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A rank-one approximation of a separated Cauchy block

Example

Take $x_{1} = 0, x_{2} = 1$ in $[0, 1]$ and $y_{1} = 2, y_{2} = 3$ in $[2, 3]$ . For $K_{i j} = 1 / (y_{j} - x_{i})$ and rank order $r = 1$ , the first geometric term is ${\tilde{K}}_{i j} = 1 / y_{j}$ . It gives a rank-one approximation with an exactly computable nonzero residual.

Facts & Assumptions

Given: The four points and the order $r = 1$ .

[F1]

A separated Cauchy block has a geometric rank bound: for $a = 1, b = 2, c = 1$ , the rank-one approximation has entries $1 / y_{j}$ and each residual entry has absolute value at most $1 / 2$ .

[F2]

Finite tensor arrays and the Frobenius inner product: the Frobenius norm is the square root of the sum of squared entry moduli.

Verification

1.1givenF1algebra

The exact block is $K = (\begin{array}{cc} 1 / 2 & 1 / 3 \\ 1 & 1 / 2 \end{array})$ . The rank-one approximation is $\tilde{K} = (\begin{array}{cc} 1 / 2 & 1 / 3 \\ 1 / 2 & 1 / 3 \end{array}) = (1, 1)^{T} (1 / 2, 1 / 3)$ , so its rank is one.

2.1F1F2step 1.1algebra∎

The residual is $K - \tilde{K} = (\begin{array}{cc} 0 & 0 \\ 1 / 2 & 1 / 6 \end{array})$ , hence $∥ K - \tilde{K} ∥_{F} = \sqrt{1 / 4 + 1 / 36} = \sqrt{10} / 6 \leq 1 = \sqrt{2 \cdot 2} (1 / 2) / (2 - 1)$ by [F2]. This attains the stated finite block bound at the entry $(2, 1)$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Steffen Börm, Lars Grasedyck, and Wolfgang Hackbusch, Hierarchical Matrices, MiS Lecture Note 21 (2003; revised June 2006), §§1.2–1.3 (standard reference, not scraped)