Question

Consider the linear system of equations \( Ax = b \) with
\[ A = \begin{pmatrix} 3 & 1 & 1 \\ 1 & 4 & 1 \\ 2 & 0 & 3 \end{pmatrix}, \quad b = \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix}. \] Which of the following statements are TRUE?

• A. The Jacobi iterative matrix is \( \begin{pmatrix} 0 & 1/4 & 1/3 \\ 1/3 & 0 & 1/3 \\ 2/3 & 0 & 0 \end{pmatrix} \)
• B. The Jacobi iterative method converges for any initial vector
• C. The Gauss-Seidel iterative method converges for any initial vector
• D. The spectral radius of the Jacobi iterative matrix is less than 1

Hint:

For iterative methods like Jacobi and Gauss-Seidel, check if the matrix is diagonally dominant or positive definite to guarantee convergence. Also, ensure the spectral radius is less than 1 for the method to converge.

Answer 1

The Correct Option is B, C, D

Given the matrix \( A \), we need to analyze the iterative methods and determine which statements are true. (A) The Jacobi iterative matrix is \( \begin{pmatrix} 0 & 1/4 & 1/3
1/3 & 0 & 1/3
2/3 & 0 & 0 \end{pmatrix} \):
This statement is incorrect. The Jacobi method does not have the form provided in the option. The Jacobi iterative matrix is typically derived from the decomposition of the matrix \( A \) into its diagonal and non-diagonal components. (B) The Jacobi iterative method converges for any initial vector:
This statement is true. The Jacobi iterative method converges for any initial vector when the matrix \( A \) is diagonally dominant. The given matrix \( A \) satisfies this condition, meaning the Jacobi method will converge for any initial vector. (C) The Gauss-Seidel iterative method converges for any initial vector:
This statement is also true. The Gauss-Seidel method has better convergence properties compared to the Jacobi method and converges for any initial vector when the matrix \( A \) is diagonally dominant or positive definite, which is the case here. (D) The spectral radius of the Jacobi iterative matrix is less than 1:
This statement is true. The spectral radius of an iterative matrix determines the convergence of the method. For the Jacobi method, the spectral radius of the matrix is less than 1, indicating that the method will converge. Thus, the correct statements are (B), (C), and (D).