Early arithmetic methods for approximation in astronomy, engineering, and measurement

Iterative procedures appear implicitly (e.g., root-finding ideas akin to later methods)

Development of calculus enables systematic approximation of derivatives and integrals

Early numerical quadrature and series-based approximations used in mechanics and astronomy

Increasing attention to approximation error and convergence in computations

Interpolation and approximation theory mature (polynomial interpolation, least squares)

Numerical solutions for differential equations emerge for physics and engineering problems

Linear algebra computations develop (elimination methods, conditioning concepts begin)

Numerical analysis becomes identifiable as a toolkit: algorithms + error/efficiency concerns

Electronic computing makes algorithmic efficiency and stability central concerns

Formal concepts expand: stability, consistency, convergence, floating-point rounding error

Standard families of methods become widely used

Large-scale linear algebra and eigenvalue problems (iterative methods, Krylov subspace methods)

Finite element and finite volume methods mature for PDEs in engineering and physics

Error estimation and adaptivity become practical (adaptive mesh refinement, adaptive step sizes)

High-performance computing shapes algorithm design (vectorization, parallelism)

Emphasis on scalability for massive data and simulations (parallel and distributed algorithms)

Structure-exploiting methods (sparse, low-rank, randomized numerical linear algebra)

Uncertainty quantification and probabilistic numerics influence modeling and decision-making

Interaction with machine learning

The study and design of algorithms to obtain approximate solutions to mathematical problems

Central pillars

Typical problem classes