数值分析作业-第四次作业

4th Homework of Numerical Analysis

第四次作业截图

1

$$\left(\begin{array}{ccc}3& -1& 1\\ 1& -8& -2\\ 1& 1& 5\end{array}\right)\left(\begin{array}{c}u\\ v\\ w\end{array}\right)=\left(\begin{array}{c}-2\\ 1\\ 4\end{array}\right)$$

• Jacobi

$$\begin{array}{rl}\left(\begin{array}{c}{u}_{k+1}\\ v_{k+1}\\ w_{k+1}\end{array}\right)& ={\left(\begin{array}{ccc}3& & \\ & -8& \\ & & 5\end{array}\right)}^{-1}\left(\begin{array}{ccc}0& 1& -1\\ -1& 0& 2\\ -1& -1& 0\end{array}\right)\left(\begin{array}{c}{u}_k\\ v_k\\ w_k\end{array}\right)+{\left(\begin{array}{ccc}3& & \\ & -8& \\ & & 5\end{array}\right)}^{-1}\left(\begin{array}{c}-2\\ 1\\ 4\end{array}\right)\\ & =\left(\begin{array}{ccc}0& 1/3& -1/3\\ 1/8& 0& -1/4\\ -1/5& -1/5& 0\end{array}\right)\left(\begin{array}{c}{u}_k\\ v_k\\ w_k\end{array}\right)+\left(\begin{array}{c}-2/3\\ -5/40\\ 4/5\end{array}\right)\end{array}$$

• Gauss-Seidel

$$\begin{array}{rl}\left(\begin{array}{c}{u}_{k+1}\\ v_{k+1}\\ w_{k+1}\end{array}\right)& ={\left(\begin{array}{ccc}3& & \\ 1& -8& \\ 1& 1& 5\end{array}\right)}^{-1}\left(\begin{array}{ccc}0& 1& -1\\ 0& 0& 2\\ 0& 0& 0\end{array}\right).\left(\begin{array}{c}{u}_k\\ v_k\\ w_k\end{array}\right)+{\left(\begin{array}{ccc}3& & \\ 1& -8& \\ 1& 1& 5\end{array}\right)}^{-1}\left(\begin{array}{c}-2\\ 1\\ 4\end{array}\right)\\ & =\left(\begin{array}{ccc}0& 1/3& -1/3\\ 1/8& 1/24& -7/24\\ -9/40& -11/40& 1/8\end{array}\right).\left(\begin{array}{c}{u}_k\\ v_k\\ w_k\end{array}\right)+\left(\begin{array}{c}-2/3\\ -5/24\\ 39/40\end{array}\right)\end{array}$$

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2

比较后发现三者对于第一题的原始系数矩阵不收敛，对于重新组织后的严格行对角占优矩阵收敛。截图中第一个为原始矩阵的迭代结果，第二个为重新组织后的矩阵迭代结果。

Jacobi

Jacobi iteraion snat

Gauss-Seidel

Gauss-Seidel

SOR

(sigma = 1.25)

SOR with sigma = 1.25, maxiteration = 1000

(sigma = 0.7)

SOR with sigma = 0.7, maxiteration = 1000

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Jacobi 和 高斯消元法比较

本例中使用的 $A,x_0,b$:

所用矩阵定义，order=12

这里是12阶的矩阵，这里对 $order$ 取 $10, 100, 1000, 10000$ 比较不同方法的速度和内存占用

| Order | 10 | 100 | 1000 | |:----------------:|:---:|:---:|:----:|:-----:| | Memory of Jacobi /KB | 88.0 | 1700.0 | 12924 | | Memory of Gauss /KB | 618.0 | 1764.0 | 27348.0 | | | Time of Jacobi /ms | 4.00 | 27.28 | 69.72 | | Time of Gauss /ms | 47.34 | 21.13 | 1780.40 | | | | | |

两种方法的内存占用比较

type: bar
labels: [10, 100, 1000]
series:
  - title: MoJ
    data: [88, 1700, 12924]
  - title: MoG
    data: [618, 1764, 27348]
tension: 0.2
width: 80%
labelColors: false
fill: false
beginAtZero: false

两种方法的时间占用比较

type: bar
labels: [10, 100, 1000]
series:
  - title: ToJ
    data: [4, 27, 69]
  - title: ToG
    data: [47, 21, 1780]
tension: 0.2
width: 80%
labelColors: false
fill: false
beginAtZero: false

可以看出，在实例中的稀疏矩阵中使用Jacobi等迭代方法具有明显的内存和时间优势。