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The Stein-Rosenberg theorem, proved in 1948, states that under certain premises, the Jacobi method and the Gauss-Seidel method are either both convergent, or both divergent. If they are convergent, then the Gauss-Seidel is asymptotically faster than the Jacobi method.
Statement
Let
A
=
(
a
i
j
)
∈
R
n
×
n
{\displaystyle A=(a_{ij})\in \mathbb {R} ^{n\times n}}
. Let
ρ
(
X
)
{\displaystyle \rho (X)}
be the spectral radius of a matrix
X
{\displaystyle X}
. Let
T
J
=
D
−
1
(
L
+
U
)
{\displaystyle T_{J}=D^{-1}(L+U)}
and
T
1
=
(
D
−
L
)
−
1
U
{\displaystyle T_{1}=(D-L)^{-1}U}
be the matrix splitting for the Jacobi method and the Gauss-Seidel method respectively.
Theorem: If
a
i
j
≤
0
{\displaystyle a_{ij}\leq 0}
for
i
≠
j
{\displaystyle i\neq j}
and
a
i
i
>
0
{\displaystyle a_{ii}>0}
for
i
=
1
,
…
,
n
{\displaystyle i=1,\ldots ,n}
. Then, one and only one of the following mutually exclusive relations is valid:
ρ
(
T
J
)
=
ρ
(
T
1
)
=
0
{\displaystyle \rho (T_{J})=\rho (T_{1})=0}
.
0
<
ρ
(
T
1
)
<
ρ
(
T
J
)
<
1
{\displaystyle 0<\rho (T_{1})<\rho (T_{J})<1}
.
1
=
ρ
(
T
J
)
=
ρ
(
T
1
)
{\displaystyle 1=\rho (T_{J})=\rho (T_{1})}
.
1
<
ρ
(
T
J
)
<
ρ
(
T
1
)
{\displaystyle 1<\rho (T_{J})<\rho (T_{1})}
.
Proof and applications
The proof uses the Perron-Frobenius theorem for non-negative matrices. Its proof can be found in Richard S. Varga's 1962 book Matrix Iterative Analysis.
In the words of Richard Varga:
the Stein-Rosenberg theorem gives us our first comparison theorem for two different iterative methods. Interpreted in a more practical way, not only is the point Gauss-Seidel iterative method computationally more convenient to use (because of storage requirements) than the point Jacobi iterative matrix, but it is also asymptotically faster when the Jacobi matrix
T
J
{\displaystyle T_{J}}
is non-negative
Employing more hypotheses, on the matrix
A
{\displaystyle A}
, one can even give quantitative results. For example, under certain conditions one can state that the Gauss-Seidel method is twice as fast as the Jacobi iteration.
References
Kata Kunci Pencarian:
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