எந்த கணிதப் பிரச்சனையும் தீர்க்கவும்

படிப்படியாக

1. f(x) = - x + \cos{\left(x \right)},\quad x_0 = 1

   Newton's method: x_{n+1} = x_n − f(x_n)/f′(x_n) — follow the tangent line down to the axis, repeat.

2. f'(x) = - \sin{\left(x \right)} - 1

   Differentiate once; the same derivative serves every iteration.

3. x_{1} = 1.0 - \frac{-0.45969772}{-1.84147096} = 0.75036387

   Iteration 1.

4. x_{2} = 0.75036387 - \frac{-0.01892307}{-1.68190503} = 0.73911289

   Iteration 2.

5. x_{3} = 0.73911289 - \frac{-4.646 \cdot 10^{-5}}{-1.67363262} = 0.73908513

   Iteration 3.

6. x_{4} = 0.73908513 - \frac{0.0}{-1.67361212} = 0.73908513

   Iteration 4.

7. x_{5} = 0.73908513 - \frac{0}{-1.67361212} = 0.73908513

   Iteration 5.

8. \text{converged}

   The iterates have stopped changing to 12 decimal places.

9. x \approx 0.73908513

   Quadratic convergence: the number of correct digits roughly doubles each step.