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Numerical-Methods · Root-Finding

Newton's method on x^2 - 5

x^{2} - 5

Passo a passo

  1. f(x) = x^{2} - 5,\quad x_0 = 2

    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) = 2 x

    Differentiate once; the same derivative serves every iteration.

  3. x_{1} = 2.0 - \frac{-1.0}{4.0} = 2.25

    Iteration 1.

  4. x_{2} = 2.25 - \frac{0.0625}{4.5} = 2.23611111

    Iteration 2.

  5. x_{3} = 2.23611111 - \frac{0.0001929}{4.47222233} = 2.23606798

    Iteration 3.

  6. x_{4} = 2.23606798 - \frac{0.0}{4.47213554} = 2.23606798

    Iteration 4.

  7. x_{5} = 2.23606798 - \frac{0.0}{4.47213554} = 2.23606798

    Iteration 5.

  8. \text{converged}

    The iterates have stopped changing to 12 decimal places.

  9. x \approx 2.23606798

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

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x \approx 2.23606798