Ko zaporedne potence matrike
T
{\displaystyle \mathbf {T} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
Regularna razdelitev nesingularne matrike
A
{\displaystyle \mathbf {A} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
A
{\displaystyle \mathbf {A} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
iterativna metoda konvergira za vsak začetni vektor , če je
T
{\displaystyle \mathbf {T} \!\,}
T
{\displaystyle \mathbf {T} \!\,}
Matrika
T
{\displaystyle \mathbf {T} \!\,}
n
×
n
{\displaystyle n\times n\!\,}
lim
k
→
∞
(
T
k
)
i
j
=
0
,
{\displaystyle \lim _{k\to \infty }(\mathbf {T} ^{k})_{ij}=0\!\,,}
(1 )
za vsak
i
=
1
,
2
,
…
,
n
{\displaystyle i=1,2,\ldots ,n\!\,}
j
=
1
,
2
,
…
,
n
{\displaystyle j=1,2,\ldots ,n\!\,}
[ 1] :404 [ 2] :14 [ 3] :13
Naj je matrika:
T
=
[
1
4
1
2
0
1
4
]
.
{\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{bmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{bmatrix}}\!\,.\end{aligned}}}
Če se izračunajo njene zaporedne potence, sledi:
T
2
=
[
1
16
1
4
0
1
16
]
,
T
3
=
[
1
64
3
32
0
1
64
]
,
T
4
=
[
1
256
1
32
0
1
256
]
,
T
5
=
[
1
1024
5
512
0
1
1024
]
,
{\displaystyle {\begin{aligned}&\mathbf {T} ^{2}={\begin{bmatrix}{\frac {1}{16}}&{\frac {1}{4}}\\[4pt]0&{\frac {1}{16}}\end{bmatrix}},\quad \mathbf {T} ^{3}={\begin{bmatrix}{\frac {1}{64}}&{\frac {3}{32}}\\[4pt]0&{\frac {1}{64}}\end{bmatrix}},\quad \mathbf {T} ^{4}={\begin{bmatrix}{\frac {1}{256}}&{\frac {1}{32}}\\[4pt]0&{\frac {1}{256}}\end{bmatrix}},\quad \mathbf {T} ^{5}={\begin{bmatrix}{\frac {1}{1024}}&{\frac {5}{512}}\\[4pt]0&{\frac {1}{1024}}\end{bmatrix}}\!\,,\end{aligned}}}
T
6
=
[
1
4096
3
1024
0
1
4096
]
,
{\displaystyle {\begin{aligned}\mathbf {T} ^{6}={\begin{bmatrix}{\frac {1}{4096}}&{\frac {3}{1024}}\\[4pt]0&{\frac {1}{4096}}\end{bmatrix}}\!\,,\end{aligned}}}
in v splošnem:
T
k
=
[
(
1
4
)
k
k
2
2
k
−
1
0
(
1
4
)
k
]
.
{\displaystyle {\begin{aligned}\mathbf {T} ^{k}={\begin{bmatrix}({\frac {1}{4}})^{k}&{\frac {k}{2^{2k-1}}}\\[4pt]0&({\frac {1}{4}})^{k}\end{bmatrix}}\!\,.\end{aligned}}}
Ker velja:
lim
k
→
∞
(
1
4
)
k
=
0
{\displaystyle \lim _{k\to \infty }\left({\frac {1}{4}}\right)^{k}=0\!\,}
in:
lim
k
→
∞
k
2
2
k
−
1
=
0
,
{\displaystyle \lim _{k\to \infty }{\frac {k}{2^{2k-1}}}=0\!\,,}
je
T
{\displaystyle \mathbf {T} \!\,}
ρ
(
T
)
=
1
/
4
{\displaystyle \rho (\mathbf {T} )=1/4\!\,}
ρ
{\displaystyle \rho \!\,}
spektralni polmer matrike
T
{\displaystyle \mathbf {T} \!\,}
lastna vrednost matrike
T
{\displaystyle \mathbf {T} \!\,}
Naj je
T
{\displaystyle \mathbf {T} \!\,}
n
×
n
{\displaystyle n\times n\!\,}
T
{\displaystyle \mathbf {T} \!\,}
lim
k
→
∞
‖
T
k
‖
=
0
{\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0\!\,}
lim
k
→
∞
‖
T
k
‖
=
0
{\displaystyle \lim _{k\to \infty }\|\mathbf {T} ^{k}\|=0\!\,}
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1\!\,}
lim
k
→
∞
T
k
x
=
0
{\displaystyle \lim _{k\to \infty }\mathbf {T} ^{k}\mathbf {x} =\mathbf {0} \!\,}
x
{\displaystyle \mathbf {x} \!\,}
[ 1] :404 [ 2] :14, 63 [ 3] :122 [ 3] :13
Splošna iterativna metoda vključuje postopek, ki pretvori sistem linearnih enačb :
A
x
=
b
{\displaystyle \mathbf {Ax} =\mathbf {b} }
(2 )
v ekvivalentni sistem oblike:
x
=
T
x
+
c
{\displaystyle \mathbf {x} =\mathbf {Tx} +\mathbf {c} }
(3 )
za neko matriko
T
{\displaystyle \mathbf {T} \!\,}
c
{\displaystyle \mathbf {c} \!\,}
x
(
0
)
{\displaystyle \mathbf {x} ^{(0)}\!\,}
x
(
k
+
1
)
=
T
x
(
k
)
+
c
{\displaystyle \mathbf {x} ^{(k+1)}=\mathbf {Tx} ^{(k)}+\mathbf {c} }
(4 )
za vsak
k
≥
0
{\displaystyle k\geq 0\!\,}
[ 1] :406 [ 3] :61 Za poljubni začetni vektor
x
(
0
)
∈
R
n
{\displaystyle \mathbf {x} ^{(0)}\in \mathbb {R} ^{n}\!\,}
{
x
(
k
)
}
k
=
0
∞
{\displaystyle \lbrace \mathbf {x} ^{\left(k\right)}\rbrace _{k=0}^{\infty }\!\,}
4
k
≥
0
{\displaystyle k\geq 0\!\,}
c
≠
0
{\displaystyle \mathbf {c} \neq 0\!\,}
3 če in samo če je
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1\!\,}
T
{\displaystyle \mathbf {T} \!\,}
[ 1] :412 [ 2] :62–63
Razdelitev matrike je izraz, ki predstavlja dano matriko kot vsoto ali razliko matrik. V sistemu linearnih enačb (2
A
{\displaystyle \mathbf {A} \!\,}
A
{\displaystyle \mathbf {A} \!\,}
B
{\displaystyle \mathbf {B} \!\,}
C
{\displaystyle \mathbf {C} \!\,}
A
=
B
−
C
,
{\displaystyle \mathbf {A} =\mathbf {B} -\mathbf {C} \!\,,}
(5 )
tako, da se lahko enačba (2 4 5
A
{\displaystyle \mathbf {A} \!\,}
B
−
1
≥
0
{\displaystyle \mathbf {B} ^{-1}\geq 0\!\,}
C
≥
0
{\displaystyle \mathbf {C} \geq 0\!\,}
B
1
{\displaystyle \mathbf {B} ^{1}\!\,}
C
{\displaystyle \mathbf {C} \!\,}
5 regularna razdelitev matrike
A
{\displaystyle \mathbf {A} \!\,}
A
−
1
≥
0
{\displaystyle \mathbf {A} ^{-1}\geq 0\!\,}
ρ
(
T
)
<
1
{\displaystyle \rho (\mathbf {T} )<1\!\,}
T
{\displaystyle \mathbf {T} \!\,}
4 [ 4] :122–123 [ 3] :89
Matrika
T
{\displaystyle \mathbf {T} \!\,}
n
×
n
{\displaystyle n\times n\!\,}
polkonvergentna matrika , če limita:
lim
k
→
∞
T
k
{\displaystyle \lim _{k\to \infty }\mathbf {T} ^{k}\!\,}
(6 )
obstaja.[ 5] :699 Če je
A
{\displaystyle \mathbf {A} \!\,}
2
b
{\displaystyle \mathbf {b} \!\,}
A
{\displaystyle \mathbf {A} \!\,}
4 2
x
(
0
)
∈
R
n
{\displaystyle \mathbf {x} ^{(0)}\!\,\in \mathbb {R} ^{n}\!\,}
T
{\displaystyle \mathbf {T} \!\,}
5 polkonvergentna razdelitev matrike
A
{\displaystyle \mathbf {A} \!\,}
[ 5] :700
Burden, Richard L. ; Faires, John Douglas (1993), Numerical Analysis izd.), Boston: Prindle, Weber and Schmidt , COBISS 375380 , ISBN 978-0-53-493219-0 OCLC 258508981 Isaacson, Eugene ; Keller, Herbert Bishop (1994), Analysis of Numerical Methods , New York: Dover , COBISS 4820252 , ISBN 978-0-486-68029-3 OCLC 912154846 Meyer, Carl Dean mlajši ; Plemmons, Robert James (september 1977), »Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems«, SIAM Journal on Numerical Analysis 14 (4): 699– 705, Bibcode :1977SJNA...14..699M , doi :10.1137/0714047 , ISSN 0036-1429 Varga, Richard Steven (1960), »Factorization and Normalized Iterative Methods«, v Langer, Rudolf Ernest (ur.), Boundary Problems in Differential Equations , Madison: University of Wisconsin Press , str. 121– 142, COBISS 6782553 , LCCN 60-60003 Varga, Richard Steven (1962), Matrix Iterative Analysis , Englewood Cliffs, New Jersey: Prentice–Hall , Bibcode :1962mia..book.....V , COBISS 21167638 , LCCN 62-21277
![{\displaystyle {\begin{aligned}&\mathbf {T} ={\begin{bmatrix}{\frac {1}{4}}&{\frac {1}{2}}\\[4pt]0&{\frac {1}{4}}\end{bmatrix}}\!\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6079ff935a4819c57377cd05adbbc7e1ddcc6e82)
![{\displaystyle {\begin{aligned}&\mathbf {T} ^{2}={\begin{bmatrix}{\frac {1}{16}}&{\frac {1}{4}}\\[4pt]0&{\frac {1}{16}}\end{bmatrix}},\quad \mathbf {T} ^{3}={\begin{bmatrix}{\frac {1}{64}}&{\frac {3}{32}}\\[4pt]0&{\frac {1}{64}}\end{bmatrix}},\quad \mathbf {T} ^{4}={\begin{bmatrix}{\frac {1}{256}}&{\frac {1}{32}}\\[4pt]0&{\frac {1}{256}}\end{bmatrix}},\quad \mathbf {T} ^{5}={\begin{bmatrix}{\frac {1}{1024}}&{\frac {5}{512}}\\[4pt]0&{\frac {1}{1024}}\end{bmatrix}}\!\,,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91ddc71dd50939d92c6b967dbb3c90aa6dd02e9c)
![{\displaystyle {\begin{aligned}\mathbf {T} ^{6}={\begin{bmatrix}{\frac {1}{4096}}&{\frac {3}{1024}}\\[4pt]0&{\frac {1}{4096}}\end{bmatrix}}\!\,,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa5bc33a6c4956b4d52095c63380842443c6c21)
![{\displaystyle {\begin{aligned}\mathbf {T} ^{k}={\begin{bmatrix}({\frac {1}{4}})^{k}&{\frac {k}{2^{2k-1}}}\\[4pt]0&({\frac {1}{4}})^{k}\end{bmatrix}}\!\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe7030eff0876e7b831451d1ecb0af19f6309d4)
