Source: https://guthlabs.ai/docs/instruments/numeric-root-finder-newton-bisection-secant/
instrument / verified-clean
# Numeric Root Finder (Newton/Bisection/Secant)
The current ledger marks this instrument verified clean within the source-bound verification corpus.
Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**
## Inspectable records
[01**Overview**Purpose and lifecycle]()[02**Mechanism**Method and assumptions]()[03**Implementation**Binding and validation]()[04**Qualification**Evidence and errata]()
## Ledger state
The current ledger marks this instrument verified clean within the source-bound verification corpus.
**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)
**Tranche**: tranche-3
**Evidence ID**: t3-009
**Ledger rows**: 1
No ledger note is attached.
## Review findings
NEEDS_REVIEW because the named methods are recognizable and Newton's formula is correct, but the exact secant formula, bisection update rules, stopping criterion, bracket requirements, endpoint behavior, and failure handling are not fully defined. The test vectors use fixed iteration counts except for the exact-endpoint case so their arithmetic is independently checkable.
### Formula and scope findings
- Newton's formula is standard and correct when f'(xₙ) is nonzero.
- Bisection is described only as midpoint halving; the required sign-based bracket update, endpoint-root handling, and stopping rule are unspecified.
- The secant formula is not written explicitly. The standard form should be xₙ₊₁ = xₙ - f(xₙ)(xₙ - xₙ₋₁)/(f(xₙ)-f(xₙ₋₁)).
- The tolerance criterion is unspecified: it could apply to |f(x)|, successive x differences, bracket width, or a combination.
### Unit assumptions
- x, initial guesses, bracket endpoints, and x-based tolerances use the same units.
- f(x) has an output unit, while f'(x) has units of f per x; therefore f(x)/f'(x) has x-units.
- For secant calculations, (f(xₙ)-f(xₙ₋₁))/(xₙ-xₙ₋₁) has derivative units, so the secant correction has x-units.
- The test vectors use dimensionless x and f values except where explicitly stated; no implicit degree/radian or unit conversion is involved.
- If tolerance is a residual tolerance, it has f-units; if it is a step or bracket-width tolerance, it has x-units.
## Boundary hardening
**BREAK_CLAIMED** · SELF_FLAGGED
Newton's method receives f(x)=x^3 with initial guess x=0, causing the update 0-0/0 to be undefined. The instrument must reject this input or explicitly report an undefined result.
## Errata and corrections
No applied erratum is attached to this unique-name record.
## Inspectable example
4 review vectors are recorded. One source-bound sample is shown below.
````
{
"inputs": {
"method": "Newton",
"function": "f(x)=x^2-2",
"derivative": "f'(x)=2x",
"initial_guess": 1,
"iterations": 3
},
"expected_outputs": {
"iteration_trace": [
{
"n": 0,
"x": 1,
"f_x": -1
},
{
"n": 1,
"x": 1.5,
"f_x": 0.25
},
{
"n": 2,
"x": 1.4166666667,
"f_x": 0.0069444444
},
{
"n": 3,
"x": 1.4142156863,
"f_x": 0.0000060073
}
],
"root_estimate": 1.4142156863,
"iteration_count": 3
},
"working": "Newton: x₁ = 1 - (-1)/2 = 1 + 0.5 = 1.5. Then f(1.5)=1.5²-2=2.25-2=0.25, so x₂ = 1.5 - 0.25/3 = 1.5 - 0.0833333333 = 1.4166666667. Then f(1.4166666667) = 1.4166666667² - 2 = 0.0069444444 and f'(x₂)=2(1.4166666667)=2.8333333333, so x₃ = 1.4166666667 - 0.0069444444/2.8333333333 = 1.4166666667 - 0.0024509804 = 1.4142156863."
}
````
## Related workflows
No source-defined multi-instrument workflow membership is recorded.
## Sources and evidence
These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.
- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`
- `evidence-tranche3/tranche3-results.json`
- `BOUNDARY_HAZARD_CATALOG.json`