Question

If \( A = [a_{ij}] \) is the coefficient matrix for a system of algebraic equations, then a sufficient condition for convergence of Gauss-Seidel iteration method is:

• A. \( A \) is strictly diagonally dominant
• B. \( |a_{ii}| = 1 \)
• C. \( \det(A) \neq 0 \)
• D. \( \det(A)>0 \)

Hint:

A sufficient condition for Gauss-Seidel iteration convergence is: \[ |a_{ii}|>\sum\limits_{j \neq i} |a_{ij}|. \] This ensures strict diagonal dominance.

Answer 1

The Correct Option is A

Step 1: Condition for convergence. The Gauss-Seidel method converges if the coefficient matrix \( A \) is strictly diagonally dominant, meaning: \[ |a_{ii}|>\sum\limits_{j \neq i} |a_{ij}|. \] 
Step 2: Evaluating given options. 
- Option (A) is correct as strict diagonal dominance ensures convergence. 
- Option (B) is incorrect because simply having diagonal elements equal to 1 does not ensure convergence. 
- Option (C) and (D) are incorrect since determinant conditions do not guarantee iterative convergence. 
Step 3: Selecting the correct option. Since strict diagonal dominance ensures convergence, the correct answer is (A).