Answered: For the linear system we have the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

For the linear system we have the following equations:

7x₁ − 2x₂ + x₃ = 17
x₁ − x₂ + x₃ + 6x₄ = 10
x₁ − 9x₂ + 3x₃ − x₄ = 13
2x₁ + 10x₃ + x₄ = 15

A) Check if the system of equations is dominant
B) Solve by the Jacobi method and the Gauss-Seidel method
C) For calculations use up to 4 figures after the decimal point
D) Iterate 7 times for both methods.

Expert Solution

Step 1

We are given the following linear system.

7x₁ − 2x₂ + x₃ = 17
x₁ − x₂ + x₃ + 6x₄ = 10
x₁ − 9x₂ + 3x₃ − x₄ = 13
2x₁ + 10x₃ + x₄ = 15

For (A),

We first need to check if the system of equations is dominant.

The above system can be written as $(\begin{matrix} 7 & - 2 & 1 & 0 \\ 1 & - 1 & 1 & 6 \\ 1 & - 9 & 3 & - 1 \\ 2 & 0 & 10 & 1 \end{matrix}) (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}) = (\begin{matrix} 17 \\ 10 \\ 13 \\ 15 \end{matrix})$ .

Now,

$| 7 | \geq | - 2 | + | 1 | \Rightarrow 7 > 3 | - 1 | \leq | 1 | + | 1 | + | 6 | \Rightarrow 1 < 8 | 3 | \leq | 1 | + | - 9 | + | - 1 | \Rightarrow 3 < 11 | 1 | \leq | 2 | + | 0 | + | 10 | \Rightarrow 1 < 12$