如何计算Newtons Method
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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.
公式
分步指南
- 1xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ)
- 2f'(x) is approximated numerically: f'(x) ≈ [f(x+h) − f(x−h)] / 2h, h = 10⁻⁷
- 3Converges when |xₙ₊₁ − xₙ| < 10⁻¹⁰
- 4May fail if f'(x₀) ≈ 0 or if starting guess is far from a root
- 5Multiple roots require different initial guesses
例题解析
常见问题
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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