Numerical analysis

Numerical analysis studies algorithms that approximate solutions to mathematical problems and assesses their accuracy, stability, and computational cost.

Truncation error: The error introduced when an exact mathematical process is replaced by a finite approximation. It measures the discrepancy created by approximating an infinite or continuous process.
Gaussian elimination: A sequence of row operations that solves linear systems by reducing their coefficient matrix to triangular form. It is a basic direct method for computing solutions of linear equations.
Numerical linear algebra: The study of algorithms for computing with matrices and linear systems, including their accuracy and efficiency. Large matrix problems are among the field's most common computational workloads.
Symbolic computation: Computer-based manipulation of mathematical expressions in exact or symbolic form. It seeks exact expressions, whereas numerical analysis often returns finite-precision approximations.
Isaac Newton: An English mathematician and physicist whose work included methods for solving equations and approximating functions. Newton's root-finding method remains a central example of numerical iteration.
Round-off error: The difference between an exact number and its finite-precision representation in computation. Finite machine precision adds error even when an algorithm's formulas are exact.
Newton's method: An iterative method that approximates a root by using local linearizations of a function. It illustrates how local information can produce rapidly converging numerical solutions.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Numerical methods approximate solutions when differential equations lack usable closed forms.
Direct method: A numerical method that aims to produce a solution through a finite prescribed sequence of operations. Direct solvers contrast with iterative methods that progressively improve an estimate.
Joseph-Louis Lagrange: An Italian-French mathematician who developed influential methods in analysis, mechanics, and interpolation. Lagrange interpolation became a foundational tool for constructing numerical approximations.

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