In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Let F ( x , y ) = 0 be a well-posed problem, i.e. F : X × Y → R is a real or complex functional relationship, defined on the cross-product of some input data set X and an output set Y , such that exists a locally lipschitz function g : X → Y called resolvent, which has the property that for every root ( x , y ) of F , y = g ( x ) . We define numerical method for the approximation of F ( x , y ) = 0 , the sequence of problems
{ M n } n ∈ N = { F n ( x n , y n ) = 0 } n ∈ N , with F n : X n × Y n → R , x n ∈ X n and y n ∈ Y n for every n ∈ N . The problems of which the method consists need not be well-posed. If they are, the method is said to be stable or well-posed.
Necessary conditions for a numerical method to effectively approximate F ( x , y ) = 0 are that x n → x and that F n behaves like F when n → ∞ . So, a numerical method method is called consistent if and only if the sequence of functions { F n } n ∈ N pointlwise converges to F on the set S of its solutions:
lim F n ( x , y ) = F ( x , y ) = 0 , ∀ ( x , y ) ∈ S . When F n = F , ∀ n ∈ N on S the method is said to be strictly consistent.
Denote by ℓ n a sequence of admissable perturbations of x ∈ X for some numerical method M (i.e. x + ℓ n ∈ X n ∀ n ∈ N ) and with y n ( x + ℓ n ) ∈ Y n the value such that F n ( x + ℓ n , y n ( x + ℓ n ) ) = 0 . A condition which the method has to satisfy to be a meaningful tool for solving the problem F ( x , y ) = 0 is convergence:
∀ ε > 0 , ∃ n 0 ( ε ) > 0 , ∃ δ ε , n 0 such that ∀ n > n 0 , ∀ ℓ n : ∥ ℓ n ∥ < δ ε , n 0 ⇒ ∥ y n ( x + ℓ n ) − y ∥ ≤ ε . One can easily prove that the point-wise convergence of { y n } n ∈ N to y implies the convergence of the associated method.
