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Paano Kalkulahin si Newtons Method

Ano ang Newtons Method?

Newton's Method (Newton-Raphson method) is an iterative algorithm for finding roots of a differentiable function f(x) — that is, values of x where f(x) = 0. Starting from an initial guess x₀, it repeatedly applies the update formula to converge to a root. When it converges, it converges quadratically — the number of correct decimal places roughly doubles with each iteration.

Pormula

xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)

Step-by-Step na Gabay

  1. 1xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
  2. 2f'(x) is approximated numerically: f'(x) ≈ [f(x+h) − f(x−h)] / 2h, h = 10⁻⁷
  3. 3Converges when |xₙ₊₁ − xₙ| < 10⁻¹⁰
  4. 4May fail if f'(x₀) ≈ 0 or if starting guess is far from a root
  5. 5Multiple roots require different initial guesses

Mga Nalutas na Halimbawa

Input
f(x) = x³ − x − 2, x₀ = 1.5
Resulta
Root ≈ 1.5213797
Converges in ~5 iterations
Input
f(x) = x² − 2, x₀ = 1.0
Resulta
Root ≈ 1.4142136 (√2)
Newton's method can compute square roots

Mga madalas itanong

What is Newtons Method?

Newton's Method (Newton-Raphson method) is an iterative algorithm for finding roots of a differentiable function f(x) — that is, values of x where f(x) = 0. Starting from an initial guess x₀, it repeatedly applies the update formula to converge to a root

How accurate is the Newtons Method calculator?

The calculator uses the standard published formula for newtons method. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Newtons Method calculator use?

This calculator works with inches, watts. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Newtons Method calculator use?

The core formula is: xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ). Each step in the calculation is shown so you can verify the result manually.

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