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Focused engineering tools for fast, well-defined problems.
19 calculators · Engineering Mathematics & Numerical Methods
Engineering Mathematics & Numerical Methods
19 calculatorsDomain overview →
Root Finding
- 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 & Matrix Computation
- 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
- 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
- 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
- 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.