Bisection Method
Animated root-finding with interval zoom, iteration table, and multiple examples.
ExploreNumerical Methods
Root-finding, interpolation, least-squares data fitting, and finite differences.
Interactive tools for exploring key numerical methods — from root finding and interpolation to data fitting and finite differences.
Root Finding
- Bisection Method — animated root-finding with interval zoom, iteration table, and multiple examples
- Newton's Method — animate the iterative root-finding process and observe convergence behavior
- Fixed Point Iteration — visualize cobweb diagrams for fixed-point iteration, convergence, and chaos
Interpolation & Data Fitting
- Polynomial Interpolation — construct interpolating polynomials through given data points
- Polynomial Least Squares — fit polynomials to data and analyze residuals
- Rational Function Least Squares — approximate data with rational functions
Differentiation
- Finite Differences — generate finite difference stencils with truncation error estimates
Numerical Linear Algebra
Cross-listed from Linear Algebra — the algorithmic core of numerical methods for solving linear systems, least-squares problems, and eigenvalue problems.
- PA = LU Factorization — Gaussian elimination with partial and scaled partial pivoting
- QR Factorization — orthogonal-triangular decomposition for stable solves
- Cholesky Factorization — efficient decomposition for symmetric positive definite matrices
- Gram-Schmidt Orthogonalization — build an orthonormal basis step-by-step
- Singular Value Decomposition (SVD) — full singular value decomposition
- Least Squares via QR / Normal Equations / SVD — three numerical routes to the same overdetermined system
- Eigenvalues and Eigenvectors — compute the eigenproblem of a matrix
Accessibility
All tools are designed to meet WCAG 2.1 AA standards. They include keyboard navigation, screen reader support, high-contrast mode, visible focus indicators, and ARIA labels throughout.
Tool Descriptions
Root Finding
Newton's Method Visualizer
Animate the iterative root-finding process and observe convergence behavior.
ExploreFixed Point Iteration
Visualize fixed-point iteration one segment at a time. Watch convergence, period-doubling cascades, and chaos unfold. Classify stability and compute the Lyapunov exponent live.
ExploreInterpolation & Data Fitting
Polynomial Interpolation
Construct and visualize interpolating polynomials through given data points.
ExplorePolynomial Least Squares Data Fitting
Fit polynomials to data using least squares and analyze residuals.
ExploreRational Function Least Squares Data Fitting
Approximate data with rational functions via least squares optimization.
ExploreDifferentiation, Integration & ODEs
Finite Difference Generator
Generate finite difference stencils with truncation error estimates.
ExploreNumerical Integration
Approximate definite integrals using Left, Right, Midpoint, Trapezoidal, Simpson, and Gauss–Legendre quadrature, plus adaptive Simpson and Romberg extrapolation. Visualize rectangles, trapezoids, and parabolas with signed-area shading; compare convergence rates across methods; and compute subdivisions needed for a given tolerance.
ExploreTaylor Series Visualizer
Plot a function together with its Taylor (or Maclaurin) polynomial about any point. Coefficients are computed by exact differentiation and shown as simplified fractions; step the degree up one at a time with the “Add next degree” button to watch the approximation converge. The expansion point is marked on the graph.
ExploreInitial Value Problem Methods
Compare numerical methods for first-order initial value problems \(y'=f(t,y)\): forward and backward Euler, improved Euler (Heun), the trapezoidal rule, Adams–Bashforth and Adams–Moulton multistep methods, Taylor methods up to order 4, and Runge–Kutta methods up to RK4. Tick several methods to overlay them, vary the step size live, and study accuracy, convergence order, and absolute-stability regions on stiff and non-stiff examples.
ExploreNumerical Linear Algebra
Cross-listed from Linear Algebra — the algorithmic core of numerical methods for solving linear systems, least-squares problems, and eigenvalue problems.
Matrix Factorizations
PA = LU Factorization
Compute the LU decomposition with partial and scaled partial pivoting — the numerical workhorse for solving \(Ax = b\) via Gaussian elimination.
Compute LUQR Factorization
Compute the orthogonal-triangular decomposition \(A = QR\) — the numerically stable route for least-squares and the building block for QR-based eigenvalue algorithms.
ComputeCholesky Factorization
Compute the Cholesky decomposition \(A = LL^T\) for symmetric positive definite matrices — roughly twice as fast as LU when applicable.
ComputeGram-Schmidt Orthogonalization
Apply the Gram-Schmidt process step-by-step to obtain an orthonormal basis — the constructive backbone of QR factorization.
CalculateSingular Value Decomposition (SVD)
Compute the full SVD \(A = U\Sigma V^T\) — the most numerically robust decomposition for rank, pseudo-inverse, and least squares.
Compute SVDLeast Squares & Eigenvalue Solvers
Least Squares Solver using QR Factorization
Solve the least-squares problem \(\min_x \|Ax - b\|_2\) via QR decomposition — the recommended numerical approach for well-conditioned problems.
ComputeLeast Squares Solver using Normal Equations
Solve the least-squares problem via the normal equations \(A^TA\,x = A^Tb\) — the classical approach, fast but with squared condition number.
ComputeLeast Squares Solver using SVD
Solve the least-squares problem via SVD — the most numerically stable route, handling rank-deficient and ill-conditioned systems gracefully.
Compute SolutionEigenvalues and Eigenvectors
Compute eigenvalues and eigenvectors of a matrix — the core numerical problem behind diagonalization, PCA, and spectral methods.
Compute