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Nearly Optimal Approximation of Matrix Functions by the Lanczos Method
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Proves that Lanczos-FA is nearly instance optimal (up to a factor depending on the rational denominator degree and condition numbers, not on k) for rational functions with poles outside the spectrum interval, and near spectrum optimal for the matrix square root and inverse square root.
Problem
Lanczos-FA is observed to outperform specialized matrix-function algorithms in practice, but existing theory only gives loose uniform polynomial approximation bounds for it; near-instance-optimality guarantees were known only for f(x)=1/x and a few other cases.
Approach
Decomposes the Lanczos-FA error for a rational function r = n/m with real poles outside the spectrum interval into a telescoping sum of errors of simpler rational functions, each measured in an A_j-norm where A_j = ±(A - z_j I) is definite, and relates these to the best polynomial approximation of r in the 2-norm; extends to f via a rational approximant and to x^{-1/2}, x^{1/2} via comparison to x
Key results
- For rational functions with real poles outside the spectrum interval, Lanczos-FA error is within q·∏κ(A_j) times the best Krylov error, with no dependence on the iteration count k.
- For A^{-1/2}b, Lanczos-FA satisfies near spectrum optimality with C = 3κ(A)/√(πk) and c = 1/2; for A^{1/2}b, C = 3κ(A)^2/k^{3/2} and c = 1/2.
- In experiments, Lanczos-FA performs nearly as well as the instance-optimal Krylov approximation and outperforms the rational and slanczos methods for the matrix sign function.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lanczos-FA instead of the specialized rational method | Matrix sign computation · Matrix sign test problems | approximation error | Helped |
| Use Lanczos-FA instead of slanczos | Matrix sign computation · Matrix sign test problems | approximation error | Helped |
| Apply Lanczos-FA to rational approximations rather than directly to the function | Lanczos-FA for A^-0.4 · A^-0.4, two-cluster spectrum | approximation error | Mixed |
| Apply Lanczos-FA directly to the function rather than to a fixed rational approximation | Lanczos-FA for A^-0.4 · A^-0.4, two-cluster spectrum | approximation error | Mixed |
| Use Lanczos-FA rather than the instance-optimal KSM | Lanczos-FA · Rational functions, three test matrices | approximation error | Hurt |
| Use the new rational-function bound instead of the standard uniform approximation bound | Lanczos-FA convergence analysis · Rational functions, three test matrices | bound tightness | Helped |
| Increase denominator degree q and condition number κ | Lanczos-FA on A^-q · A^-q, spectrum with varying condition number | optimality ratio: Ω(√(q · κ(A))) | Hurt |
| Use Lanczos-FA instead of the rational and slanczos specialized methods | Matrix sign computation · Matrix sign test problems | approximation error | Helped |
Surprising findings
- The optimality ratio for approximating A^{-q}b appears to grow at least as Ω(sqrt(q·κ(A))), suggesting some dependence on q is necessary.
- Lanczos-OR, which is exactly optimal for rational functions in a different norm, can perform poorly in the Euclidean norm on high-degree rational functions.
Limitations
- The bound has exponential dependence on the rational denominator degree q (κ(A)^q), limiting practicality.
- Analysis is in exact arithmetic; finite-precision behavior is not covered.
- Does not extend directly to poles inside the spectrum interval or to Stieltjes functions.
Experimental setup
Benchmarks: synthetic test matrices with uniform, geometric, and clustered spectra
Metrics: 2-norm error optimality ratio versus best Krylov approximation
Baselines: Lanczos-FA uniform polynomial bound rational method (Hale-Higham-Trefethen) slanczos Lanczos-OR BRASIL rational approximation
Builds on: Lanczos-FA conjugate gradient Lanczos-OR MINRES
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