1. **Problem Statement:** We are given a system of linear equations (SPL): $$\begin{cases} 3x - y + z = 5 \\ x + 4y + 2z = 6 \\ 2x - y + 5z = 7 \end{cases}$$ We need to determine if this system is convergent and then perform the first 3 iterations of the Gauss-Seidel method including the error. 2. **Convergence Check:** The Gauss-Seidel method converges if the coefficient matrix is diagonally dominant or if it satisfies certain spectral radius conditions. The coefficient matrix is: $$A = \begin{bmatrix} 3 & -1 & 1 \\ 1 & 4 & 2 \\ 2 & -1 & 5 \end{bmatrix}$$ Check diagonal dominance for each row: - Row 1: $|3| \geq |-1| + |1| \Rightarrow 3 \geq 2$ (True) - Row 2: $|4| \geq |1| + |2| \Rightarrow 4 \geq 3$ (True) - Row 3: $|5| \geq |2| + |-1| \Rightarrow 5 \geq 3$ (True) Since all rows satisfy diagonal dominance, the system is convergent. 3. **Gauss-Seidel Iteration Formula:** For each variable, update using: $$x^{(k+1)} = \frac{1}{a_{11}} \left(b_1 - a_{12}y^{(k)} - a_{13}z^{(k)}\right)$$ $$y^{(k+1)} = \frac{1}{a_{22}} \left(b_2 - a_{21}x^{(k+1)} - a_{23}z^{(k)}\right)$$ $$z^{(k+1)} = \frac{1}{a_{33}} \left(b_3 - a_{31}x^{(k+1)} - a_{32}y^{(k+1)}\right)$$ 4. **Initial Guess:** Start with $x^{(0)}=0$, $y^{(0)}=0$, $z^{(0)}=0$. 5. **Iteration 1:** $$x^{(1)} = \frac{1}{3}(5 + y^{(0)} - z^{(0)}) = \frac{5}{3} = 1.6667$$ $$y^{(1)} = \frac{1}{4}(6 - x^{(1)} - 2z^{(0)}) = \frac{1}{4}(6 - 1.6667) = 1.0833$$ $$z^{(1)} = \frac{1}{5}(7 - 2x^{(1)} + y^{(1)}) = \frac{1}{5}(7 - 3.3334 + 1.0833) = 0.9499$$ 6. **Iteration 2:** $$x^{(2)} = \frac{1}{3}(5 + y^{(1)} - z^{(1)}) = \frac{1}{3}(5 + 1.0833 - 0.9499) = 1.7111$$ $$y^{(2)} = \frac{1}{4}(6 - x^{(2)} - 2z^{(1)}) = \frac{1}{4}(6 - 1.7111 - 1.8998) = 0.8473$$ $$z^{(2)} = \frac{1}{5}(7 - 2x^{(2)} + y^{(2)}) = \frac{1}{5}(7 - 3.4222 + 0.8473) = 0.8850$$ 7. **Iteration 3:** $$x^{(3)} = \frac{1}{3}(5 + y^{(2)} - z^{(2)}) = \frac{1}{3}(5 + 0.8473 - 0.8850) = 1.6541$$ $$y^{(3)} = \frac{1}{4}(6 - x^{(3)} - 2z^{(2)}) = \frac{1}{4}(6 - 1.6541 - 1.7700) = 0.8939$$ $$z^{(3)} = \frac{1}{5}(7 - 2x^{(3)} + y^{(3)}) = \frac{1}{5}(7 - 3.3082 + 0.8939) = 0.9171$$ 8. **Error Calculation:** Use absolute difference between iterations for each variable: - Between iteration 1 and 2: - $|x^{(2)} - x^{(1)}| = |1.7111 - 1.6667| = 0.0444$ - $|y^{(2)} - y^{(1)}| = |0.8473 - 1.0833| = 0.2360$ - $|z^{(2)} - z^{(1)}| = |0.8850 - 0.9499| = 0.0649$ - Between iteration 2 and 3: - $|x^{(3)} - x^{(2)}| = |1.6541 - 1.7111| = 0.0570$ - $|y^{(3)} - y^{(2)}| = |0.8939 - 0.8473| = 0.0466$ - $|z^{(3)} - z^{(2)}| = |0.9171 - 0.8850| = 0.0321$ **Final answer:** The system is convergent. The first three Gauss-Seidel iterations are: $$\begin{cases} (x^{(1)}, y^{(1)}, z^{(1)}) = (1.6667, 1.0833, 0.9499) \\ (x^{(2)}, y^{(2)}, z^{(2)}) = (1.7111, 0.8473, 0.8850) \\ (x^{(3)}, y^{(3)}, z^{(3)}) = (1.6541, 0.8939, 0.9171) \end{cases}$$ Errors between iterations are as calculated above.