Given the system of linear equations: x + y/2 = 9 3x + y = 10 What can be said…
2006
Given the system of linear equations:
x + y/2 = 9
3x + y = 10
What can be said about the Gauss-Seidel iterative method for solving the above system, using the normalized iterative arrangement?
- A.
It will converge
- B.
It will diverge
- C.
It will neither converge nor diverge
- D.
It is not applicable
Attempted by 2 students.
Show answer & explanation
Correct answer: A
Use the normalized arrangement also used for the related norm calculation:
From 3x + y = 10:
x = 10/3 - y/3.
From x + y/2 = 9, multiply by 2:
2x + y = 18,
y = 18 - 2x.
For Gauss-Seidel, use the newest value of x in the second equation:
x^(k+1) = 10/3 - y^(k)/3,
y^(k+1) = 18 - 2x^(k+1).
Substitute x^(k+1) into the second equation:
y^(k+1) = 18 - 2(10/3 - y^(k)/3)
= 34/3 + (2/3)y^(k).
So the error is multiplied by 2/3 in each iteration. Since |2/3| < 1, the Gauss-Seidel method converges.
Therefore, the correct answer is: It will converge.