Manuscript En08-B@1 · En08-B.tex · section · 23cac52a

A deterministic rounding convention

Explicit routing and complete quantum transport › A deterministic rounding convention · lines 493–511

En08-B@1#eq:rounding, En08-B@1#eq:rounding-error

Define coordinatewise nearest-lattice rounding by

(Q_{j} z)_{i} = h_{j} ⌊ \frac{z _{i}}{h _{j}} + \frac{1}{2} ⌋, z \in R^{d} .

Ties are rounded towards the larger coordinate. This map is coordinatewise monotone, fixes $Λ_{j}$ , and satisfies

∥ Q_{j} z - z ∥_{2} \leq ε_{j}, ε_{j} = \frac{κ h _{j}}{2} = \frac{κ ℓ}{2 j ^{2}} .

Displayed formulas in this passage

En08-B@1#eq:rounding · line 496
$(Q_{j} z)_{i} = h_{j} ⌊ \frac{z _{i}}{h _{j}} + \frac{1}{2} ⌋, z \in R^{d} .$
En08-B@1#eq:rounding-error · line 504
$∥ Q_{j} z - z ∥_{2} \leq ε_{j}, ε_{j} = \frac{κ h _{j}}{2} = \frac{κ ℓ}{2 j ^{2}} .$