Newton's method
Problem 3.777 · hard Video
Use Newton's method on \( \displaystyle f(x) = x^{3} - 2 x + 3 \) with \( \displaystyle x_0 = 2 \). Write the iteration \( \displaystyle x_{n+1} = F(x_n) \) and find \( \displaystyle x_1 \) and \( \displaystyle x_2 \) exactly.
- \[ \frac{d}{d x} \left(x^{3} - 2 x + 3\right) = 3 x^{2} - 2 \]f′(x).✓ Proved
- \[ x - \frac{x^{3} - 2 x + 3}{3 x^{2} - 2} = \frac{2 x^{3} - 3}{3 x^{2} - 2} \]Newton's formula x − f(x)/f′(x), simplified.✓ Proved
- \[ \left. \frac{2 x^{3} - 3}{3 x^{2} - 2} \right|_{\substack{ x=2 }} = \frac{13}{10} \]x₁ = F(x₀).✓ Proved
- \[ \left. \frac{2 x^{3} - 3}{3 x^{2} - 2} \right|_{\substack{ x=\frac{13}{10} }} = \frac{209952 \cdot 2^{\frac{28}{29}} \cdot 3^{\frac{10}{87}} \cdot 5^{\frac{8}{29}} \cdot 7^{\frac{3}{29}}}{1953125} \]x₂ = F(x₁).≈ Checked numerically
Answer \( x_1 = \frac{13}{10},\ x_2 = \frac{209952 \cdot 2^{\frac{28}{29}} \cdot 3^{\frac{10}{87}} \cdot 5^{\frac{8}{29}} \cdot 7^{\frac{3}{29}}}{1953125} \approx 0.454072 \)
Lines: 3 proved, 1 checked numerically. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: numeric agreement only, at 12 of 12 sampled points |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | two Newton steps in 30-digit floating point |
Reviewers
gpt-oss:20b: fail (error) — The expression for x₂ is incorrect; substituting x₁=13/10 into the Newton iteration yields a rational number, not a product of fractional powers of primes as shown.qwen3.6:27b-mlx: fail (error) — The calculation for x_2 is mathematically incorrect. Substituting x_1 = 13/10 into the iteration function yields a rational number (13585/2997), but the solution provides a complex irrational expression involving fractional exponents, likely due to a computational error or hallucination.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-11-03 — The calculation for x_2 is mathematically incorrect. Substituting x_1 = 13/10 into the iteration function yields a rational number (13585/2997), but the solution provides a complex irrational expression involving fractional exponents, likely due to a computational error or hallucination.gpt-oss:20b: fail (error) 2026-11-03 — The expression for x₂ is incorrect; substituting x₁=13/10 into the Newton iteration yields a rational number, not a product of fractional powers of primes as shown.qwen3.6:27b-mlx: fail (error) 2026-11-02 — The calculation for x_2 is numerically incorrect. Substituting x_1 = 1.3 into the iteration formula yields approximately 0.752, whereas the stated answer is approximately 0.454. The CAS output for line 4 appears to be corrupted or incorrect.gpt-oss:20b: inconclusive 2026-11-02 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The Newton iteration for \(x_2\) should be a rational number \(697/1535\approx0.454\). The solution’s symbolic expression for \(x_2\) is incorrect and doeqwen3.6:27b-mlx: fail (error) 2026-11-02 — The algebraic simplification of the Newton iteration formula in line 2 is incorrect. The correct simplification of x - (x^3 - 2x + 3)/(3x^2 - 2) is (2x^3 - 3)/(3x^2 - 2) only if the numerator calculation x(3x^2 - 2) - (x^3 - 2x + 3) = 3x^3 - 2x - x^3 + 2x - 3 = 2x^3 - 3 is correct, which it is. However, the stated answer for x_2 is numerically incorrect and contains irrational terms (roots) which cannot arise from applying a rational function to a rational number. The exact value of x_2 should be a rational number. Specifically, F(13/10) = (2*(13/10)^3 - 3) / (3*(13/10)^2 - 2) = (2*2197/1000 - 3) / (3*169/100 - 2) = (4394/1000 - 3000/1000) / (507/100 - 200/100) = (1394/1000) / (307/100) = 1394/3070 = 697/1535. The provided x_2 is completely wrong.gpt-oss:20b: pass 2026-11-02
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/newtons_method, checked 2026-11-03 with SymPy 1.14.0.