Manuscript Em04-A@2 · Em04-A-v2.tex · section · f5b28abc

Composite trapezoidal approximation

Compressed oscillator position measurements › Composite trapezoidal approximation · lines 1173–1209

Em04-A@2#eq:trapezoid, Em04-A@2#eq:trapbudget

Let

ϕ_{n} (x) = \frac{H _{n} ( x ) e ^{- x^{2} /2}}{π ^{1/4} 2 ^{n} n !}, H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}} .

Set

v_{d} (x) = (ϕ_{0} (x), \dots, ϕ_{d - 1} (x))^{T}, G_{d} (x) = v_{d} (x) v_{d} (x)^{†} .

For real $x$ , $G_{d} (x) \geq 0$ , and $E_{d} (B) = \int_{B} G_{d} (x) d x$ .

Fix $L > 0$ . Suppose $B \cap [- L, L]$ is a finite union of intervals, partitioned into cells $I_{ℓ} = [a_{ℓ}, b_{ℓ}]$ of widths $h_{ℓ} = b_{ℓ} - a_{ℓ}$ . Boundary points have zero position probability and may be assigned consistently to either bin. Define the exact-arithmetic trapezoidal matrix

Q_{d}^{trap} (B; L) = ℓ \sum \frac{h _{ℓ}}{2} (G_{d} (a_{ℓ}) + G_{d} (b_{ℓ})),

and

η_{d}^{trap} (B; L) = \frac{1}{12} ℓ \sum h_{ℓ}^{3} x \in I_{ℓ} sup ∥ G_{d}^{''} (x) ∥_{\infty} .

This rule retains positive weights and half-weight endpoints. It is the quadrature to be checked against the reported simulator.

Displayed formulas in this passage

Em04-A@2#eq:trapezoid · line 1194
$Q_{d}^{trap} (B; L) = ℓ \sum \frac{h _{ℓ}}{2} (G_{d} (a_{ℓ}) + G_{d} (b_{ℓ})),$
Em04-A@2#eq:trapbudget · line 1201
$η_{d}^{trap} (B; L) = \frac{1}{12} ℓ \sum h_{ℓ}^{3} x \in I_{ℓ} sup ∥ G_{d}^{''} (x) ∥_{\infty} .$
unlabeled · line 1176
$ϕ_{n} (x) = \frac{H _{n} ( x ) e ^{- x^{2} /2}}{π ^{1/4} 2 ^{n} n !}, H_{n} (x) = (- 1)^{n} e^{x^{2}} \frac{d ^{n}}{d x ^{n}} e^{- x^{2}} .$
unlabeled · line 1182
$v_{d} (x) = (ϕ_{0} (x), \dots, ϕ_{d - 1} (x))^{T}, G_{d} (x) = v_{d} (x) v_{d} (x)^{†} .$

Composite trapezoidal approximation

Displayed formulas in this passage

Nearby in this section