Noniterative

Noniterative describes algorithms that compute a solution in a finite number of independent steps without requiring iterative refinement or repeated updating of estimates. This contrasts with iterative methods, which start from an initial guess and repeatedly refine the result until a convergence criterion is met.

In numerical analysis, noniterative (direct) methods solve problems in a fixed sequence of operations. Examples include Gaussian elimination with back substitution, LU decomposition, Cholesky factorization for symmetric positive-definite systems, and QR factorization used for least-squares problems. When these methods are applied with finite-precision arithmetic, they yield a solution with a predictable number of arithmetic steps, with accuracy limited by round-off errors rather than by convergence tolerance.

Direct approaches are common in solving linear systems, computing exact solutions for well-posed problems, and obtaining closed-form or near-closed-form results in algebra and statistics. For least-squares problems, QR decomposition or solving normal equations via factorization provides a noniterative route to the best-fit solution.

Advantages of noniterative methods include determinism, predictable execution time for a given problem size, and often strong numerical stability when pivoting or appropriate factorization is used. However, they can be expensive in time and memory for very large problems due to dense factorization, and their memory footprint grows more quickly than that of many iterative methods.

In practice, the choice between noniterative and iterative methods depends on problem size, structure, needed accuracy, and available computational resources. See also iterative methods, direct methods, numerical linear algebra.