Question

Consider the linear system \( A x = b \), where \( A = [a_{ij}] \), \( i, j = 1, 2, 3 \), and \( a_{ii} \neq 0 \) for \( i = 1, 2, 3 \), is a matrix with entries in \( \mathbb{R} \). For

\( D = \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} \), let

\[ D^{-1} A = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix}, \quad D^{-1}b = \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix}. \]

Consider the following two statements:

S1: The approximation of \( x \) after one iteration of the Jacobi scheme with initial vector \( x_0 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \) is \( x_1 = \begin{bmatrix} 5 \\ -1 \\ -1 \end{bmatrix} \).

S2: There exists an initial vector \( x_0 \) for which the Jacobi iterative scheme diverges.

Then, which one of the following is correct?

• A. S1 is TRUE and S2 is FALSE
• B. S2 is TRUE and S1 is FALSE
• C. both S1 and S2 are TRUE
• D. neither S1 nor S2 is TRUE

Hint:

In the Jacobi iterative method, always check the spectral radius of the iteration matrix to determine if the method converges. The approximation after one iteration can be computed using the formula \( x_1 = D^{-1} (b - (A - D) x_0) \).

Answer 1

The Correct Option is C

We are given the system \( A x = b \), the matrix \( D^{-1} A \), and \( D^{-1} b \). We need to evaluate the two statements \( S1 \) and \( S2 \).

Step 1: Evaluate S1

The Jacobi iterative method is given by:

\[ x_1 = D^{-1} (b - (A - D) x_0). \]

Substitute \( D^{-1} A \), \( D^{-1} b \), and \( x_0 \):

\[ x_1 = \begin{bmatrix} 1 & 1 & -2 \\ 3 & 1 & 2 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 4 \\ 4 \\ 1 \end{bmatrix} - \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \]

Calculating this, we find:

\[ x_1 = \begin{bmatrix} 5 \\ -1 \\ -1 \end{bmatrix}. \]

Thus, S1 is TRUE.

Step 2: Evaluate S2

For the Jacobi method to converge, the spectral radius of the iteration matrix \( D^{-1} (A - D) \) must be less than 1. If the spectral radius is greater than or equal to 1, the method will diverge.

By examining the matrix \( D^{-1} (A - D) \), we find that it is possible for the method to diverge for certain initial vectors \( x_0 \). Therefore, S2 is TRUE.

Thus, the correct answer is:

\[ \boxed{{C. \text{both S1 and S2 are TRUE}}} \]