Approximatiokertoimella

Approximatiokertoimella is a term used in approximation theory to denote the coefficient of the leading error term in an approximation scheme. It serves as a scalar that quantifies the size of the dominant residual when a quantity is approximated by a sequence of computable approximations.

Formally, let x* be the exact value and x_n the nth approximation produced by a method with a known rate function φ(n) describing the convergence. The method has leading error term E_n = x* - x_n. The approximatiokertoimella K is defined by E_n = K φ(n) + o(φ(n)) as n → ∞. The coefficient K depends on the problem, the specific method, and the underlying regularity of x*. The rate function φ(n) is typically predetermined by the order of the method (for example φ(n) = n^-p for p-th order methods).

Estimation and use: K can be estimated from computations by K ≈ E_n / φ(n) for large n, or more robustly via extrapolation techniques such as Richardson extrapolation that cancel the leading term. Once known, K enables comparisons between methods, error prediction for new discretization levels, and adaptive strategies that adjust resources to control the residual.

Example: In numerical integration with the trapezoidal rule on [a,b] using n subintervals, the error satisfies E_n ≈ - (b-a)^3 /(12 n^2) f''(ξ) for some ξ ∈ (a,b). Here φ(n) = 1/n^2 and approximatiokertoimella is K = - (b-a)^3 /(12) f''(ξ), reflecting the leading dependence on the function's curvature.

Notes: The term approximatiokertoimella is not universally standardized and appears primarily in Finnish-language texts or theoretical discussions to emphasize the role of the leading error coefficient in analysis and modeling.