The secant method
Problem 3.637 · medium
Use the secant method on \( \displaystyle f(x) = x^{3} - 2 \) 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
- \[ 2 - \frac{\left. x^{3} - 2 \right|_{\substack{ x=2 }}}{- \left. x^{3} - 2 \right|_{\substack{ x=1 }} + \left. x^{3} - 2 \right|_{\substack{ x=2 }}} = \frac{8}{7} \]x2 ≈ 1.142857.✓ Proved
- \[ \frac{8}{7} + \frac{\left(2 - \frac{8}{7}\right) \left. x^{3} - 2 \right|_{\substack{ x=\frac{8}{7} }}}{- \left. x^{3} - 2 \right|_{\substack{ x=2 }} + \left. x^{3} - 2 \right|_{\substack{ x=\frac{8}{7} }}} = \frac{75}{62} \]x3 ≈ 1.209677.✓ Proved
Answer \( x_2 = \frac{8}{7} \approx 1.142857,\ x_3 = \frac{75}{62} \approx 1.209677 \)
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 applies the secant method formula and the algebraic steps are verified by the CAS. The results match the stated answer.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly applies the secant method formula and the algebraic steps are verified by the CAS. The results 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 applies it accurately to compute x_2 and x_3. The algebraic steps are verified as correct.gpt-oss:20b: fail (error) 2026-10-14 — The third step incorrectly uses x_2 and x_0 (or 2) instead of x_2 and x_1 in the secant formula, so the derivation of x_3 is invalid.qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies the secant method formula and computes the iterations accurately.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.