Approximation Function for Intensive Calculations

This function provides a lookup-based approximation for calculations that are computationally intensive. Once computed, it stores the results in an environment and uses linear interpolation for new data points to speed up subsequent computations.

Usage

fun_approx(u, u_lb = -100, u_ub = 100, resol = 10000, fun = W_kernel)

Arguments

u

A vector of values where the function should be evaluated.

u_lb

Lower bound for the precomputed range. Defaults to -10.

u_ub

Upper bound for the precomputed range. Defaults to 10.

resol

The resolution or number of sample points in the precomputed range. Defaults to 1000.

fun

A function for which the approximation is computed. Defaults to the W function.

Value

A vector of approximated function values corresponding to u.

Details

The fun_approx function works by initially creating a lookup table of function values based on the range specified by u_lb and u_ub and the resolution resol. This precomputation only happens once for a given set of parameters (u_lb, u_ub, resol, and fun). Subsequent calls to fun_approx with the same parameters use the lookup table to find the closest precomputed points to the requested u values and then return an interpolated result.

Linear interpolation is used between the two closest precomputed points in the lookup table. This ensures a smooth approximation for values in between sample points.

This function is especially useful for computationally intensive functions where recalculating function values is expensive or time-consuming. By using a combination of precomputation and interpolation, fun_approx provides a balance between accuracy and speed.