Newton-Raphson Quadratic Convergence Root-Finder (x³ − a·x − b = 0) Calculator

Compute Newton-Raphson root iterations x_{k+1} = x_k − (x_k³ − a·x_k − b)/(3x_k² − a).

Converged Root x* (After 6 Iterations)
2.095 x*
Derivative Slope f′(x*) at Root 11.161 slope
Verification x*³ (Equals a·x* + b) 9.189 x³
Absolute Residual |f(x*)| 0 error
Inputs
⚡ Scenarios:
a
Coefficient a in x³ − a·x − b = 0.
b
Constant b in x³ − a·x − b = 0.
x₀
Starting guess x₀.
iterations
Number of Newton-Raphson iterations to run.
Results & Graph
Converged Root x* (After 6 Iterations)
2.095 x*
Derivative Slope f′(x*) at Root
11.161 slope
Verification x*³ (Equals a·x* + b)
9.189 x³
Absolute Residual |f(x*)|
0 error

Converged Root x* (After 6 Iterations)

Graph 1 • Newton-Raphson Root Solver • Domain Architecture & Live Telemetry Maps Coeff Linear A, Constant Term B, Initial Guess X0 through Newton-Raphson Root Solver Formula.
Converged Root x* (After 6 Ite: 2.095 x* Newton-Raphson Root Solver • Invariants & Metrics Converged Root x* (After 6 I 2.095 x* Derivative Slope f′(x*) at R 11.161 slope Verification x*³ (Equals a·x 9.189 x³ Absolute Residual |f(x*)| 0 error
Graph 2 • Converged Root x* (After 6 Iterations) Across Coeff Linear A Sweep (40% → 160%) Evaluates x_{n+1} = x_n − f(x_n) / f′(x_n), where f(x) = x³ − a·x − b and f′(x) across 7 operating points (current input highlighted).
0 x* 1.27 x* 2.55 x* 1.86 0.800a 1.94 1.20a 2.02 1.60a 2.10 2a 2.17 2.40a 2.24 2.80a 2.31 3.20a
Graph 3 • Joint Sensitivity: Coeff Linear A × Constant Term B Compares Converged Root x* (After 6 Iterations) when Constant Term B is shifted +25% (solid) vs. −25% (dashed).
0 x* 1.27 x* 2.54 x* --- Constant Term B @ 75% 2.09 -30% 2.15 -15% 2.20 +0% 2.25 +15% 2.31 +30%
Summary: Solving x³ − 2x − 5 = 0 from x₀=2 converges to x* = 2.094551 with residual |f(x*)| = 8.88e-16.
Summary: Solving x³ − 2x − 5 = 0 from x₀=2 converges to x* = 2.094551 with residual |f(x*)| = 8.88e-16.
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Formula & Mathematical Methodology: Newton-Raphson Root Solver Formula

xn+1 = xn − f(xn) / f′(xn), where f(x) = x³ − a·x − b and f′(x) = 3x² − a
x_{n+1} = x_n − f(x_n) / f′(x_n), where f(x) = x³ − a·x − b and f′(x) = 3x² − a
  • coeffLinearA: Coeff Linear A (a)
  • constantTermB: Constant Term B (b)
  • initialGuessX0: Initial Guess X0 (x₀)
  • iterationsCount: Iterations Count (iterations)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Newton-Raphson Root Solver

Compute Newton-Raphson root iterations x_{k+1} = x_k − (x_k³ − a·x_k − b)/(3x_k² − a).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: x_{n+1} = x_n − f(x_n) / f′(x_n), where f(x) = x³ − a·x − b and f′(x) = 3x² − a.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Newton-Raphson Root Solver via x_{n+1} = x_n − f(x_n) / f′(x_n), where f(x) = x³ − a·x − b and f′(x) = 3x² − a.

Frequently Asked Questions

How does the Newton-Raphson Quadratic Convergence Root-Finder (x³ − a·x − b = 0) Calculator compute the solution?

It evaluates x_{n+1} = x_n − f(x_n) / f′(x_n), where f(x) = x³ − a·x − b and f′(x) = 3x² − a directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

Newton-Raphson Quadratic Convergence Root-Finder (x³ − a·x − b = 0) Calculator: inputs, method, and how to read the result

Compute Newton-Raphson root iterations x_{k+1} = x_k − (x_k³ − a·x_k − b)/(3x_k² − a).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Coeff Linear A: Coefficient a in x³ − a·x − b = 0.
  • Constant Term B: Constant b in x³ − a·x − b = 0.
  • Initial Guess X0: Starting guess x₀.
  • Iterations Count: Number of Newton-Raphson iterations to run.

What the calculation includes

With the default values shown, Converged Root x* (After 6 Iterations) is 2.095 x*. The calculation follows Newton-Raphson Root Solver Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Coeff Linear A from 2 a to 3 a changes Converged Root x* (After 6 Iterations) from 2.095 x* to 2.279 x*. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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