Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stability of the Lanczos Method for Matrix Function Approximation   The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n
\times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the
method's error after $k$ iterations is bounded by the error of the best
degree-$k$ polynomial uniformly approximating $f(x)$ on the range
$[\lambda_{min}(A), \lambda_{max}(A)]$. However, despite decades of work, it
has been unclear if this powerful guarantee holds in finite precision.
  We resolve this problem, proving that when $\max_{x \in [\lambda_{min},
\lambda_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic
guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our
proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability
of the classic Chebyshev recurrence to bound the stability of any polynomial
approximating $f(x)$.
  We also study the special case of $f(A) = A^{-1}$, where stronger guarantees
hold. In exact arithmetic Lanczos performs as well as the best polynomial
approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full
eigenvalue range. In seminal work, Greenbaum gives an approach to extending
this bound to finite precision: she proves that finite precision Lanczos and
the related CG method match any polynomial approximating $1/x$ in a tiny range
around each eigenvalue [Gre89].
  For $A^{-1}$, this bound appears stronger than ours. However, we exhibit
matrices with condition number $\kappa$ where exact arithmetic Lanczos
converges in $polylog(\kappa)$ iterations, but Greenbaum's bound predicts
$\Omega(\kappa^{1/5})$ iterations. It thus cannot offer significant improvement
over the $O(\kappa^{1/2})$ bound achievable via our result. Our analysis raises
the question of if convergence in less than $poly(\kappa)$ iterations can be
expected in finite precision, even for matrices with clustered, skewed, or
otherwise favorable eigenvalue distributions.

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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con