During a Jacobi iteration sweep, component x₁^(k+1) has just been computed. When computing x₂^(k+1), which value of x₁ is used?
AThe freshly computed x₁^(k+1) from this sweep
BThe old value x₁^(k) from the previous iteration
CThe average of x₁^(k) and x₁^(k+1)
DWhichever value accelerates convergence
Jacobi's defining rule is that all components of x^(k+1) are computed simultaneously using only values from the previous iterate x^(k). No freshly computed component is used within the same sweep. This 'all-old' update is what distinguishes Jacobi from Gauss-Seidel, which immediately uses fresh values as they are computed. It also enables parallel computation of all components, since they are mutually independent within a sweep.
Question 2 Multiple Choice
A linear system Ax = b has a coefficient matrix A where each diagonal entry is strictly larger in magnitude than the sum of the absolute values of all other entries in its row. What can you conclude about Jacobi iteration on this system?
AJacobi will diverge because the diagonal dominates and suppresses off-diagonal corrections
BJacobi is guaranteed to converge because A is diagonally dominant
CConvergence cannot be determined without computing the spectral radius explicitly
DJacobi will converge only if A is also symmetric
Diagonal dominance — |aᵢᵢ| > Σⱼ≠ᵢ |aᵢⱼ| for every row — is a sufficient condition for Jacobi convergence. It guarantees the spectral radius of the iteration matrix D⁻¹(L+U) is less than 1. Intuitively, when the diagonal entry dominates, each update is controlled primarily by the correct term bᵢ/aᵢᵢ, and the off-diagonal perturbations are small enough that the iteration self-corrects. Note this is sufficient but not necessary — Jacobi can converge for some non-dominant matrices too.
Question 3 True / False
The Jacobi method can be parallelized more easily than Gauss-Seidel because all component updates within a single sweep are independent of each other.
TTrue
FFalse
Answer: True
True. Because Jacobi only uses values from the previous iteration (x^(k)) to compute all components of x^(k+1), every component update is independent — no component depends on another freshly-computed component in the same sweep. This makes Jacobi trivially parallelizable. Gauss-Seidel, by contrast, immediately uses freshly computed values, creating data dependencies that prevent straightforward parallel execution.
Question 4 True / False
If the Jacobi method fails to converge for a given linear system, then Gauss-Seidel will also fail to converge on the same system.
TTrue
FFalse
Answer: False
False. Jacobi and Gauss-Seidel have different convergence properties and different iteration matrices. It is possible for Jacobi to diverge while Gauss-Seidel converges on the same system (and in rare cases, the reverse). The two methods share the decomposition A = D + L + U but use different update rules, resulting in different spectral radii for their respective iteration matrices. Diagonal dominance guarantees convergence for both, but outside that condition the methods can behave differently.
Question 5 Short Answer
Why does the Jacobi method use only values from the previous iteration when computing updates, and what practical advantage does this 'all-old' rule provide?
Think about your answer, then reveal below.
Model answer: The all-old update rule means each component x_i^(k+1) = (b_i − Σ_{j≠i} a_{ij} x_j^(k)) / a_{ii} depends only on the previous iterate, not on freshly computed values from the current sweep. This makes all n component updates mutually independent within a single sweep, enabling them to be computed simultaneously on parallel hardware. The trade-off is slower convergence compared to Gauss-Seidel, which uses fresh values immediately and typically halves the iteration count for the same accuracy.
The design reflects a deliberate choice: sacrifice convergence speed for parallelism and implementation simplicity. In large sparse systems — common in scientific computing — the ability to distribute all n updates across many processors at once can more than compensate for the extra iterations required compared to Gauss-Seidel.