Numeric Root Finder (Newton/Bisection/Secant)

Inspectable records

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