Domain
Engineering Mathematics & Numerical Methods
Numerical methods on Anemodia: root finding by bisection, Newton–Raphson, secant and fixed-point iteration, interpolation by polynomial and spline, numerical differentiation and quadrature, linear systems, determinants, matrix inverses, LU factorisation, matrix conditioning and eigenvalues, and fixed-step Euler and RK4 solvers for initial-value problems.
Numerical methods for engineering computation, with explicit assumptions, conditioning, approximation error, convergence behaviour and transparent numerical diagnostics where applicable. Every tool below runs on the same typed numerical layer and one safe expression parser. Iterative methods state their stopping rule and report iterations, evaluations and residuals; direct methods such as interpolation report the data, the interval and the structure of the approximation instead. In neither case is a failure rounded away. Further numerical methods are added to this domain over time; every tool listed here is complete and available now.
Root Finding
Solve f(x) = 0 in one variable: bracketed bisection, Newton–Raphson with analytic or numerical derivatives, the derivative-free secant method and fixed-point iteration x = g(x).
- Bisection Method CalculatorBracketed root of f(x) = 0 by repeated halving, with the sign-change requirement, bracket history and guaranteed error bound.
- Newton–Raphson CalculatorRoot of f(x) = 0 by the tangent update x − f(x)/f'(x), with an analytic or finite-difference derivative and explicit near-zero-derivative failure.
- Secant Method CalculatorDerivative-free root of f(x) = 0 from two starting values, with the secant update, step history and denominator failure reported.
- Fixed-Point Iteration CalculatorIteration x = g(x) with cobweb geometry, an optional residual f(x), and the local |g'(x*)| contraction diagnostic.
Linear Algebra
Enter a matrix or a system and find out whether the answer can be trusted: Gaussian elimination with partial pivoting, the determinant, the inverse with both-sided residual verification, LU factorisation, conditioning from singular values, and eigenvalues and eigenvectors including complex pairs.
- Linear System SolverSolve Ax = b by Gaussian elimination with partial pivoting: unique, infinitely many or no solution classified by rank, with the residual and the full elimination trace.
- LU Decomposition CalculatorLU factorization with partial pivoting: P, L and U, the row swaps, pivot growth and the verified factorisation residual ‖PA − LU‖_F.
- Matrix Determinant CalculatorCalculate det(A) for a real square matrix from a numerically stable pivoted factorisation, with the sign, the logarithmic magnitude, row-swap parity and singularity diagnostics.
- Matrix Inverse CalculatorCalculate A⁻¹ for a real square matrix by solving A X = I against one pivoted factorisation, verified from both sides and reported with the conditioning of A.
- Matrix Condition Number CalculatorMatrix conditioning from singular values: κ₂ = σmax/σmin with κ₁ and κ∞, the reciprocal condition number, and why a small residual is not the same as a well-conditioned problem.
- Eigenvalues & Eigenvectors CalculatorEigenvalues and eigenvectors of a real square matrix, including complex conjugate pairs and repeated or defective eigenvalues, each validated by its own residual.
Interpolation & Approximation
Read a value between tabulated points: the straight line through two of them, the single global polynomial through all of them, and the natural cubic spline that stays smooth without pretending to know more than the data does.
- Linear Interpolation CalculatorInterpolate between two known points along the straight line joining them, with the fraction t, the slope and an explicit interpolation-or-extrapolation verdict.
- Lagrange Interpolation CalculatorThe unique polynomial of degree ≤ n − 1 through n points, evaluated in stable barycentric form, with degree, data span and extrapolation reported.
- Cubic Spline CalculatorNatural cubic spline through your data: piecewise cubic with C² continuity, S″ = 0 at both ends, segment coefficients and an explicit extrapolation verdict.
Numerical Differentiation & Integration
First derivatives by forward, backward and central differences, and definite integrals by the composite trapezoidal rule, composite Simpson's rule and Gauss-Legendre quadrature.
- Numerical Derivative CalculatorEstimate f'(x) by forward, backward or central difference with an automatic or manual step h, and see the stencil values and the step actually used.
- Trapezoidal Rule CalculatorDefinite integral by the composite trapezoidal rule, with the step size, node table, evaluation count and the O(h²) validity conditions.
- Simpson's Rule CalculatorDefinite integral by composite Simpson's 1/3 rule, with the even-n requirement, the 1-4-2-4-1 coefficients and the O(h⁴) validity conditions.
- Gaussian Quadrature CalculatorDefinite integral by Gauss-Legendre quadrature of order 2 to 8, with mapped nodes and weights, per-node contributions and the degree of exactness.
Ordinary Differential Equations
Scalar first-order initial-value problems y′ = f(t, y) marched with explicit fixed-step methods: Euler at one derivative evaluation per step, classical RK4 at four, with step refinement to see the order each one actually achieves. Systems of equations, adaptive stepping and stiff solvers are not covered.
- Euler Method CalculatorSolve a scalar first-order initial-value problem with the explicit Euler method: signed fixed step, derived end time t₀ + N·h, full node table and an optional supplied reference.
- Runge-Kutta 4 (RK4) CalculatorSolve a scalar first-order initial-value problem with classical fourth-order Runge-Kutta: four stages per step, derived end time, node table and an optional supplied reference.