IGNOU BMTE 144 2026 - Set up the Gauss-Seidel iteration scheme in matrix form for solving th

Question

Set up the Gauss-Seidel iteration scheme in matrix form for solving the system of equations

$$\begin{bmatrix} 1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 6 \\ 4 \end{bmatrix}.$
Show that the method is convergent and hence find its rate of convergence.

09 Jan 2026

Answer :

Word Count : 589

We are asked to set up the Gauss-Seidel iteration for the system: [ \begin{bmatrix} 1 & 1 & 1 \ 4 & 3 & -1 \ 3 & 5 & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} ======================================= \begin{bmatrix} 1 \ 6 \ 4 \end{bmatrix}. ] --- The system of equations is: 1. (x + y + z = 1) 2. (4x + 3y - z = 6) 3. (3x + 5y + 3z = 4) --- ### Step 1: Rewrite each variable in terms of the others (for Gauss-Seidel) From Equation 1: [ x = 1 - y - z ] From Equation 2: [ 4x + 3y - z = 6 \implies z = 4x + 3y - 6 ] From Equation 3: [ 3x + 5y + 3z = 4 \implies z = \frac{4 - 3x - 5y}{3} ] For Gauss-Seidel, we prefer an iteration in the form: [ x^{(k+1)} = f_1(y^{(k)}, z^{(k)}) ] [ y^{(k+1)} = f_2(x^{(k+1)}, z^{(k)}) ] [ z^{(k+1)} = f_3(x^{(k+1)}, y^{(k+1)}) ] --- #### Solving for Gauss-Seidel scheme Equation 1 → solve for (x): [ x = 1 - y - z ] Equation 2

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