fnorm: Function Norm

Description

The fnorm function calculates several different types of function norms for depending on the argument p.

Usage

fnorm(f, g, x1, x2, p = 2, npoints = 100)

Arguments

f, g

functions given by name or string.

x1, x2

endpoints of the interval.

Numeric scalar or Inf, -Inf; default is 2.

npoints

number of points to be considered in the interval.

Value

Numeric scalar (or Inf), or NA if one of these functions returns NA.

Details

fnorm returns a scalar that gives some measure of the distance of two functions f and g on the interval [x1, x2].

It takes npoints equidistant points in the interval, computes the function values for f and g and applies Norm to their difference.

Especially p=Inf returns the maximum norm, while fnorm(f, g, x1, x2, p = 1, npoints) / npoints would return some estimate of the mean distance.

Examples

Run this code

xp <- seq(-1, 1, length.out = 6)
yp <- runge(xp)
p5 <- polyfit(xp, yp, 5)
f5 <- function(x) polyval(p5, x)
fnorm(runge, f5, -1, 1, p = Inf)                  #=> 0.4303246
fnorm(runge, f5, -1, 1, p = Inf, npoints = 1000)  #=> 0.4326690

# Compute mean distance using fnorm:
fnorm(runge, f5, -1, 1, p = 1, 1000) / 1000       #=> 0.1094193

# Compute mean distance by integration:
fn <- function(x) abs(runge(x) - f5(x))
integrate(fn, -1, 1)$value / 2                    #=> 0.1095285