Numerical analysis

Numerical analysis studies how mathematical problems are represented, approximated, and computed, and how error moves through those steps. Its core questions include what a floating-point answer means, when a residual controls the solution, how discretization error depends on stability and regularity, and which assumptions make an iteration converge. The subject connects direct and iterative linear algebra with interpolation, quadrature, ordinary differential equations, partial differential equations, finite elements, spectral methods, and randomized computation.

Each method is presented with its input class, arithmetic or mesh model, norm, regularity, boundary conditions, and error claim. Exact-arithmetic identities remain separate from finite-precision stability; continuous well-posedness remains separate from discrete convergence; estimator values remain separate from certified error; and sampling variability remains separate from solver, approximation, and model bias. Existing Linear Algebra, PDE, probability, and functional-analysis results provide their established hypotheses. Numerical methods do not turn a small residual, a CFL condition, or an observed rate into a theorem without the matching proof.

Executable methods specify finite budgets, stopping and failure statuses, and the arithmetic or random-draw model. Termination and computational cost are proved separately from convergence and accuracy. Geometric integration retains the domain of every intermediate drift or kick; time-dependent finite elements retain the full projection residual. Reference-cell estimates on polygonal meshes require their own local trace and scaling proofs. Mathematical designs developed in physics can inform shared mathematical suppliers, while physical modeling assumptions remain attributed contextual applications.