式・未知数を入れ替えることにより ガウス・ザイデル法の収束の速さが変わる例
$\dps{
\left\{
\begin{array}{l}
3 x + 2 y + 2 z = 7 \\
2 x + 5 y - 4 z = 3 \\
x + y + 7 z = 9 \\
\end{array}
\right.
}$
を解きます。
k = 0 : (x,y,z) = ( 0.0000000000, 0.0000000000, 0.0000000000)
k = 1 : (x,y,z) = ( 2.3333333333, -0.3333333333, 1.0000000000)
k = 2 : (x,y,z) = ( 1.8888888889, 0.6444444444, 0.9238095238)
k = 3 : (x,y,z) = ( 1.2878306878, 0.8239153439, 0.9840362812)
k = 4 : (x,y,z) = ( 1.1280322499, 0.9360161250, 0.9908502322)
...
k = 24 : (x,y,z) = ( 1.0000000014, 0.9999999993, 0.9999999999)
k = 25 : (x,y,z) = ( 1.0000000005, 0.9999999997, 1.0000000000)
k = 26 : (x,y,z) = ( 1.0000000002, 0.9999999999, 1.0000000000)
k = 27 : (x,y,z) = ( 1.0000000001, 1.0000000000, 1.0000000000)
k = 28 : (x,y,z) = ( 1.0000000000, 1.0000000000, 1.0000000000)
同じ方程式を、式の順番・未知数の順番を全て逆にして
$\dps{
\left\{
\begin{array}{l}
7 x + y + z = 9 \\
- 4 x + 5 y + 2 z = 3 \\
2 x + 2 y + 3 z = 7 \\
\end{array}
\right.
}$
として解きます。
k = 0 : (x,y,z) = ( 0.0000000000, 0.0000000000, 0.0000000000)
k = 1 : (x,y,z) = ( 1.2857142857, 1.6285714286, 0.3904761905)
k = 2 : (x,y,z) = ( 0.9972789116, 1.2416326531, 0.8407256236)
k = 3 : (x,y,z) = ( 0.9882345319, 1.0542973761, 0.9716453947)
k = 4 : (x,y,z) = ( 0.9962938899, 1.0083769540, 0.9968861040)
...
k = 12 : (x,y,z) = ( 1.0000000027, 1.0000000143, 0.9999999887)
k = 13 : (x,y,z) = ( 0.9999999996, 1.0000000042, 0.9999999975)
k = 14 : (x,y,z) = ( 0.9999999998, 1.0000000008, 0.9999999996)
k = 15 : (x,y,z) = ( 0.9999999999, 1.0000000001, 1.0000000000)
k = 16 : (x,y,z) = ( 1.0000000000, 1.0000000000, 1.0000000000)
随分ステップ数が少なくなりました。