Principem iteračních metod je postupné zpřesňování počátečního
odhadu řešení. Iterační metody lze užívat pouze v případě, že odhady
konvergují k přesnému řešení. Iterační proces se ukončuje v případě,
že bylo dosaženo požadované přesnosti. Soustavu
A x=b
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyqaiaaykW7caWH4bGaaCypaiaahkgaaaa@3B14@
rozepsanou
|
[
a
11
a
12
⋯
a
1n
a
21
a
22
⋯
a
2n
⋮
a
n1
a
n2
⋯
a
nn
][
x
1
x
2
⋮
x
n
]=[
b
1
b
2
⋮
b
n
]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6E84@
|
(1)
|
převedeme na ekvivalentní tvar vhodný pro iteraci:
x=H x+g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEaiaah2dacaWHibGaaGPaVlaahIhacaWHRaGaaC4zaaaa@3CD5@
.
Iterační metody dělíme na metody stacionární, které mají
iterační matici
H
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisaaaa@36DE@
a vektor
g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4zaaaa@36FD@
konstantní a na metody nestacionární. Většina
iteračních metod je stacionárních - matice
H
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisaaaa@36DE@
i vektor
g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaC4zaaaa@36FD@
se tedy po dobu výpočtu nemění. Pro analýzu i
výpočet je to výhodné. U nestacionárních metod však lze urychlit konvergenci
iteračních procesů.
Pak se jednobodová lineární stacionární iterační metoda dá
charakterizovat předpisem:
x
(i+1)
=H
x
i
+g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaadMgacqGHRaWkcaaIXaGaaiykaaaakiaah2dacaWHibGaaGPaVlaahIhadaahaaWcbeqaaiaadMgaaaGccaWHRaGaaC4zaaaa@4215@
Matice
H
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisaaaa@36DE@
je iterační matice. Pokud je splněno kriterium
konvergence: jestliže je některá z norem matice
H
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCisaaaa@36DE@
menší než jedna, pak metoda prosté iterace
konverguje. Posloupnost iterací
x
(i+1)
=H×
x
i
+g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaadMgacqGHRaWkcaaIXaGaaiykaaaakiaah2dacaWHibGaaC41aiaahIhadaahaaWcbeqaaiaadMgaaaGccaWHRaGaaC4zaaaa@41EA@
,
pro
i=1,2,…
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIMaYsaaa@3BDB@
vede k řešení soustavy. Počáteční iterace
x
(0)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaaicdacaGGPaaaaaaa@394E@
(zahájení výpočtu) je v případě
konvergentního procesu libovolná; často se užívá nulový vektor. Ukončení
výpočtu provádíme mnohdy podmínkou
‖
x
k
−
x
k−1
‖
z
<δ
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacaWG4bWaaWbaaSqabeaacaWGRbaaaOGaeyOeI0IaamiEamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaaakiaawMa7caGLkWoadaWgaaWcbaGaamOEaaqabaGccqGH8aapcqaH0oazaaa@43EF@
,
kde
‖
x
k
−
x
k−1
‖
z
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacaWG4bWaaWbaaSqabeaacaWGRbaaaOGaeyOeI0IaamiEamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaaakiaawMa7caGLkWoadaWgaaWcbaGaamOEaaqabaaaaa@413C@
je jedna z vektorových norem a číslo
δ>0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2exLMBb50ujbqeguuDJXwAKbacfaGae8Npa4JaaGimaaaa@3E33@
.
Jiný způsob je stanovení podmínky, kolik výpočtů máme provést.
V současnosti se užívají různé iterační formule: Jacobiova,
Gaussova-Saidlova, atd.
Postačující podmínkou konvergence je vztah
‖ H ‖≤q<1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacaWGibaacaGLjWUaayPcSdGaeyizImQaamyCaiabgYda8iaaigdaaaa@3E6B@
Pak posloupnost
{
x
(k)
}
k=0
∞
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWH4bWaaWbaaSqabeaacaGGOaGaam4AaiaacMcaaaaakiaawUhacaGL9baadaqhaaWcbaGaam4Aaiabg2da9iaaicdaaeaacaaMc8UaaGPaVlaaykW7cqGHEisPaaaaaa@44AE@
určená ze vztahu
x
(k+1)
=H×
x
k
+g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaaakiabg2da9iaahIeacaWHxdGaaCiEamaaCaaaleqabaGaam4AaaaakiabgUcaRiaahEgaaaa@425C@
konverguje při libovolné volbě vektoru
x
(0)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaaicdacaGGPaaaaaaa@394E@
.
i
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36DC@
-tá rovnice v (1)
je
a
i1
x
1
+
a
i2
x
2
+⋯+
a
in
x
n
+=
b
i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyyamaaBaaaleaacaWGPbGaaGOmaaqabaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey4kaSIaeS47IWKaey4kaSIaamyyamaaBaaaleaacaWGPbGaamOBaaqabaGccaWG4bWaaSbaaSqaaiaad6gaaeqaaOGaey4kaSIaeyypa0JaamOyamaaBaaaleaacaWGPbaabeaaaaa@4D15@
,
i=1,2,…,n
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIMaYscaGGSaGaamOBaaaa@3D7E@
Pokud
a
i,1
≠0
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGPbGaaiilaiaaigdaaeqaaOGaeyiyIKRaaGimaaaa@3BE4@
,
získáme
x
i
=
1
a
ii
(
b
i
−
∑
j=1
j≠i
n
a
ij
x
j
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaadggadaWgaaWcbaGaamyAaiaadMgaaeqaaaaakiaacIcacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0YaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadIhadaWgaaWcbaGaamOAaaqabaaaeaqabeaacaWGQbGaeyypa0JaaGymaaqaaiaadQgacqGHGjsUcaWGPbaaaeaacaWGUbaaniabggHiLdGccaGGPaaaaa@4FC9@
,
i=1,2,…,n
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIMaYscaGGSaGaamOBaaaa@3D7E@
Z
i
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36DC@
-té rovnice jsme spočítali
i
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36DC@
-tou neznámou
Jacobiova iterační formuli pro
k=1,2,…
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x
i
(k+1)
=
1
a
ii
(
b
i
−
∑
j=1
j≠i
n
a
ij
x
j
(k)
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaDaaaleaacaWGPbaabaGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaaakiabg2da9maalaaabaGaaGymaaqaaiaadggadaWgaaWcbaGaamyAaiaadMgaaeqaaaaakiaacIcacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0YaaabCaeaacaWGHbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaadIhadaqhaaWcbaGaamOAaaqaaiaacIcacaWGRbGaaiykaaaaaqaabeqaaiaadQgacqGH9aqpcaaIXaaabaGaamOAaiabgcMi5kaadMgaaaqaaiaad6gaa0GaeyyeIuoakiaacMcaaaa@55FA@
,
i=1,2,…,n
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiabg2da9iaaigdacaGGSaGaaGOmaiaacYcacqWIMaYscaGGSaGaamOBaaaa@3D7E@
maticově
x
(k+1)
=H×
x
k
+g
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaCaaaleqabaGaaiikaiaadUgacqGHRaWkcaaIXaGaaiykaaaakiabg2da9iaahIeacaWHxdGaaCiEamaaCaaaleqabaGaam4AaaaakiabgUcaRiaahEgaaaa@425C@
Existuje efektivnější algoritmus: Gaussova-Seidelova metoda. Ta
je ve srovnání s Jacobiho metodou odlišná v tom, že vypočtenou iteraci
první neznámé ihned použijeme k výpočtu druhé neznámé.
Podrobně se s postupem výpočtu seznámíte ve studijním
článku „Výukový program v Delphi“.