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Explicit root margins at $a = 100$ : endpoint signs $25$ and $43$ , and $d = 2 / 13$ for $η = 1$

Example

Take $a = 100$ in Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ , so that $R = 13$ , $l = 7$ , and suppose the defect bound is $η = 1$ . Then:

for all $u, w \in [- 13, 13]$ the unperturbed quadratic $v^{2} - 100 - u - w$ satisfies

$v^{2} - 100 - u - w ∣_{v = 7} \leq 49 - 100 + 26 = - 25, v^{2} - 100 - u - w ∣_{v = 13} \geq 169 - 100 - 26 = 43,$

so the signs at the endpoints are strictly opposite with margins $25$ and $43$ ; 2. the derivative bound is $d = (1 + η) / (2 l - η) = (1 + 1) / (14 - 1) = 2 / 13$ , and

$2 d = \frac{4}{13} < 1, d (\frac{1}{2} + 2) = \frac{5}{13} < 1, \frac{d}{1 - d / 2} = \frac{2 / 13}{12 / 13} = \frac{1}{6} < \frac{2}{3};$

the root margins are at least $24 / 27 > 3 / 4$ and $42 / 27 > 3 / 4$ , since the slope of $P_{u, w}$ is bounded by $2 R + η = 27$ on $[7, 13]$ .

Facts & Assumptions

Given: The parameter $a = 100$ , $R = 13$ , $l = 7$ , the defect bound $η = 1$ , the perturbed recurrence data of Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ and the inverse-branch estimates of A $C^{2}$ map $C^{1}$ -close to $F$ has a unique inverse branch for its first component, with $O (ε)$ derivative bounds.

[L1]

Perturbed root data (Root bounds and $C^{1}$ dependence for the perturbed scalar recurrence $v^{2} - a - u - w + e (u, v, w) = 0$ ): with $η$ as the defect bound, the root map satisfies $∣ \partial_{u} M_{s} ∣, ∣ \partial_{w} M_{s} ∣ \leq d = (1 + η) / (2 l - η)$ and every root $v = M_{s} (u, w)$ satisfies $s v - l \geq \frac{3}{4}$ and $R - s v \geq \frac{3}{4}$ .

[L2]

Elementary arithmetic (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Absolute value in an ordered field, Basic properties of the absolute value, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): $7^{2} = 49$ , $13^{2} = 169$ , $2 \cdot 13 = 26$ , $14 - 1 = 13$ , $4 / 13 < 1$ , $5 / 13 < 1$ , $1 / 6 < 2 / 3$ , and $24 / 27 > 3 / 4$ , $42 / 27 > 3 / 4$ .

Verification

technique · direct; evaluate the quadratic at the two endpoints in the worst case and substitute the numerical data into the general bounds

1.1givenL2algebra

Endpoint signs: for $u, w \in [- 13, 13]$ the sum $u + w$ ranges over $[- 26, 26]$ , so $v^{2} - 100 - u - w$ at $v = 7$ is at most $49 - 100 + 26 = - 25 < 0$ , and at $v = 13$ is at least $169 - 100 - 26 = 43 > 0$ .

1.2givenL1L2algebra

Derivative bound and cone constants: substituting $l = 7$ , $η = 1$ into $d = (1 + η) / (2 l - η)$ gives $d = 2 / 13$ , and the displayed inequalities follow from [L2].

2.1givenstep 1.1L1L2algebra∎

Margins: with $η = 1$ and $2 R + η = 27$ , the endpoint margins guaranteed by the root lemma are $24 / 27$ and $42 / 27$ : the endpoint values of $P_{u, w}$ have magnitudes at least $25 - 1 = 24$ and $43 - 1 = 42$ , and $∣ P_{u, w}^{'} ∣ \leq 27$ on $[7, 13]$ . Thus the root distances are at least $24 / 27 > 3 / 4$ and $42 / 27 > 3 / 4$ , consistent with the general margins of [L1].

Remarks

Every number is evaluated symbolically at the worst case $a = 100$ and $η = 1$ ; larger $a$ or smaller $η$ only improve the margins.

Explicit root margins at $a = 100$ : endpoint signs $25$ and $43$ , and $d = 2 / 13$ for $η = 1$

Example

Facts & Assumptions

Verification

Remarks

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Dependency tree · 2 levels

Sources

Explicit root margins at a=100a=100: endpoint signs 2525 and 4343, and d=2/13d=2/13 for η=1η=1

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Explicit root margins at $a = 100$ : endpoint signs $25$ and $43$ , and $d = 2 / 13$ for $η = 1$