Newton-Raphson Method Calculator
Newton-Raphson follows the tangent line to the axis each step: x³ − 2x − 5 = 0 from x₀ = 2 converges to x ≈ 2.09455148154.
Use x, ^ for powers, and sin, cos, tan, exp, ln, log, sqrt, abs, pi, e. 2x and xsin(x) are understood.
Leave blank to use a numerical derivative.
Stop when successive values or |f(x)| fall below this.
2.09455148154233
Full answer, then cut to fewer decimals (not rounded):
- 2.09455148154233
- 2.0945514815423
- 2.094551481542
- 2.09455148154
- 2.0945514815
- 2.094551481
- 2.09455148
- 2.0945514
- 2.094551
- 2.09455
- 2.0945
- 2.094
- 2.09
- 2.0
- f(root)
- -8.88178 × 10⁻¹⁶
- Iterations
- 4
- Converged
- yes
| n | xₙ | f(xₙ) | f′(xₙ) | xₙ₊₁ |
|---|---|---|---|---|
| 0 | 2 | -1 | 10 | 2.1 |
| 1 | 2.1 | 0.061 | 11.23 | 2.0945681211 |
| 2 | 2.0945681211 | 0.000185723 | 11.161647 | 2.0945514817 |
| 3 | 2.0945514817 | 1.73977 × 10⁻⁹ | 11.161438 | 2.09455148154 |
| 4 | 2.09455148154 | -8.88178 × 10⁻¹⁶ |
Step-by-step solution
- xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
- Start at x₀ = 2
- Converged to x ≈ 2.09455148154 after 4 iterations
Newton-Raphson Method Formula
xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
quadratic convergence near a simple root
Worked Example
f(x) = x³ − 2x − 5 starting at x₀ = 2:
- xₙ₊₁ = xₙ − f(xₙ) ÷ f′(xₙ)
- Start at x₀ = 2
- Converged to x ≈ 2.09455148154 after 4 iterations
Iterations: 4 · Converged: yes
| n | xₙ | f(xₙ) | f′(xₙ) | xₙ₊₁ |
|---|---|---|---|---|
| 0 | 2 | -1 | 10 | 2.1 |
| 1 | 2.1 | 0.061 | 11.23 | 2.0945681211 |
| 2 | 2.0945681211 | 0.000185723 | 11.161647 | 2.0945514817 |
| 3 | 2.0945514817 | 1.73977 × 10⁻⁹ | 11.161438 | 2.09455148154 |
| 4 | 2.09455148154 | -8.88178 × 10⁻¹⁶ |
GATE tip
GATE questions often ask for x₁ or x₂ only. Read the first rows of the table: x₁ = x₀ − f(x₀)/f′(x₀). For √a, Newton’s iteration simplifies to xₙ₊₁ = (xₙ + a/xₙ)/2.
More Numerical Methods Calculators
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Bisection Method Calculator
c = (a + b)/2; regula falsi c = (a·f(b) − b·f(a)) ÷ (f(b) − f(a))
keep the half where f changes sign
Example: f(x) = x³ − x − 2 on [1, 2]: Root x ≈ 1.5213804245.
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Secant Method Calculator
xₙ₊₁ = xₙ − f(xₙ)(xₙ − xₙ₋₁) ÷ (f(xₙ) − f(xₙ₋₁))
order of convergence ≈ 1.618
Example: f(x) = x² − 612 starting from 10 and 30: Root x ≈ 24.7386337537.
-
Lagrange Interpolation Calculator
P(x) = Σ yᵢ Π (x − xⱼ)/(xᵢ − xⱼ)
degree ≤ n − 1 through n points
Example: The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5: P(2.5) = 15.625.
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Newton Interpolation Calculator
P(x) = f[x₀] + f[x₀,x₁](x − x₀) + f[x₀,x₁,x₂](x − x₀)(x − x₁) + …
f[xᵢ,…,xᵢ₊ₖ] = (f[xᵢ₊₁,…] − f[xᵢ,…]) ÷ (xᵢ₊ₖ − xᵢ)
Example: The points (1, 1), (2, 8), (3, 27), (4, 64) at x = 2.5: P(2.5) = 15.625.
Frequently asked questions
When does Newton-Raphson fail?
When f′(x) is zero or tiny near the guess, when the guess is far from the root, or near a repeated root where convergence slows to linear. Try a different starting value.
What is the order of convergence of Newton-Raphson?
Quadratic (order 2) for a simple root: the number of correct digits roughly doubles each iteration.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Newton-Raphson Method Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/newton-raphson-calculator/
IEEE
Scientific Converters, "Newton-Raphson Method Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/newton-raphson-calculator/