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Twice is enough for dangerous eigenvalues

Andrew Horning, Yuji Nakatsukasa

TL;DR

A simple restart strategy is demonstrated that recovers full precision in the target eigenpairs in the context of Arnoldi with shift-and-invert enhancement and is shown to be stable even when an eigenvalue is located near a pole of the filter.

Abstract

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

Twice is enough for dangerous eigenvalues

TL;DR

A simple restart strategy is demonstrated that recovers full precision in the target eigenpairs in the context of Arnoldi with shift-and-invert enhancement and is shown to be stable even when an eigenvalue is located near a pole of the filter.

Abstract

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

Paper Structure

This paper contains 16 sections, 9 theorems, 63 equations, 9 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Subspace iteration with rational filters
  3. Principal angles between subspaces
  4. Dangerous eigenvalues
  5. Accuracy of the computed orthonormal basis
  6. Twice is enough
  7. A well-conditioned basis
  8. Convergence and stability
  9. One-step refinement bounds
  10. Stability and stagnation
  11. Non-normal matrices
  12. First iteration
  13. Iterating with orthonormal bases
  14. Iterating with approximate eigenvectors
  15. Restarting Arnoldi
  16. ...and 1 more sections

Key Result

Theorem 1

\newlabelthm:1step_standard0 Let normal $A\in\mathbb{C}^{n\times n}$ and $r:\Lambda\rightarrow\mathbb{C}$ satisfy eqn:eigendecompositioneqn:eig_index, respectively, and let $\mathcal{S}_{j}={\rm span}(Q_{j})$ in eqn:ratSI, for $j\geq 0$. If $V_1^*Q_0$ has full rank, then for each $v_i\in\mathcal{V Moreover, each $P_\mathcal{V}s_i^{(j)}=v_i$, where $P_\mathcal{V}=V_1V_1^*$ is the spectral projecto

Figures (9)

  • Figure 1: The residuals for two approximate eigenpairs of a real-symmetric $100\times 100$ matrix at iterations $k=2,\ldots,50$ of Arnoldi (left) and iterations $k=1,\ldots,25$ of subspace iteration (right), both with shift-and-invert enhancement. The approximate eigenpairs correspond to a dangerous eigenvalue (black) with $|z-\lambda_1|=10^{-12}$ and a second target eigenvalue (red) with $|z-\lambda_2|\approx 0.1$. \newlabelfig:arnoldi_fsi0
  • Figure 1: The eigenvalues of a $100\times 100$ real-symmetric matrix overlaid on a complex color plot of the magnitude of a rational approximation to the characteristic function on $[10,15]$. A dangerous eigenvalue is located at distance $d=10^{-10}$ from the pole at $z=10$. \newlabelfig:exp1_setup0
  • Figure 1: The structure of the iterates $\hat{X}_2$ and $\hat{Q}_2$ after the second iteration of subspace iteration with a rational filter. On the left, the eigenvector coordinates of the $10$th column of the computed orthonormal basis color-coded for dangerous component (red), remaining target components (blue), and unwanted components (black). On the right, the eigenvector coordinates of the $1$st (circles) and $10$th (triangles) columns of the computed basis $\hat{X}_2$ with the same color code used in the left panel. \newlabelfig:exp2_results0
  • Figure 1: The dynamics of perturbed subspace iteration from \ref{['eqn:discrete_dynamics']}. In the left panel, the solid line is the graph of $\Phi(\eta)$ and its fixed points (green circles) are marked at the intersections $\Phi(\eta_\pm)=\eta_\pm$. If $\tan\theta_1(\mathcal{\hat{S}}_0,\mathcal{V})$ falls between the two fixed points (green circles), then the $\tan\theta_1(\mathcal{\hat{S}}_k,\mathcal{V})$ must converge geometrically to a threshold near the lower fixed point (see \ref{['thm:perturbed_convergence']}). In the right panel, the iterated map $\phi_k(\eta_0)$ (circles) is compared with the upper bound in \ref{['thm:perturbed_convergence']} (dashed line) for $k=0,\ldots,14$. For this experiment, $\eta_0=100$, $\epsilon_1=10^{-5}$, $\epsilon_2=10^{-14}$, and $\rho=10^{-4}$. \newlabelfig:stability0
  • Figure 1: Dangerous eigenvalues of a non-normal matrix. The eigenvalues and rational filter are identical to the setup displayed in \ref{['fig:exp1_setup']}, however, this matrix has non-orthogonal eigenvectors and the dangerous eigenvalue has been moved to distance $d=10^{-13}$ from the pole at $z=10$. On the left, the maximum residual of $10$ target eigenpairs after each iteration of \ref{['eqn:ratSI']} (blue squares), a variant of subspace iteration based on Schur vectors saad2011numerical (red triangles), and a variant based on approximate eigenvectors, described in \ref{['alg:FSI_w_RR']} (black circles). On the right, the condition number of the iterates $\hat{Z}_kD_k$ ($D_k$ scales the columns of $\hat{Z}_k$ to have unit norm) decreases in step with residuals from \ref{['alg:FSI_w_RR']}, at a rate of about $u/d$ per iteration. \newlabelfig:exp3_results0
  • ...and 4 more figures

Theorems & Definitions (18)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Proof 1
  • Proposition 1
  • Proof 2
  • Theorem 1: Twice-is-enough
  • Proof 3
  • Lemma 1
  • Proof 4
  • ...and 8 more