finite difference instrument
| Definition | An estimate of a derivative from two evaluations of the function at points a small step apart. Equation 0.8. Also finite-difference. |
|---|---|
| Example | For C(x) equal to x squared, evaluating at 0.8 plus and minus 0.6 gives (1.96 − 0.04) over 1.2, exactly 1.6. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id finite-difference, kind instrument. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.8. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/FiniteDifference.lean |
| Reviewed | not yet reviewed; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.8.
\[g_j\;\approx\;\frac{C(x+h\,e_j)-C(x-h\,e_j)}{2h},\qquad j=1,\dots,d.\]
Conditions
- An estimate of a derivative from two evaluations of the function at points a small step apart. The central difference recovers the derivative of a quadratic exactly at every step size, while the one-sided difference is off by the curvature times the step, which is why doing it in both directions is more accurate.
- Doing it once per coordinate costs d evaluations and in both directions costs 2d, which chapter 11 shows is the price at one operating point for a probe confined to the directions it chooses. The measurement instrument for that is the blind probe.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Chapter 0 section 0.5 of Data Mining as Observation,
equation 0.8, with the budget law measured in
readscope/README.md:118-141 and
readscope/CALIBRATION.md:419-434.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/FiniteDifference.lean,
theorems central_quad, forward_quad,
forward_error, central_cost, at
observation-data-mining f3914f0; what the check covers is stated in the
book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 1, 2, 4, 6, 7, 11, 12, 14.
Related
sensitivity; read operator; blind probe; budget; budget cliff.
See also
Book equations stated beside the entry’s terms, not defining it: 0.9, 11.4.
Ledger rows that cite the entry’s records without naming it: GO-1.
Sources-table rows that share a record with the entry without naming it: chapter 11 section 11.7.
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.