Numerical methods calculator: bisection, Newton, secant, Simpson, Gauss, Romberg, Lagrange and spline interpolation, with tables

What it does

A teaching tool for a first course in numerical methods or numerical analysis. Type a function and watch each method work, step by step, next to the others.

• Roots: solve f(x) = 0 by bisection, false position (regula falsi), Newton-Raphson (the derivative is worked out for you) or the secant method. You get the full iteration table, why it stopped, the estimated order of convergence, a graph of f with the numbered iterates, and a log-scale plot of the error of all four methods side by side.
• Integrals: the midpoint, trapezoid and Simpson rules with n intervals, Gauss-Legendre with 2 to 20 points and the Romberg table, each with its error against an exact value (worked out symbolically when possible) or a high-accuracy reference. A log-log plot shows the error against the number of function values, so you can read off the orders h² and h⁴.
• Interpolation: through points you type, or points sampled from a function (equal or Chebyshev spacing): the Lagrange polynomial (by the barycentric formula), the Newton divided-difference table with the Newton and expanded forms, and the natural cubic spline with each piece’s coefficients. When a function is given, an error plot compares the polynomial and the spline.

How to use it

1. Pick Roots, Integrals or Interpolation.
2. Type the function in x, such as x^3 - 2x - 5, cos(x) - x or e^x. Angles are in radians.
3. For roots, bisection and false position need an interval [a, b] where f changes sign; Newton starts from x₀ (the second point is used by the other methods for the comparison table); the secant method needs two starting points. Choose the tolerance for the change between steps.
4. For integrals, give a, b, the number of intervals n (even for Simpson) and the number of Gauss points.
5. Download the table as CSV, or the whole result as an image, PDF or Word file.

Worked examples

Newton’s method on f(x) = x³ − 2x − 5 from x₀ = 2 (Newton’s own example) gives 2.1, 2.094568121, 2.094551482 and stops after 5 iterations at x ≈ 2.094551482. With a tolerance of 10⁻¹⁰ the secant method takes 7 iterations, false position 22 and bisection 34: the estimated orders come out near 2, 1.6 and 1.

For ∫₀¹ eˣ dx = e − 1 ≈ 1.718281828 with n = 8, the trapezoid rule gives 1.720518592, Simpson 1.718284155 and 3-point Gauss 1.718281004; Romberg reaches the exact value to 15 digits with 33 function values.

The points (0, 1), (1, 3), (2, 2), (4, 5), (5, 4) give the polynomial −0.2x⁴ + 1.98333x³ − 6.05x² + 6.26667x + 1, which is 2.7 at x = 3; the natural spline gives 3.32172.

Runge’s example (the Runge button) samples 1/(1 + 25x²) at 11 equally spaced points on [−1, 1]: the polynomial swings wildly near the ends (largest error about 1.92), while Chebyshev spacing cuts it to about 0.11 and the spline to about 0.022.

Tips

• If Newton’s method “runs away” (try atan(x) from 1.5), start nearer the root or bracket it with bisection first.
• Bisection’s error bound (b − a)/2 tells you in advance how many steps you need: each step gains one binary digit.
• A high-degree polynomial through many equally spaced points is usually a bad idea; use a spline or Chebyshev points.

Frequently asked questions

Why does Simpson beat the trapezoid rule so easily? Its error falls like h⁴ instead of h², so halving h cuts the error 16 times instead of 4.

What does it not do? Systems of equations, complex roots, ordinary differential equations, adaptive integration and other spline end conditions are not included.

Is my work sent anywhere? No. Everything runs on your device.

Last updated 2026-11-09.