Numerical

Type an expression such as sin(x) or x^2-2 and instantly plot its graph over a chosen x range. Maximum, minimum, zeros, and sample point values are shown alongside.

Plot two expressions f(x) and g(x) on the same axes in distinct colors, and automatically find their max, min, and intersection points.

Just paste your (x, y) data points to instantly draw a line-and-point chart. See the point count plus the minimum, maximum, and average of y at the same time.

Paste two sets of (x, y) data points and get f(x) and g(x) drawn as color-coded overlaid line graphs, with each series' point count and min/max.

Solve f(x)=0 with the bisection method. Enter a function and an interval to get the approximate root, f(root), the iteration count, and the error, plus a per-step table of intervals and midpoints and a function graph.

A numerical tool that solves your equation f(x)=0 with the false position method (linear interpolation, regula falsi). Enter an interval and a tolerance to get the approximate root, f(root), the iteration count, and a table and chart of the iteration process.

Enter f(x) and its derivative f'(x) as expressions and use Newton's method to find an approximate root of f(x)=0. Includes an iteration table and a convergence chart.

Solve f(x) = 0 with Newton's method. Just enter the equation, initial guess, tolerance, and max iterations - the derivative is approximated automatically by numerical differentiation. Shows the approximate root, f(root), iteration count, error, plus a step table and a graph.

Solve f(x)=0 with Halley's method. Enter the expression, an initial guess, a tolerance, and the iteration limit to get the approximate root, f(root), iteration count, error, an iteration table, and a convergence chart.

Expand a real number such as π or √2 into a simple continued fraction [a0; a1, a2, …] and list each convergent with its approximation error.

Expand a real number into a regular continued fraction and list every fraction approximation (convergent p/q) until a target accuracy or maximum denominator is met. Expressions such as pi or the golden ratio are supported.

Enter the a and b sequences to numerically evaluate the generalized continued fraction b0 + a1/(b1 + a2/(b2 + …)) with backward recurrence. See each level's partial approximation and its convergence in a table and graph.

Approximate eˣ, log(1+x), arctan(x), and tan(x) with continued-fraction expansions, and watch convergence and the error against the true value in a chart and table as you add more terms.

Solve dy/dx = f(x, y) numerically with the 2nd-order Runge-Kutta (midpoint) method, showing the approximate value at the end point, a table of the numerical solution, and a solution-curve graph.

Enter the right-hand side of dy/dx=f(x,y) plus the initial values x0 and y0, the step size h, and the end point xEnd to solve the ODE numerically with the classic 4th-order Runge-Kutta method (RK4). Shows the approximate solution y at the end point, a table of k1–k4 and y at each step, and a graph of the solution curve.

Convert y''=f(x,y,y') into a first-order system and solve it numerically with the 2nd-order Runge-Kutta method. Get y and y' at the end point, plus a solution curve and a table of every step.

Solve y''=f(x,y,y') numerically with the 4th-order Runge-Kutta method. Enter the initial conditions and step size to get a table of x, y, y' at each step plus a graph of the solution curve.

Enter dy/dx = f(x, y) with an initial condition and step size, and Euler's method computes the numerical solution. Shows the y value at the end point, a step-by-step table, and a graph of the solution curve.

Solve second-order ordinary differential equations of the form y''=f(x, y, y') numerically by converting them to a first-order system and applying Euler's method. Just enter the initial conditions and step size to get the approximate y at the endpoint, a graph of the solution, and a table of (x, y, y') at each step.

Approximate the definite integral of f(x) using the trapezoidal rule. Split [a, b] into n subintervals and view the result, step size, and each sample point.

Approximate a definite integral ∫f(x)dx with Simpson's 1/3 rule. Just enter f(x), the interval, and an even number of subdivisions.

Numerically integrate f(x) with both the trapezoidal and Simpson's rules, then compare the two approximations.

Enter a function f(x) and an interval [a, b] to compute the definite integral ∫f(x)dx over a finite interval with high accuracy using the double exponential (DE / tanh-sinh) quadrature, which handles endpoint singularities well.

Enter f(x), a lower limit a, and the number of nodes to evaluate the half-infinite integral over (a, ∞) accurately with the double exponential (DE) formula.

Numerically evaluate a definite integral over the half-infinite interval (−∞, b) using the double-exponential (DE) formula. Just enter f(x), the upper limit b, and the number of nodes.

Numerically integrate ∫f(x)dx over the whole real line with the double exponential (DE) transform. Just enter f(x) and the number of nodes.

Enter f(x) and an interval [a,b] to compute the definite integral with tanh-sinh (double exponential) quadrature—robust even for endpoint singularities.

List the abscissas x_k and weights w_k of Tanh-Sinh (double exponential) quadrature from a step size and node count. Enter an integrand to also get the approximate integral over (-1, 1).

Numerically evaluate definite integrals with Gauss quadrature: Legendre, Chebyshev, Laguerre, or Hermite. Enter f(x) and the order to get the integral plus nodes and weights.

Compute the nodes and weights of Gaussian quadrature rules (Legendre, Chebyshev, Laguerre, Hermite) for any order, and numerically integrate any f(x) with Gauss–Legendre.

Enter f(x), the interval [a, b] and the order n to evaluate the definite integral ∫f(x)dx with high accuracy using Gauss-Legendre quadrature. Shows the nodes, weights and function values.

Enter a degree n to compute the Gauss-Legendre quadrature nodes x_i and weights w_i on the standard interval [-1, 1] and list them in a table.

Enter an integrand f(x) and order n to compute the weighted integral ∫[-1,1] f(x)/√(1−x²) dx by Gauss-Chebyshev quadrature of the first kind, with the nodes, weights, and function values listed.

Enter the order n to list the Gauss-Chebyshev (first kind) quadrature nodes x_i = cos((2i−1)π/2n) and weights w_i = π/n.

Enter an integrand f(x) and order n to evaluate the weighted integral ∫[-1,1] f(x)√(1−x²)dx using second-kind Gauss-Chebyshev quadrature. Nodes and weights are shown too.

Enter the order n to list the nodes x_i=cos(iπ/(n+1)) and weights w_i=π/(n+1)·sin²(iπ/(n+1)) of the second-kind Gauss-Chebyshev quadrature in a table.

Enter an integrand f(x) and an order n to numerically evaluate the semi-infinite integral ∫₀^∞ f(x)e^(−x)dx by Gauss-Laguerre quadrature.

Enter the order n to compute the nodes (zeros of the Laguerre polynomial Lₙ) and weights of Gauss-Laguerre quadrature in a table. Useful for evaluating ∫₀^∞ e^(-x) f(x) dx numerically.

Enter an integrand f(x) and order n to numerically evaluate ∫[-∞,∞] f(x)e^(-x²)dx by Gauss-Hermite quadrature, using the roots of the Hermite polynomial as nodes with standard weights.

Enter the order n to list the Gauss-Hermite quadrature nodes (roots of the Hermite polynomial Hₙ) and weights for the weight function e^(-x²) in a table.

Enter an integrand f(x), the order n, and the weight exponents α and β to evaluate ∫₋₁¹ f(x)(1−x)^α(1+x)^β dx by Gauss-Jacobi quadrature. Nodes and weights are computed with the Golub-Welsch method (eigendecomposition of a tridiagonal matrix).

Compute the nodes and weights of Gauss-Jacobi quadrature for the weight (1-x)^a (1+x)^b from the degree n and parameters a and b, shown in a table.

Enter the integrand f(x), the interval [a, b], and the order n to evaluate the definite integral with endpoint-inclusive Gauss-Lobatto quadrature.

Enter the number of points n to list the Gauss-Lobatto quadrature nodes and weights, including the endpoints at minus one and plus one.

Enter an integrand f(x) and an interval [a, b] to evaluate the definite integral with the 15-point Kronrod rule and estimate the error from the 7-point Gauss value.

List the nodes, Kronrod weights, and Gauss weights of the 15-point Gauss-Kronrod rule on the standard interval.

Enter f(x), the interval a to b, and a refinement level k to compute a high-accuracy definite integral using Romberg integration: successive trapezoidal refinement with Richardson extrapolation.

Enter f(x), the interval a to b, and a level count to build the Romberg table R(i,j) and show the best estimate R(n,n).

Compute the gamma function Γ(s), the integral of t^(s-1) e^(-t) from 0 to infinity. Enter a positive real s to get Γ(s), lnΓ(s), its link to factorials, and a direct double-exponential integration reference value.

Compute the Beta function B(a,b)=∫_0^1 t^(a-1)(1-t)^(b-1)dt both by Gauss-Legendre numerical integration and the gamma form Γ(a)Γ(b)/Γ(a+b), with lnB and the agreement error.

Enter any one of the side length, height, perimeter, or area of an equilateral triangle and get all the other values at once. It also computes the inradius and circumradius.