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Explicit constants at $a = 100$ : the contraction factor, the fixed points $r_{\pm}$ and the constant-code solutions

Example

Take $a = 100$ in The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, so that

$R = \sqrt{100} + 3 = 13, L = \sqrt{a - 2 R} = \sqrt{74}, \sqrt{a + 2 R} = \sqrt{126} .$

Then $126 < 169 = 13^{2}$ , so $\sqrt{126} < R = 13$ ; $74 > 1$ , so $L > 1$ ; and the contraction constant of $T_{s}$ on $B_{13}$ is

$\frac{1}{L} = \frac{1}{\sqrt{74}} < 1.$

The constant codes $\overset{ˉ}{u}$ ( $u = \pm 1$ ) are solved by the constant sequences $x_{n} = r_{u}$ with

$11 < r_{+} = 1 + \sqrt{101} < 12, - 10 < r_{-} = 1 - \sqrt{101} < - 9,$

the two displayed bounds hold because $100 < 101 < 121$ ; and both satisfy the exact identity $r_{u}^{2} = 100 + 2 r_{u}$ . Every coded sequence has $∣ x_{n} ∣ \geq \sqrt{74}$ , and its sign is the corresponding symbol.

Facts & Assumptions

Given: The parameter $a = 100$ , the constants $R = 13$ , $L = \sqrt{74}$ , and the numbers $r_{\pm} = 1 \pm \sqrt{101}$ .

[L1]

Model data (The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution, The two-sign bilateral shift space, its coding map, and the ball of bounded bilateral real sequences): for $a \geq 100$ , $R = \sqrt{a} + 3$ , $L = \sqrt{a - 2 R}$ , the operator $T_{s}$ is a contraction of $B_{R}$ with constant $1 / L$ ; its unique fixed point satisfies $sign x_{n} (s) = s_{n}$ , $∣ x_{n} (s) ∣ \geq L$ and $x_{n + 1} (s) + x_{n - 1} (s) = x_{n} (s)^{2} - a$ .

[L2]

Square roots and squares (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, Integer powers $a^{m}$ , Basic properties of the absolute value): $13^{2} = 169$ , $(\sqrt{101})^{2} = 101$ , $\sqrt{74} > 1$ because $74 > 1$ , and $\sqrt{126} < 13$ because $126 < 169$ and both sides are nonnegative.

Verification

technique · direct; evaluate the defining inequalities and identities at the explicit constants

1.1givenL2algebra

Endpoint inequality: $a + 2 R = 100 + 26 = 126$ and $R^{2} = 169$ , so $a + 2 R < R^{2}$ ; by [L2] $\sqrt{126} < 13 = R$ , which is the self-mapping inequality of The square-root recurrence $T_{s} (x)_{n} = s_{n} \sqrt{a + x_{n - 1} + x_{n + 1}}$ is a contraction of $B_{R}$ with a unique coded solution. Similarly $a - 2 R = 100 - 26 = 74 > 1$ , so $L = \sqrt{74} > 1$ and $1 / L < 1$ .

1.2givenL1L2algebra

The fixed points: for $r_{\pm} = 1 \pm \sqrt{101}$ one has, using [L2], $r_{\pm}^{2} = 1 \pm 2 \sqrt{101} + 101 = 102 \pm 2 \sqrt{101} = 100 + 2 (1 \pm \sqrt{101}) = 100 + 2 r_{\pm} .$ Hence $(T_{\overset{ˉ}{u}} \tilde{x})_{n} = u \sqrt{100 + r_{u} + r_{u}} = u \sqrt{100 + 2 r_{u}} = u \sqrt{r_{u}^{2}} = r_{u}$ by [L2] and $sign r_{u} = u$ , so the constant sequence ${\tilde{x}}_{n} = r_{u}$ is a fixed point of $T_{\overset{ˉ}{u}}$ ; by uniqueness it is $x (\overset{ˉ}{u})$ .

2.1givenstep 1.2L1L2algebra∎

Bounds: $∣ r_{\pm} ∣ \geq \sqrt{101} - 1 > 9 > \sqrt{74} = L$ and $∣ r_{\pm} ∣ < 12 < 13 = R$ , consistent with the general bounds of [L1]; the strict bounds $9 > \sqrt{74}$ and $\sqrt{101} < 11$ follow from $74 < 81$ and $101 < 121$ .

Remarks

All verifications are exact algebraic identities involving $\sqrt{101}$ , $\sqrt{74}$ and $\sqrt{126}$ , with rational comparisons for the inequalities; no numerical integration, simulation or measurement is involved.

Explicit constants at $a = 100$ : the contraction factor, the fixed points $r_{\pm}$ and the constant-code solutions

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Explicit constants at a=100a=100: the contraction factor, the fixed points r±r±​ and the constant-code solutions

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Explicit constants at $a = 100$ : the contraction factor, the fixed points $r_{\pm}$ and the constant-code solutions