
janta vaareegaa sankalanam

sankhya vislaeshanhaloo, cascade summation anie kooda piluvabade jatha vaareegaa summation anedi parimita-khachitamaina floating-point sankhyala kramaanni sankalanam chese oka saanketikata, idhi amayakamga kramamlo mottaanni koodabettadamtho poliste perukupoyina round-off errarnu gananeeyamgaa taggistundi. saadhaaranamgaa inca chinna round-off lopaalanu kaligi unde kahan sankalanam vanti itara paddhatulu unnappatikee, jatha vaareegaa sankalanam daadaapu antha manchidi (lagarithmic karakam dwara matrame bhinnamgaa untundi), ayithe chaalaa takkuva ganana vyayaanni kaligi untundi-idhi amayaka sankalanam vale daadaapu adhe kharchunu (mariyu sarigga adhe sankhyalo ankaganita karyakalapalu) kaligi undela amalu cheyavachu.
mukhyamgaa, "n" sankhyala "xn" kramam yokka jatha vaareegaa sankalanam kramaanni rendu bhagaluga vibhajinchadam dwara, prati sagam mottaanni jodinchadam dwara mariyu rendu mottaalanu jodinchadam dwara panichestundihi oka vibhajana mariyu jayinche algorithm. daani chettha-kesu roundoff lopaalu garishtamgaa "o" (ε logg "n") gaa lakshanarahitamgaa perugutaayi, ikkada ε anedi yantram yokka khachitatvam (krinda charchinchinatlugaa sthira paristhiti sankhyanu oohinchi). polchi choosthe, mottaanni kramamlo koodabettukune amayaka saanketikata ("i" = 1 kosam prati "xi" nii oka samayamlo okati jodinchadam,..., n") roundoff lopaalu kaligi undhi, ivi "o" (ε "n") gaa chettagaa perugutaayi. kahan sankalanam "n" nundi swatantramgaa, sumaruga "o" (ε) yokka chettha-kesu doshaanni kaligi untundi, conny aneka retlu ekkuva ankaganita karyakalapalu avasaram. roundoff lopaalu yaadrucchikamgaa unte, mariyu mukhyamgaa yaadruchchika sankethaalu unte, avi yaadruchchika nadakanu yerparustaayi mariyu dosha perugudala janta vaareegaa sankalanam kosam suutram _ 1 sagatuku tagginchabadutundi.
aneka vegavantamaina forier transform (fft) algorithamlalo chaalaa saaruupya punaraavruta kuurpu kanipistundi, mariyu aaa fft yokka adhe nemmadigaa roundoff perukupovadaniki idhi baadhyata vahistundi.
algorithm.
sudocode loe, ≤0 podavu shreni kosam jatha vaareegaa sankalanam algorithm vraayavachuhu
 "s" = jatha vaareegaa ("x" [1.."enn"))
 n" ≤ "" n "" bays cases ayiteh taginanta chinna shreni kosam amayaka sankalanam "
 "s" = 0
 i" = 1 nundi "n" kosam"
 "s" = "" s "" + "" x "" ["" i "" "]
 lekapothe "vibhajinchandi mariyu jayinchandih varusalooni rendu bhaagaalanu punaraavrutamgaa sameekarinchandi"
 "m" = naela ("n"/2)
 "s" = jatha vaareegaa ("x" [1.."m"]) + jatha vaareegaa ("x" ["m" + 1.."enn"))
 unte muginchandi
kontamandiki taginanta chinnadigaa, eee algorithm oka bays kesga naiv loop-aadhaarita sammeshanku maarutundi, dheeni error bound o (nε). mottam mottamlo chettha-kesu lopam undhi, idhi icchina paristhiti sankhya kosam pedda "n" kosam "o" (ε logg "n") gaa lakshanarahitamgaa perugutundi (krinda chudandi).
eee vidhamaina algorithamlo (saadhaaranamgaa algorithamlanu vibhajinchi, jayinchadam kosam), punaraavrutam yokka ovarheadnu tirigi chellinchadaaniki pedda bays kesnu upayoginchadam manchidi. n" = 1 ayithe, appudu prati inputku sumaruga oka recursive subrutain coll untundi, conny marinta saadhaaranamgaa recursion sarigga "n" = "n" vadda aagithe prati "n"/2 inputlaku (sumaruga) oka recursive coll untundi. n" nuu taginantagaa peddadigaa cheyadam dwara, punaraavrutam yokka ovarheadnu chaalaa takkuvagaa cheyavachu (khachitamgaa punaraavruta sankalanam kosam pedda bays cases yokka eee saanketikata adhika-paniteeru gala fft amalu dwara upayoginchabadutundi).
n" thoo sambandham lekunda, sarigga "n" −1 cherpulu mottam nirvahinchabadataayi, idhi amayaka sankalanam maadirigaane untundi, kabatti punaraavruta ovarheadnu chaalaa takkuvagaa chesthe, appudu janta vaareegaa sankalanam tappanisarigaa amayaka sankalanam kosam adhe ganana kharchunu kaligi untundi.
eee aalochanalo oka vaividhyam emitante, prati punaraavruta dasalo mottaanni "bii" blaxga vibhajinchadam, prati blacknu punaraavrutamgaa sankalanam cheyadam, aapai phalitaalanu sankalanam cheyadam, dheenini daani pratipaadanadaarulu "superblock" algorithm anie pilicharu. paina perkonna jatha vaarii algorithm "b" = 2kii anugunamgaa untundi, chivari dasha "b" = "n" tappa prati dasaku.
khachitatvam.
okati "i" = 1 kosam "x" i "viluvalanu" n "viluvalanu sankalanam chestondani anukundam..., enn. khachitamaina mottamha
(anantamaina khachitatvamto lekkinchabadindi).
bays cases "n" = 1 kosam jatha vaareegaa sankalanamtho, badulugaa suutram _ 3 nuu pondutundi, ikkada error suutram _ 4 paina eee krindi vidhamgaa untundih
ikkada ε anedi upayoginchabadutunna ankaganitam yokka yantra khachitatvam (e.zii. praamaanika double khachitatva teliade binduvu kosam ε ± 10−16). saadhaaranamgaa, vaddii parimaanam saapeksha dosha suutram _ 6, anduvalla idhi paina perkonna vidhamgaa untundih
saapeksha dosham kattubadi unna vyakteekaranalo, bhinnam (σ)|"xi"|/|σ "xi"|) anedi summation samasya yokka sharatu sankhya. tappanisarigaa, paristhiti sankhya anedi elaa lekkinchabadinaa, doshaalaku sankalanam samasya yokka "antargata" sunnitatvaanni suuchistumdi. sthiramaina khachitatvamto sthiramaina algorithm dwara "prati" (venukaku sthiramaina) sankalanam paddhati yokka saapeksha lopam (i.ii. ekapaksha-khachitamaina ankaganitam upayoginchevi, ledhaa dayta aadhaaramgaa memory mariyu samaya avasaralu maare algorithamlu) eee paristhiti sankhyaku anulomaanupaatamlo undavu. anaarogyakaramaina" sankalanam samasya anedi eee nishpatti ekkuvagaa unna samasya, mariyu eee sandarbhamlo janta vaareegaa kuudika kooda pedda saapeksha lopaanni kaligi untundi. udaaharanaku, "xi" anae sankalanaalu sunna sagatutho sambandham laeni yaadruchchika sankhyalu ayithe, mottam yaadruchchika nadaka mariyu sharatu sankhya suutram _ 8kii anulomaanupaatamlo perugutundi. marovaipu, sunna kaani yaadruchchika inputla kosam sharatu sankhya suutram _ 9 gaa parimita sthiraamkaaniki lakshanarahitam. inputlu annee pratikuulamgaa lekapothe, appudu sharatu sankhya 1 avutundi.
suutram _ 10 denaminator aacharanalo samarthavantamgaa 1 anie gamaninchandi, endukante suutram _ 11 anedi 1 kante chaalaa chinnadi, endukante "n" anedi 21/ε kramamlo untundi, idhi daadaapu 101015 rettimpu khachitatvam kaligi untundi.
polchi choosthe, amaayakamaina sankalanam (kevalam kramaanugatamgaa sankhyalanu jodinchadam, prati dasalo rounding) kuu kattubadi unna saapeksha lopam formula _ 12 nuu condition sankhyatho guninchadam vantidigaa perugutundi. aacharanalo, rounding lopaalu sunna sagatutho yaadruchchika gurtunu kaligi unde avakaasam undhi, tadwara avi yaadruchchika nadakanu yerparustaayi; eee sandarbhamlo, amayaka sankalanaaniki muula sagatu varga saapeksha lopam untundi, idhi suutram _ 13 gaa perugutundi mariyu jatha vaareegaa sankalanamlo oka lopam untundi, idhi suutram _ 1 gaa perugutundi.
software amalu.
jatala vaareegaa sankalanam anedi nampee mariyu julia technical-computing languagelo default summation algorithm, ikkada rendu sandarbhaallo idhi amayaka sankalanamtho polchadagina vegaanni kaligi unnatlu kanugonabadindi (pedda bays kesnu upayoginchadam valla).
itara software amalu karyakramalalo sii sharp longuage kosam hetch pisi sharp library mariyu dii loe praamaanika library sankalanam unnaayi.

