Direct method

A numerical method designed to obtain a solution through a finite prescribed sequence of operations, rather than by repeated approximation toward a limit.

Gaussian elimination: A finite sequence of row operations that reduces a linear system to triangular form for solution by back substitution. It is the standard direct procedure for solving a nonsingular system of linear equations.
Iterative method: A numerical procedure that repeatedly updates an approximation, aiming to converge toward a solution. Unlike a direct method, it generally approaches rather than reaches a solution through a fixed finite sequence.
Linear system: A set of linear equations that can be written as a matrix equation involving unknown variables. Direct methods are especially prominent in computing solutions to finite-dimensional linear systems.
Matrix factorization: A representation of a matrix as a product of simpler matrices with useful algebraic or computational properties. Many direct solvers work by factoring a matrix into forms that are easier to solve with.
LU decomposition: A factorization of a matrix into lower- and upper-triangular matrices, often with a permutation matrix. It expresses elimination as reusable matrix factors, especially when solving systems with multiple right-hand sides.
Jacobi method: An iterative algorithm for linear systems that updates each variable using values from the previous iteration. It illustrates how a solver can trade finite elimination for repeated, parallelizable updates.
Least squares: A method for estimating unknowns by minimizing the sum of squared residuals in an overdetermined system. QR factorization provides a direct route to least-squares solutions without forming normal equations.
Floating-point arithmetic: A system for representing and calculating with real numbers using finite precision and a bounded exponent range. Rounding means a finite sequence of operations may not yield an exact solution in computer arithmetic.
Cholesky decomposition: A factorization of a symmetric positive-definite matrix as the product of a lower-triangular matrix and its transpose. It gives a specialized direct solver that exploits symmetry and positive definiteness.
Gauss–Seidel method: An iterative algorithm for linear systems that uses newly computed values immediately within each iteration. Its convergence depends on repeated updates, unlike the prescribed elimination steps of a direct solver.

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