Source: https://guthlabs.ai/docs/instruments/numeric-root-finder-newton-bisection-secant/implementation/

instrument implementation / implementation-contract

# Numeric Root Finder (Newton/Bisection/Secant) implementation

A source-bound implementation contract for Numeric Root Finder (Newton/Bisection/Secant); deployment and executable qualification remain separate.

Status: **VERIFIED_CLEAN**Lifecycle: **implementation-contract**

## Binding contract

This record describes what an executable surface must bind and validate. It does not claim that a public calculator, package, or production route is deployed.

**Record owner**: Guth Instruments

**Lifecycle**: verified-clean

**Evidence ID**: t3-009

**Effect**: Computation or source lookup only

## Input and validation requirements

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.

## Required corrections

No binding erratum is attached to this unique-name record. Implementers must still preserve the declared units, domain, and failure behavior.

## Inspectable interface fixture

The sample below is evidence for interface and validation tests, not a production implementation.

````
{
  "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."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/numeric-root-finder-newton-bisection-secant/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/numeric-root-finder-newton-bisection-secant/mechanism/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/numeric-root-finder-newton-bisection-secant/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche3/tranche3-results.json`