Use the iterative method to solve [K]{U} = {F} where. [latex][K]=\begin{bmatrix} 2 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0\\ 0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & -1 & 2 \end{bmatrix}[/latex] and [latex]\left\{F\right\} =\begin{Bmatrix} 5 \\ 2 \\ 2 \\ 2 \\5 \end{Bmatrix} [/latex]

Let [latex]\left\{U\right\} ^{(0)} [/latex] = [1 1 1 1 1] and the terminate tolerance ε = 0.0001. Using Eq. (3.99) to execute the iterative process, the process to obtain the final solutions is [latex]U^{(k+ 1)}_{i} =\frac{1}{k_{ii} } \left[F_{i}-\sum\limits_{j=1;j\neq i}^{N}{K_{ij}U^{(k)}_{j} } \right] , (i=1,2,...N)[/latex] (3.99) Step [latex]\left\{x_{i} \right\} ^{(k)} [/latex] Convergence 0 [1 1 1 1 1] 1 [3.0000 2.0000 2.0000 2.0000 3.0000] 3.5000 10 [6.5762 8.6270 9.1523 8.6270 6.5762] 0.7750 20 [7.6621 10.4369 11.3242 10.4369 7.6621] 0.1839 30 [7.9198 10.8664 11.8396 10.8664 7.9198] 0.0436 40 [7.9810 10.9683 11.9619 10.9683 7.9810] 0.0104 50 [7.9955 10.9925 11.9910 10.9925 7.9955] 0.0025 60 [7.9989 10.9982 11.9979 10.9982 7.9989] 5.8325e-04 70 [7.9997 10.9996 11.9995 10.9996 7.9997] 1.3841e-04 73 [7.9998 10.9997 11.9997 10.9997 7.9998] 8.9898e-05

Question 3.6: Use the iterative method to solve [K]{U} = {F} where. [K]=[2...

Practical Guide to Digital Manufacturing

Use the iterative method to solve [K]{U} = {F} where.

$[K]=\begin{bmatrix} 2 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0\\ 0 & -1 & 2 & -1 & 0\\ 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & -1 & 2 \end{bmatrix}$ and $\left\{F\right\} =\begin{Bmatrix} 5 \\ 2 \\ 2 \\ 2 \\5 \end{Bmatrix}$

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Let $\left\{U\right\} ^{(0)}$ = [1 1 1 1 1] and the terminate tolerance ε = 0.0001.

Using Eq. (3.99) to execute the iterative process, the process to obtain the final solutions is

$U^{(k+ 1)}_{i} =\frac{1}{k_{ii} } \left[F_{i}-\sum\limits_{j=1;j\neq i}^{N}{K_{ij}U^{(k)}_{j} } \right] , (i=1,2,…N)$ (3.99)

Step	$\left\{x_{i} \right\} ^{(k)}$	Convergence
0	[1 1 1 1 1]
1	[3.0000 2.0000 2.0000 2.0000 3.0000]	3.5000
10	[6.5762 8.6270 9.1523 8.6270 6.5762]	0.7750
20	[7.6621 10.4369 11.3242 10.4369 7.6621]	0.1839
30	[7.9198 10.8664 11.8396 10.8664 7.9198]	0.0436
40	[7.9810 10.9683 11.9619 10.9683 7.9810]	0.0104
50	[7.9955 10.9925 11.9910 10.9925 7.9955]	0.0025
60	[7.9989 10.9982 11.9979 10.9982 7.9989]	5.8325e-04
70	[7.9997 10.9996 11.9995 10.9996 7.9997]	1.3841e-04
73	[7.9998 10.9997 11.9997 10.9997 7.9998]	8.9898e-05