Question Bank • QB_EM_ADV_079
MSQ • 2 Marks
Section 1: Engineering Mathematics
Numerical Methods
Subtopic: Solutions of linear and non-linear algebraic equations
Question Statement:
For solving a linear algebraic system $Ax = b$ using iterative techniques, which of the following statements is/are TRUE?
Options:
-
(A) The Gauss–Seidel method updates coordinate values immediately as soon as they are computed within the current iteration
-
(B) Both Jacobi and Gauss–Seidel iterations are guaranteed to converge for any initial vector if $A$ is strictly diagonally dominant
-
(C) The Jacobi method computes all new components using only values from the previous iteration step
-
(D) The asymptotic rate of convergence of the Gauss–Seidel method is generally about twice as fast as the Jacobi method for tridiagonal systems
Step-by-Step Solution & Explanation
1. **Immediate Coordinate Update (Option A)**: Gauss–Seidel uses newly calculated $x_j^{(k+1)}$ for $j < i$ during step $i$ of iteration $k+1$ (True).
2. **Diagonal Dominance (Option B)**: By the Gershgorin circle theorem, strict diagonal dominance ($|a_{ii}| > \sum_{j \neq i} |a_{ij}|$) guarantees the spectral radius $\rho(T) < 1$ for both Jacobi and Gauss–Seidel iteration matrices (True).
3. **Simultaneous Jacobi Step (Option C)**: Jacobi iteration evaluates $x^{(k+1)} = D^{-1}(b - (L + U)x^{(k)})$, using exclusively previous-step values (True).
4. **Relative Convergence Rate (Option D)**: For consistently ordered matrices (e.g. tridiagonal), $\rho(T_{GS}) = \rho(T_J)^2$, so Gauss–Seidel requires roughly half as many iterations (True).
Practice In Authentic CBT Exam Interface
Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.
Launch CBT Mock Test