr algoritam pinthudarunnu.
mattu vakabhedangal.
qr algoritam" ennathinte oru vakabhedamaaya "golab-kahan-rainsh" algoritam aarambhikkunnath oru pothu maatriksine dvibhujaakrithiyilekku kuraykkunnathil ninnaan. ekavachana moolyangalude kanakkukoottalinulla "qr algoritam"-nte eee vakabhedam aadyam vivarichathu. ekavachana moolyangal valare cheruthaanenkil chila parishkkaranangalode lapac subruitin dbdsqr eee aavarthana reethi nadappilaakkunnu. veettammamaarude prathiphalanangal upayogichulla aadya ghattavum uchithamenkil, qr vighadanavum upayogichu, ithu eaka moolya vighadanathinte kanakkukoottalinulla djsvd dinacharya roopappeduthunnu. anubandha kanverjans phalangal upayogichu ananthamaaya alavukalilum qr algoritam nadappilaakkaam.