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Exact minimum formula for a fixed polynomial eigenpair's weighted backward error

Statement

Under the hypotheses of Coefficientwise weighted Frobenius backward error for a polynomial eigenpair, put $S : = \sum_{j = 0}^{m} ω_{j}^{2} ∣ α ∣^{2 j} ∣ β ∣^{2 (m - j)} .$ Then $S > 0$ , the infimum is attained, and $η_{F} (P, [α : β], x; ω) = \frac{∥ P_{h} (α, β) x ∥_{2}}{∥ x ∥_{2} \sqrt{S}} .$ At a finite parameter $[λ : 1]$ this is $∥ P (λ) x ∥_{2} / (∥ x ∥_{2} \sqrt{\sum_{j} ω_{j}^{2} ∣ λ ∣^{2 j}})$ ; at infinity $[1 : 0]$ it is $∥ A_{m} x ∥_{2} / (ω_{m} ∥ x ∥_{2})$ . When $α \neq 0$ , setting $μ = β / α$ rewrites the same formula for $P_{m}^{r e v} (μ)$ with reversed coefficient weights ${\tilde{ω}}_{k} : = ω_{m - k}$ .

Facts & Assumptions

Given: A regular grade- $m$ matrix polynomial, $[α : β] \in P^{1} (C)$ , $x \neq 0$ , and weights $ω_{j} > 0$ .

[F1]

Coefficientwise weighted Frobenius backward error for a polynomial eigenpair: the error is the infimum of ${(\sum_{j} ∥ Δ A_{j} ∥_{F}^{2} / ω_{j}^{2})}^{1 / 2}$ over coefficient perturbations satisfying $\sum_{j} (A_{j} + Δ A_{j}) α^{j} β^{m - j} x = 0$ .

Proof

technique · direct minimization

1.1givenF1algebra

Since $[α : β]$ is nonzero, either $α \neq 0$ or $β \neq 0$ . If $β \neq 0$ , the $j = 0$ term in $S$ is $ω_{0}^{2} ∣ β ∣^{2 m} > 0$ ; if $β = 0$ , then $α \neq 0$ and the $j = m$ term is $ω_{m}^{2} ∣ α ∣^{2 m} > 0$ . Thus $S > 0$ and the displayed denominator is defined because also $x \neq 0$ .

2.1givenF1step 1.1algebra

We use only finite-dimensional estimates, derived as follows. For $u, v \in C^{d}$ with $v \neq 0$ , put $c : = ⟨ u, v ⟩ / ∥ v ∥_{2}^{2}$ . Direct expansion gives $∥ u - c v ∥_{2}^{2} = ∥ u ∥_{2}^{2} - ∣ ⟨ u, v ⟩ ∣^{2} / ∥ v ∥_{2}^{2} \geq 0$ , so $∣ ⟨ u, v ⟩ ∣^{2} \leq ∥ u ∥_{2}^{2} ∥ v ∥_{2}^{2}$ ; if $v = 0$ the inequality is immediate. Since $∥ u + v ∥_{2}^{2} = ∥ u ∥_{2}^{2} + ∥ v ∥_{2}^{2} + 2 Re ⟨ u, v ⟩ \leq (∥ u ∥_{2} + ∥ v ∥_{2})^{2}$ , taking nonnegative square roots gives $∥ u + v ∥_{2} \leq ∥ u ∥_{2} + ∥ v ∥_{2}$ . Applying the inner-product bound to each row of any matrix $M$ and $\overset{ˉ}{v}$ and summing gives $∥ M v ∥_{2} \leq ∥ M ∥_{F} ∥ v ∥_{2}$ . For real lists $a, b \in R^{d}$ with $b \neq 0$ , set $t = (\sum_{j} a_{j} b_{j}) / (\sum_{j} b_{j}^{2})$ ; direct expansion gives $\sum_{j} (a_{j} - t b_{j})^{2} = \sum_{j} a_{j}^{2} - (\sum_{j} a_{j} b_{j})^{2} / (\sum_{j} b_{j}^{2}) \geq 0$ , hence $(\sum_{j} a_{j} b_{j})^{2} \leq (\sum_{j} a_{j}^{2}) (\sum_{j} b_{j}^{2})$ , and the same inequality is immediate if $b = 0$ . For any feasible perturbations, $r = - \sum_{j} Δ A_{j} c_{j} x$ , where $c_{j} = α^{j} β^{m - j}$ . The derived matrix estimate, $∥ c_{j} w ∥_{2} = ∣ c_{j} ∣ ∥ w ∥_{2}$ , and repeated triangle inequality give $∥ r ∥_{2} \leq ∥ x ∥_{2} \sum_{j} ∣ c_{j} ∣ ∥ Δ A_{j} ∥_{F}$ . Apply the derived real-list inequality to $a_{j} = ω_{j} ∣ c_{j} ∣$ and $b_{j} = ∥ Δ A_{j} ∥_{F} / ω_{j}$ to obtain $∥ r ∥_{2} \leq ∥ x ∥_{2} \sqrt{\sum_{j} ω_{j}^{2} ∣ c_{j} ∣^{2}} {(\sum_{j} \frac{∥ Δ A_{j} ∥_{F}^{2}}{ω_{j}^{2}})}^{1 / 2} = ∥ x ∥_{2} \sqrt{S} {(\sum_{j} \frac{∥ Δ A_{j} ∥_{F}^{2}}{ω_{j}^{2}})}^{1 / 2}$ . Hence every feasible cost is at least $∥ r ∥_{2} / (∥ x ∥_{2} \sqrt{S})$ .

2.2givenstep 1.1constructalgebra

Define ${\hat{Δ A}}_{j} : = - \frac{ω_{j}^{2} \overline{α^{j} β^{m - j}}}{∥ x ∥_{2}^{2} S} r x^{*}$ . Then ${\hat{Δ A}}_{j} c_{j} x = - ω_{j}^{2} ∣ c_{j} ∣^{2} r / S$ , because $x^{*} x = ∥ x ∥_{2}^{2}$ . Summing over $j$ yields $\sum_{j} {\hat{Δ A}}_{j} c_{j} x = - r$ , so these perturbations are feasible.

3.1F1step 2.1step 2.2algebra∎

Directly from the entries of the rank-one matrix, $∥ r x^{*} ∥_{F}^{2} = \sum_{a, b} ∣ r_{a} \overline{x_{b}} ∣^{2} = ∥ r ∥_{2}^{2} ∥ x ∥_{2}^{2}$ . Therefore the cost of the feasible perturbations in step 2.2 is $\sum_{j} ∥ {\hat{Δ A}}_{j} ∥_{F}^{2} / ω_{j}^{2} = ∥ r ∥_{2}^{2} / (∥ x ∥_{2}^{2} S)$ . Taking square roots and combining with the lower bound in step 2.1 proves attainment and the formula. Substituting $[λ : 1]$ leaves the terms $ω_{j}^{2} ∣ λ ∣^{2 j}$ ; substituting $[1 : 0]$ leaves only $ω_{m}^{2}$ and $P_{h} (1, 0) = A_{m}$ . When $α \neq 0$ , set $μ = β / α$ and reindex with $k = m - j$ ; then $S = ∣ α ∣^{2 m} \sum_{k = 0}^{m} ω_{m - k}^{2} ∣ μ ∣^{2 k}$ and $P_{h} (α, β) = α^{m} P_{m}^{r e v} (μ)$ , so the homogeneous formula is precisely the reversed-polynomial formula with weights ${\tilde{ω}}_{k} = ω_{m - k}$ . The zero-residual case is included, with minimizer $Δ A_{j} = 0$ .

Depends on

Coefficientwise weighted Frobenius backward error for a polynomial eigenpair

Used by

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Sources

Michael Berhanu, The Polynomial Eigenvalue Problem, Chapter 3, §3.2, Theorem 3.3 (standard reference, not scraped)
Françoise Tisseur and Karl Meerbergen, The Quadratic Eigenvalue Problem, §4.2 (standard reference, not scraped)