The secant method
Problem 3.641 · medium
Use the secant method on \( \displaystyle f(x) = x^{3} - 3 \) with \( \displaystyle x_0 = 1 \) and \( \displaystyle x_1 = 2 \) to find \( \displaystyle x_2 \) and \( \displaystyle x_3 \).
- x_{n+1} = x_n − f(x_n)(x_n − x_{n−1}) / (f(x_n) − f(x_{n−1})).Reviewed
- \[ 2 - \frac{\left. x^{3} - 3 \right|_{\substack{ x=2 }}}{- \left. x^{3} - 3 \right|_{\substack{ x=1 }} + \left. x^{3} - 3 \right|_{\substack{ x=2 }}} = \frac{9}{7} \]x2 ≈ 1.285714.✓ Proved
- \[ \frac{9}{7} + \frac{\left(2 - \frac{9}{7}\right) \left. x^{3} - 3 \right|_{\substack{ x=\frac{9}{7} }}}{- \left. x^{3} - 3 \right|_{\substack{ x=2 }} + \left. x^{3} - 3 \right|_{\substack{ x=\frac{9}{7} }}} = \frac{561}{403} \]x3 ≈ 1.392060.✓ Proved
Answer \( x_2 = \frac{9}{7} \approx 1.285714,\ x_3 = \frac{561}{403} \approx 1.392060 \)
Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the iteration run in 30-digit floating point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly states the secant method formula and the algebraic steps are verified as correct by the CAS. The final values match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly states the secant method formula and the algebraic steps are verified as correct by the CAS. The final values match the stated answer.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly states the secant method formula and the algebraic steps for calculating x_2 and x_3 are verified as correct.gpt-oss:20b: fail (error) 2026-10-14 — The symbolic computation for x₃ is incorrect; the derived fraction −514500/4,838,015 does not simplify to 561/403, indicating a mistake in the algebraic manipulation.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the secant method formula and the algebraic steps for calculating x_2 and x_3 are verified as correct.gpt-oss:20b: pass 2026-10-14
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/secant_method, checked 2026-10-17 with SymPy 1.14.0.