Which of the following best explains why Krylov methods are preferred over the full QR algorithm for large sparse eigenvalue problems?

Question

BlackTom AI · Accepted Answer

Krylov methods require only matrix–vector products with A (so they preserve sparsity) and are therefore far cheaper for computing a few extreme eigenpairs of large sparse matrices.

1) Relevant knowledge points and concepts
- Sparsity: a sparse n×n matrix A has nnz(A) nonzero entries with nnz(A) ≪ n^2. A matrix–vector product Ax costs O(nnz(A)).
- Krylov (iterative) methods build a k-dimensional Krylov subspace using repeated Ax operations and typically extract a few extreme eigenvalues; cost ≈ O(k · cost(Ax)) = O(k · nnz(A)).
- The full QR (direct) approach requires first reducing A to Hessenberg (or tridiagonal for symmetric problems) with Householder transforms, costing O(n^3) flops for dense reduction, and this step fills in zeros (destroys sparsity). Subsequent QR iterations are also O(n^3) in total for all eigenvalues.

2) Step-by-step reasoning
- If only a few (say k ≪ n) eigenvalues are needed, Krylov methods run roughly k matrix–vector products plus orthogonalization and small dense eigensolves on k×k matrices → total cost O(k · nnz(A) + k^2 n).
- The QR algorithm’s Hessenberg reduction step involves operations across rows/columns that introduce fill-in; even if A was sparse, the reduction and subsequent dense operations destroy sparsity and require O(n^3) work and O(n^2) memory.

3) Sample calculation (intermediate steps)
- Example: n = 100,000, nnz(A) ≈ 10n = 1,000,000. One Ax ≈ 1e6 operations. For k = 50, Krylov cost ≈ 50 · 1e6 = 5e7 ops (feasible). QR cost scale ~ n^3 ≈ 1e15 ops (infeasible).

4) Why the correct answer is correct
- Answer b explicitly states the key practical reasons: Krylov only needs Ax (preserving sparsity) and scales like O(k·nnz(A)) to find a few extreme eigenvalues, whereas QR is O(n^3) and destroys sparsity. This matches the complexity and storage realities for large sparse problems.

5) Why the other options are incorrect
- a: Krylov methods are not universally more numerically stable than QR; stability depends on context.  
- c: Krylov methods do not always find all n eigenvalues; they target a few.  
- d: The QR algorithm does converge for sparse matrices in theory, but it is impractical because of fill-in and cost.

Conclusion: The best explanation is option b, which matches the provided correct answer.

Which of the following best explains why Krylov methods are preferred over the full QR algorithm for large sparse eigenvalue problems?单项选择题

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