An engineer runs Jacobi iteration on a diagonally dominant system with condition number 50,000. After 200 iterations, convergence is extremely slow. They conclude the large condition number is preventing convergence. What is wrong with this diagnosis?
AJacobi iteration only applies to symmetric matrices, making diagonal dominance irrelevant
BThe condition number measures sensitivity of the solution to data perturbations, not convergence speed — the spectral radius of the iteration matrix, which may be close to 1, determines convergence rate
C200 iterations is simply insufficient; diagonal dominance guarantees convergence but requires at least 1,000 iterations
DDiagonal dominance guarantees fast convergence for Jacobi, so the slow convergence must be a software error
The condition number κ(A) measures how sensitively the solution responds to small changes in input data — a property of the original system, not the iteration process. The spectral radius ρ(M) of the iteration matrix determines whether and how fast the iterative method converges. Diagonal dominance guarantees ρ(M) < 1 (convergence is assured), but the spectral radius might be 0.99 — painfully slow. The fix is to reduce ρ(M) through better methods like SOR, not to blame the condition number.
Question 2 Multiple Choice
For the iterative scheme x^(k+1) = Mx^(k) + c, which condition is BOTH necessary AND sufficient for convergence from any starting point?
AM is symmetric positive definite
BThe spectral radius ρ(M) = max|λ_i| < 1
CA is strictly diagonally dominant
DThe condition number κ(A) < 10
ρ(M) < 1 is the necessary and sufficient condition. The error satisfies e^(k+1) = Me^(k), so e^(k) = M^k e^(0). Decomposing in the eigenvector basis, each component decays as λ_i^k. All components vanish if and only if all |λ_i| < 1, i.e., ρ(M) < 1. Symmetric positive definiteness and diagonal dominance are sufficient in specific contexts but not necessary. The condition number concerns A, not the iteration matrix M, and is irrelevant to whether iteration converges.
Question 3 True / False
If the spectral radius of an iteration matrix is 0.5, the error after 20 iterations is approximately 10^(−6) of the initial error.
TTrue
FFalse
Answer: True
Error decays geometrically: after k iterations, error ≈ ρ(M)^k × initial error. With ρ = 0.5 and k = 20: 0.5^20 = 1/2^20 ≈ 9.5 × 10^(−7) ≈ 10^(−6). Compare with ρ = 0.9: 0.9^20 ≈ 0.12, barely reduced after 20 steps. The spectral radius is the right tool precisely because this geometric decay rate is what practitioners care about when choosing or tuning iterative methods.
Question 4 True / False
The spectral radius of the iteration matrix and the condition number of A both measure the same underlying property — how difficult a linear system is to solve.
TTrue
FFalse
Answer: False
They measure completely different things. The condition number κ(A) = ‖A‖·‖A^(−1)‖ measures how sensitively the solution responds to small perturbations in the input — relevant when you care about accuracy of a solution you already have. The spectral radius ρ(M) measures whether and how fast the iterative scheme converges — relevant when you care about the iteration process. A system can have a large condition number but a small spectral radius (converges fast, but the solution is sensitive to noise), or vice versa.
Question 5 Short Answer
Why is the spectral radius of the iteration matrix, rather than the condition number of A, the correct tool for predicting whether an iterative method converges?
Think about your answer, then reveal below.
Model answer: The iteration generates error vectors satisfying e^(k+1) = M·e^(k), so e^(k) = M^k·e^(0). For the error to vanish, M^k must go to zero — which happens if and only if all eigenvalues of M satisfy |λ| < 1, i.e., ρ(M) < 1. The condition number κ(A) concerns the original matrix A and measures solution sensitivity to input perturbations — a completely different question. The iteration matrix M is derived from A but is a separate object, and its eigenvalues govern the iterative dynamics entirely.
This distinction separates two concerns: (1) once you have a solution, how accurate is it given noisy input? — answered by the condition number; (2) will the iterative process reach a solution and how fast? — answered by the spectral radius of M. A common error is using the condition number as a general proxy for 'difficulty.' For iterative solvers, the spectral radius is the correct diagnostic — and it points toward the right remedies, like SOR, that actually reduce ρ(M).